Lemmas about linear ordered (semi)fields

@[simp]

theorem OrderIso.mulLeft₀_symm_apply {α : Type u_2} [LinearOrderedSemifield α] (a : α) (ha : 0 < a) (x : α) : ↑(RelIso.symm (OrderIso.mulLeft₀ a ha)) x = a⁻¹ * x

@[simp]

theorem OrderIso.mulLeft₀_apply {α : Type u_2} [LinearOrderedSemifield α] (a : α) (ha : 0 < a) (x : α) : ↑(OrderIso.mulLeft₀ a ha) x = a * x

def OrderIso.mulLeft₀ {α : Type u_2} [LinearOrderedSemifield α] (a : α) (ha : 0 < a) : α ≃o α

Equiv.mulLeft₀ as an order_iso.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem OrderIso.mulRight₀_apply {α : Type u_2} [LinearOrderedSemifield α] (a : α) (ha : 0 < a) (x : α) : ↑(OrderIso.mulRight₀ a ha) x = x * a

@[simp]

theorem OrderIso.mulRight₀_symm_apply {α : Type u_2} [LinearOrderedSemifield α] (a : α) (ha : 0 < a) (x : α) : ↑(RelIso.symm (OrderIso.mulRight₀ a ha)) x = x * a⁻¹

def OrderIso.mulRight₀ {α : Type u_2} [LinearOrderedSemifield α] (a : α) (ha : 0 < a) : α ≃o α

Equiv.mulRight₀ as an order_iso.

Equations

• One or more equations did not get rendered due to their size.

Lemmas about pos, nonneg, nonpos, neg

@[simp]

theorem inv_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 0 < a⁻¹ ↔ 0 < a

theorem inv_pos_of_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 0 < a → 0 < a⁻¹

Alias of the reverse direction of inv_pos.

@[simp]

theorem inv_nonneg {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 0 ≤ a⁻¹ ↔ 0 ≤ a

theorem inv_nonneg_of_nonneg {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 0 ≤ a → 0 ≤ a⁻¹

Alias of the reverse direction of inv_nonneg.

@[simp]

theorem inv_lt_zero {α : Type u_2} [LinearOrderedSemifield α] {a : α} : a⁻¹ < 0 ↔ a < 0

@[simp]

theorem inv_nonpos {α : Type u_2} [LinearOrderedSemifield α] {a : α} : a⁻¹ ≤ 0 ↔ a ≤ 0

theorem one_div_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 0 < 1 / a ↔ 0 < a

theorem one_div_neg {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 1 / a < 0 ↔ a < 0

theorem one_div_nonneg {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 0 ≤ 1 / a ↔ 0 ≤ a

theorem one_div_nonpos {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 1 / a ≤ 0 ↔ a ≤ 0

theorem div_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : 0 < a / b

theorem div_nonneg {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a / b

theorem div_nonpos_of_nonpos_of_nonneg {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : a ≤ 0) (hb : 0 ≤ b) : a / b ≤ 0

theorem div_nonpos_of_nonneg_of_nonpos {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 ≤ a) (hb : b ≤ 0) : a / b ≤ 0

theorem zpow_nonneg {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 0 ≤ a) (n : ℤ) : 0 ≤ a ^ n

theorem zpow_pos_of_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 0 < a) (n : ℤ) : 0 < a ^ n

Relating one division with another term.

theorem le_div_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) : a ≤ b / c ↔ a * c ≤ b

theorem le_div_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) : a ≤ b / c ↔ c * a ≤ b

theorem div_le_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hb : 0 < b) : a / b ≤ c ↔ a ≤ c * b

theorem div_le_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hb : 0 < b) : a / b ≤ c ↔ a ≤ b * c

theorem lt_div_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) : a < b / c ↔ a * c < b

theorem lt_div_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) : a < b / c ↔ c * a < b

theorem div_lt_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) : b / c < a ↔ b < a * c

theorem div_lt_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) : b / c < a ↔ b < c * a

