statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
div_lt_iff' (hc : 0 < c) : b / c < a ↔ b < c * a | by rw [mul_comm, div_lt_iff hc] | lemma | div_lt_iff' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_iff",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_le_iff (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ b * c | begin
rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div],
exact div_le_iff' h,
end | lemma | inv_mul_le_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_one_div",
"div_le_iff'",
"inv_eq_one_div",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_le_iff' (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ c * b | by rw [inv_mul_le_iff h, mul_comm] | lemma | inv_mul_le_iff' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_mul_le_iff",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_le_iff (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ b * c | by rw [mul_comm, inv_mul_le_iff h] | lemma | mul_inv_le_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_mul_le_iff",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_le_iff' (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ c * b | by rw [mul_comm, inv_mul_le_iff' h] | lemma | mul_inv_le_iff' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_mul_le_iff'",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_self_le_one (a : α) : a / a ≤ 1 | if h : a = 0 then by simp [h] else by simp [h] | lemma | div_self_le_one | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_lt_iff (h : 0 < b) : b⁻¹ * a < c ↔ a < b * c | begin
rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div],
exact div_lt_iff' h,
end | lemma | inv_mul_lt_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_one_div",
"div_lt_iff'",
"inv_eq_one_div",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_lt_iff' (h : 0 < b) : b⁻¹ * a < c ↔ a < c * b | by rw [inv_mul_lt_iff h, mul_comm] | lemma | inv_mul_lt_iff' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_mul_lt_iff",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_lt_iff (h : 0 < b) : a * b⁻¹ < c ↔ a < b * c | by rw [mul_comm, inv_mul_lt_iff h] | lemma | mul_inv_lt_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_mul_lt_iff",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_lt_iff' (h : 0 < b) : a * b⁻¹ < c ↔ a < c * b | by rw [mul_comm, inv_mul_lt_iff' h] | lemma | mul_inv_lt_iff' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_mul_lt_iff'",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pos_le_iff_one_le_mul (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ b * a | by { rw [inv_eq_one_div], exact div_le_iff ha } | lemma | inv_pos_le_iff_one_le_mul | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff",
"inv_eq_one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pos_le_iff_one_le_mul' (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ a * b | by { rw [inv_eq_one_div], exact div_le_iff' ha } | lemma | inv_pos_le_iff_one_le_mul' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff'",
"inv_eq_one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pos_lt_iff_one_lt_mul (ha : 0 < a) : a⁻¹ < b ↔ 1 < b * a | by { rw [inv_eq_one_div], exact div_lt_iff ha } | lemma | inv_pos_lt_iff_one_lt_mul | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_iff",
"inv_eq_one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pos_lt_iff_one_lt_mul' (ha : 0 < a) : a⁻¹ < b ↔ 1 < a * b | by { rw [inv_eq_one_div], exact div_lt_iff' ha } | lemma | inv_pos_lt_iff_one_lt_mul' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_iff'",
"inv_eq_one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_of_nonneg_of_le_mul (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) : a / b ≤ c | by { rcases eq_or_lt_of_le hb with rfl|hb', simp [hc], rwa [div_le_iff hb'] } | lemma | div_le_of_nonneg_of_le_mul | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff",
"eq_or_lt_of_le"
] | One direction of `div_le_iff` where `b` is allowed to be `0` (but `c` must be nonnegative) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_le_of_nonneg_of_le_div (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b / c) : a * c ≤ b | begin
obtain rfl | hc := hc.eq_or_lt,
{ simpa using hb },
{ rwa le_div_iff hc at h }
end | lemma | mul_le_of_nonneg_of_le_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"le_div_iff"
] | One direction of `div_le_iff` where `c` is allowed to be `0` (but `b` must be nonnegative) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a / b ≤ 1 | div_le_of_nonneg_of_le_mul hb zero_le_one $ by rwa one_mul | lemma | div_le_one_of_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_of_nonneg_of_le_mul",
"one_mul",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_inv_of_le (ha : 0 < a) (h : a ≤ b) : b⁻¹ ≤ a⁻¹ | by rwa [← one_div a, le_div_iff' ha, ← div_eq_mul_inv, div_le_iff (ha.trans_le h), one_mul] | lemma | inv_le_inv_of_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_inv",
"div_le_iff",
"le_div_iff'",
"one_div",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a | by rw [← one_div, div_le_iff ha, ← div_eq_inv_mul, le_div_iff hb, one_mul] | lemma | inv_le_inv | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_inv_mul",
"div_le_iff",
"le_div_iff",
"