theorem inv_mul_le_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ b * c

theorem inv_mul_le_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ c * b

theorem mul_inv_le_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ b * c

theorem mul_inv_le_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ c * b

theorem div_self_le_one {α : Type u_2} [LinearOrderedSemifield α] (a : α) : a / a ≤ 1

theorem inv_mul_lt_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) : b⁻¹ * a < c ↔ a < b * c

theorem inv_mul_lt_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) : b⁻¹ * a < c ↔ a < c * b

theorem mul_inv_lt_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) : a * b⁻¹ < c ↔ a < b * c

theorem mul_inv_lt_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) : a * b⁻¹ < c ↔ a < c * b

theorem inv_pos_le_iff_one_le_mul {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ b * a

theorem inv_pos_le_iff_one_le_mul' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ a * b

theorem inv_pos_lt_iff_one_lt_mul {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) : a⁻¹ < b ↔ 1 < b * a

theorem inv_pos_lt_iff_one_lt_mul' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) : a⁻¹ < b ↔ 1 < a * b

theorem div_le_of_nonneg_of_le_mul {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) : a / b ≤ c

One direction of div_le_iff where b is allowed to be 0 (but c must be nonnegative)

theorem mul_le_of_nonneg_of_le_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b / c) : a * c ≤ b

One direction of div_le_iff where c is allowed to be 0 (but b must be nonnegative)

theorem div_le_one_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (h : a ≤ b) (hb : 0 ≤ b) : a / b ≤ 1

Bi-implications of inequalities using inversions

theorem inv_le_inv_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : a ≤ b) : b⁻¹ ≤ a⁻¹

theorem inv_le_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a

See inv_le_inv_of_le for the implication from right-to-left with one fewer assumption.

theorem inv_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a

In a linear ordered field, for positive a and b we have a⁻¹ ≤ b ↔ b⁻¹ ≤ a. See also inv_le_of_inv_le for a one-sided implication with one fewer assumption.

theorem inv_le_of_inv_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : a⁻¹ ≤ b) : b⁻¹ ≤ a

theorem le_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : a ≤ b⁻¹ ↔ b ≤ a⁻¹

theorem inv_lt_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b⁻¹ ↔ b < a

See inv_lt_inv_of_lt for the implication from right-to-left with one fewer assumption.

theorem inv_lt_inv_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) (h : b < a) : a⁻¹ < b⁻¹

theorem inv_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b ↔ b⁻¹ < a

In a linear ordered field, for positive a and b we have a⁻¹ < b ↔ b⁻¹ < a. See also inv_lt_of_inv_lt for a one-sided implication with one fewer assumption.

theorem inv_lt_of_inv_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : a⁻¹ < b) : b⁻¹ < a

theorem lt_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : a < b⁻¹ ↔ b < a⁻¹

theorem inv_lt_one {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 1 < a) : a⁻¹ < 1

theorem one_lt_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h₁ : 0 < a) (h₂ : a < 1) : 1 < a⁻¹

theorem inv_le_one {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 1 ≤ a) : a⁻¹ ≤ 1

theorem one_le_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h₁ : 0 < a) (h₂ : a ≤ 1) : 1 ≤ a⁻¹

theorem inv_lt_one_iff_of_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h₀ : 0 < a) : a⁻¹ < 1 ↔ 1 < a

theorem inv_lt_one_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} : a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a

theorem one_lt_inv_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 1 < a⁻¹ ↔ 0 < a ∧ a < 1

theorem inv_le_one_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} : a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a

theorem one_le_inv_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 1 ≤ a⁻¹ ↔ 0 < a ∧ a ≤ 1

Relating two divisions.

theorem div_le_div_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 ≤ c) (h : a ≤ b) : a / c ≤ b / c

theorem div_le_div_of_le_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) : a / b ≤ a / c

theorem div_le_div_of_le_of_nonneg {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hab : a ≤ b) (hc : 0 ≤ c) : a / c ≤ b / c

theorem div_lt_div_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) (h : a < b) : a / c < b / c

theorem div_le_div_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) : a / c ≤ b / c ↔ a ≤ b

theorem div_lt_div_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) : a / c < b / c ↔ a < b

theorem div_lt_div_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b < a / c ↔ c < b

theorem div_le_div_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b ≤ a / c ↔ c ≤ b

theorem div_lt_div_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (b0 : 0 < b) (d0 : 0 < d) : a / b < c / d ↔ a * d < c * b

theorem div_le_div_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (b0 : 0 < b) (d0 : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b

theorem div_le_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hbd : d ≤ b) : a / b ≤ c / d

theorem div_lt_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (hac : a < c) (hbd : d ≤ b) (c0 : 0 ≤ c) (d0 : 0 < d) : a / b < c / d

theorem div_lt_div' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (hac : a ≤ c) (hbd : d < b) (c0 : 0 < c) (d0 : 0 < d) : a / b < c / d

theorem div_lt_div_of_lt_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) (hb : 0 < b) (h : b < a) : c / a < c / b

Relating one division and involving 1

theorem div_le_self {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 ≤ a) (hb : 1 ≤ b) : a / b ≤ a

theorem div_lt_self {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 1 < b) : a / b < a

theorem le_div_self {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 ≤ a) (hb₀ : 0 < b) (hb₁ : b ≤ 1) : a ≤ a / b

theorem one_le_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) : 1 ≤ a / b ↔ b ≤ a

theorem div_le_one {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) : a / b ≤ 1 ↔ a ≤ b

theorem one_lt_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) : 1 < a / b ↔ b < a

theorem div_lt_one {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) : a / b < 1 ↔ a < b

theorem one_div_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ b ↔ 1 / b ≤ a

theorem one_div_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : 1 / a < b ↔ 1 / b < a

theorem le_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : a ≤ 1 / b ↔ b ≤ 1 / a

theorem lt_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : a < 1 / b ↔ b < 1 / a

Relating two divisions, involving 1

theorem one_div_le_one_div_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : a ≤ b) : 1 / b ≤ 1 / a

theorem one_div_lt_one_div_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : a < b) : 1 / b < 1 / a

theorem le_of_one_div_le_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : 1 / a ≤ 1 / b) : b ≤ a

theorem lt_of_one_div_lt_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : 1 / a < 1 / b) : b < a

theorem one_div_le_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ 1 / b ↔ b ≤ a

For the single implications with fewer assumptions, see one_div_le_one_div_of_le and le_of_one_div_le_one_div

theorem one_div_lt_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) : 1 / a < 1 / b ↔ b < a

For the single implications with fewer assumptions, see one_div_lt_one_div_of_lt and lt_of_one_div_lt_one_div

theorem one_lt_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h1 : 0 < a) (h2 : a < 1) : 1 < 1 / a

theorem one_le_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h1 : 0 < a) (h2 : a ≤ 1) : 1 ≤ 1 / a

Results about halving.

The equalities also hold in semifields of characteristic 0.

theorem add_halves {α : Type u_2} [LinearOrderedSemifield α] (a : α) : a / 2 + a / 2 = a

theorem add_self_div_two {α : Type u_2} [LinearOrderedSemifield α] (a : α) : (a + a) / 2 = a

theorem half_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h : 0 < a) : 0 < a / 2

theorem one_half_pos {α : Type u_2} [LinearOrderedSemifield α] : 0 < 1 / 2

@[simp]

theorem half_le_self_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} : a / 2 ≤ a ↔ 0 ≤ a

@[simp]

theorem half_lt_self_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} : a / 2 < a ↔ 0 < a

theorem half_le_self {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 0 ≤ a → a / 2 ≤ a

Alias of the reverse direction of half_le_self_iff.

theorem half_lt_self {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 0 < a → a / 2 < a

Alias of the reverse direction of half_lt_self_iff.

theorem div_two_lt_of_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} : 0 < a → a / 2 < a

Alias of the reverse direction of half_lt_self_iff.

---

Alias of the reverse direction of half_lt_self_iff.

theorem one_half_lt_one {α : Type u_2} [LinearOrderedSemifield α] : 1 / 2 < 1

theorem two_inv_lt_one {α : Type u_2} [LinearOrderedSemifield α] : 2⁻¹ < 1

theorem left_lt_add_div_two {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} : a < (a + b) / 2 ↔ a < b

theorem add_div_two_lt_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} : (a + b) / 2 < b ↔ a < b

theorem add_thirds {α : Type u_2} [LinearOrderedSemifield α] (a : α) : a / 3 + a / 3 + a / 3 = a

Miscellaneous lemmas

theorem mul_le_mul_of_mul_div_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (h : a * (b / c) ≤ d) (hc : 0 < c) : b * a ≤ d * c

theorem div_mul_le_div_mul_of_div_le_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} {e : α} (h : a / b ≤ c / d) (he : 0 ≤ e) : a / (b * e) ≤ c / (d * e)

theorem exists_pos_mul_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h : 0 < a) (b : α) : ∃ c, 0 < c ∧ b * c < a

theorem exists_pos_lt_mul {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h : 0 < a) (b : α) : ∃ c, 0 < c ∧ b < c * a

theorem Monotone.div_const {α : Type u_2} [LinearOrderedSemifield α] {β : Type u_4} [Preorder β] {f : β → α} (hf : Monotone f) {c : α} (hc : 0 ≤ c) : Monotone fun x => f x / c

theorem StrictMono.div_const {α : Type u_2} [LinearOrderedSemifield α] {β : Type u_4} [Preorder β] {f : β → α} (hf : StrictMono f) {c : α} (hc : 0 < c) : StrictMono fun x => f x / c

instance LinearOrderedSemiField.toDenselyOrdered {α : Type u_2} [LinearOrderedSemifield α] : DenselyOrdered α

Equations

• @LinearOrderedSemiField.toDenselyOrdered = @LinearOrderedSemiField.toDenselyOrdered.proof_1

theorem min_div_div_right {α : Type u_2} [LinearOrderedSemifield α] {c : α} (hc : 0 ≤ c) (a : α) (b : α) : min (a / c) (b / c) = min a b / c

theorem max_div_div_right {α : Type u_2} [LinearOrderedSemifield α] {c : α} (hc : 0 ≤ c) (a : α) (b : α) : max (a / c) (b / c) = max a b / c

theorem one_div_strictAntiOn {α : Type u_2} [LinearOrderedSemifield α] : StrictAntiOn (fun x => 1 / x) (Set.Ioi 0)

theorem one_div_pow_le_one_div_pow_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a) {m : ℕ} {n : ℕ} (mn : m ≤ n) : 1 / a ^ n ≤ 1 / a ^ m

theorem one_div_pow_lt_one_div_pow_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a) {m : ℕ} {n : ℕ} (mn : m < n) : 1 / a ^ n < 1 / a ^ m

theorem one_div_pow_anti {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a) : Antitone fun n => 1 / a ^ n

theorem one_div_pow_strictAnti {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a) : StrictAnti fun n => 1 / a ^ n

theorem inv_strictAntiOn {α : Type u_2} [LinearOrderedSemifield α] : StrictAntiOn (fun x => x⁻¹) (Set.Ioi 0)

theorem inv_pow_le_inv_pow_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a) {m : ℕ} {n : ℕ} (mn : m ≤ n) : (a ^ n)⁻¹ ≤ (a ^ m)⁻¹

theorem inv_pow_lt_inv_pow_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a) {m : ℕ} {n : ℕ} (mn : m < n) : (a ^ n)⁻¹ < (a ^ m)⁻¹

theorem inv_pow_anti {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a) : Antitone fun n => (a ^ n)⁻¹

theorem inv_pow_strictAnti {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a) : StrictAnti fun n => (a ^ n)⁻¹

Results about IsGLB

theorem IsGLB.mul_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b) : IsGLB ((fun b => a * b) '' s) (a * b)

theorem IsGLB.mul_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b) : IsGLB ((fun b => b * a) '' s) (b * a)

Lemmas about pos, nonneg, nonpos, neg

theorem div_pos_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} : 0 < a / b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0

theorem div_neg_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} : a / b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b

theorem div_nonneg_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} : 0 ≤ a / b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0

theorem div_nonpos_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} : a / b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b

theorem div_nonneg_of_nonpos {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a ≤ 0) (hb : b ≤ 0) : 0 ≤ a / b

theorem div_pos_of_neg_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : 0 < a / b

theorem div_neg_of_neg_of_pos {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : 0 < b) : a / b < 0

theorem div_neg_of_pos_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : 0 < a) (hb : b < 0) : a / b < 0

Relating one division with another term

theorem div_le_iff_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) : b / c ≤ a ↔ a * c ≤ b

theorem div_le_iff_of_neg' {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) : b / c ≤ a ↔ c * a ≤ b

theorem le_div_iff_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) : a ≤ b / c ↔ b ≤ a * c

theorem le_div_iff_of_neg' {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) : a ≤ b / c ↔ b ≤ c * a

theorem div_lt_iff_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) : b / c < a ↔ a * c < b

theorem div_lt_iff_of_neg' {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) : b / c < a ↔ c * a < b

theorem lt_div_iff_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) : a < b / c ↔ b < a * c

theorem lt_div_iff_of_neg' {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) : a < b / c ↔ b < c * a

theorem div_le_one_of_ge {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (h : b ≤ a) (hb : b ≤ 0) : a / b ≤ 1

Bi-implications of inequalities using inversions

theorem inv_le_inv_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a

theorem inv_le_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a

theorem le_inv_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : a ≤ b⁻¹ ↔ b ≤ a⁻¹

theorem inv_lt_inv_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : a⁻¹ < b⁻¹ ↔ b < a

theorem inv_lt_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : a⁻¹ < b ↔ b⁻¹ < a

theorem lt_inv_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : a < b⁻¹ ↔ b < a⁻¹

Relating two divisions

theorem div_le_div_of_nonpos_of_le {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c ≤ 0) (h : b ≤ a) : a / c ≤ b / c

theorem div_lt_div_of_neg_of_lt {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) (h : b < a) : a / c < b / c

theorem div_le_div_right_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) : a / c ≤ b / c ↔ b ≤ a

theorem div_lt_div_right_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) : a / c < b / c ↔ b < a

Relating one division and involving 1

theorem one_le_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) : 1 ≤ a / b ↔ a ≤ b

theorem div_le_one_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) : a / b ≤ 1 ↔ b ≤ a

theorem one_lt_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) : 1 < a / b ↔ a < b

theorem div_lt_one_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) : a / b < 1 ↔ b < a

theorem one_div_le_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : 1 / a ≤ b ↔ 1 / b ≤ a

theorem one_div_lt_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : 1 / a < b ↔ 1 / b < a

theorem le_one_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : a ≤ 1 / b ↔ b ≤ 1 / a

theorem lt_one_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : a < 1 / b ↔ b < 1 / a

theorem one_lt_div_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} : 1 < a / b ↔ 0 < b ∧ b < a ∨ b < 0 ∧ a < b

theorem one_le_div_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} : 1 ≤ a / b ↔ 0 < b ∧ b ≤ a ∨ b < 0 ∧ a ≤ b

theorem div_lt_one_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} : a / b < 1 ↔ 0 < b ∧ a < b ∨ b = 0 ∨ b < 0 ∧ b < a

theorem div_le_one_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} : a / b ≤ 1 ↔ 0 < b ∧ a ≤ b ∨ b = 0 ∨ b < 0 ∧ b ≤ a

Relating two divisions, involving 1

theorem one_div_le_one_div_of_neg_of_le {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) (h : a ≤ b) : 1 / b ≤ 1 / a

theorem one_div_lt_one_div_of_neg_of_lt {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) (h : a < b) : 1 / b < 1 / a

theorem le_of_neg_of_one_div_le_one_div {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) (h : 1 / a ≤ 1 / b) : b ≤ a

theorem lt_of_neg_of_one_div_lt_one_div {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) (h : 1 / a < 1 / b) : b < a

theorem one_div_le_one_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : 1 / a ≤ 1 / b ↔ b ≤ a

For the single implications with fewer assumptions, see one_div_lt_one_div_of_neg_of_lt and lt_of_one_div_lt_one_div

theorem one_div_lt_one_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) : 1 / a < 1 / b ↔ b < a

For the single implications with fewer assumptions, see one_div_lt_one_div_of_lt and lt_of_one_div_lt_one_div

theorem one_div_lt_neg_one {α : Type u_2} [LinearOrderedField α] {a : α} (h1 : a < 0) (h2 : -1 < a) : 1 / a < -1

theorem one_div_le_neg_one {α : Type u_2} [LinearOrderedField α] {a : α} (h1 : a < 0) (h2 : -1 ≤ a) : 1 / a ≤ -1

Results about halving

theorem sub_self_div_two {α : Type u_2} [LinearOrderedField α] (a : α) : a - a / 2 = a / 2

theorem div_two_sub_self {α : Type u_2} [LinearOrderedField α] (a : α) : a / 2 - a = -(a / 2)

theorem add_sub_div_two_lt {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (h : a < b) : a + (b - a) / 2 < b

theorem sub_one_div_inv_le_two {α : Type u_2} [LinearOrderedField α] {a : α} (a2 : 2 ≤ a) : (1 - 1 / a)⁻¹ ≤ 2

An inequality involving 2.

Results about IsLUB

theorem IsLUB.mul_left {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsLUB s b) : IsLUB ((fun b => a * b) '' s) (a * b)

theorem IsLUB.mul_right {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsLUB s b) : IsLUB ((fun b => b * a) '' s) (b * a)

Miscellaneous lemmmas

theorem mul_sub_mul_div_mul_neg_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) : (a * d - b * c) / (c * d) < 0 ↔ a / c < b / d

theorem mul_sub_mul_div_mul_nonpos_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) : (a * d - b * c) / (c * d) ≤ 0 ↔ a / c ≤ b / d

theorem div_lt_div_of_mul_sub_mul_div_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) : (a * d - b * c) / (c * d) < 0 → a / c < b / d

Alias of the forward direction of mul_sub_mul_div_mul_neg_iff.

theorem mul_sub_mul_div_mul_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) : a / c < b / d → (a * d - b * c) / (c * d) < 0

Alias of the reverse direction of mul_sub_mul_div_mul_neg_iff.

theorem div_le_div_of_mul_sub_mul_div_nonpos {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) : (a * d - b * c) / (c * d) ≤ 0 → a / c ≤ b / d

Alias of the forward direction of mul_sub_mul_div_mul_nonpos_iff.

theorem mul_sub_mul_div_mul_nonpos {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) : a / c ≤ b / d → (a * d - b * c) / (c * d) ≤ 0

Alias of the reverse direction of mul_sub_mul_div_mul_nonpos_iff.

theorem exists_add_lt_and_pos_of_lt {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (h : b < a) : ∃ c, b + c < a ∧ 0 < c

theorem le_of_forall_sub_le {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (h : ∀ (ε : α), ε > 0 → b - ε ≤ a) : b ≤ a

theorem mul_self_inj_of_nonneg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (a0 : 0 ≤ a) (b0 : 0 ≤ b) : a * a = b * b ↔ a = b

theorem min_div_div_right_of_nonpos {α : Type u_2} [LinearOrderedField α] {c : α} (hc : c ≤ 0) (a : α) (b : α) : min (a / c) (b / c) = max a b / c

theorem max_div_div_right_of_nonpos {α : Type u_2} [LinearOrderedField α] {c : α} (hc : c ≤ 0) (a : α) (b : α) : max (a / c) (b / c) = min a b / c

theorem abs_inv {α : Type u_2} [LinearOrderedField α] (a : α) : |a⁻¹| = |a|⁻¹

theorem abs_div {α : Type u_2} [LinearOrderedField α] (a : α) (b : α) : |a / b| = |a| / |b|

theorem abs_one_div {α : Type u_2} [LinearOrderedField α] (a : α) : |1 / a| = 1 / |a|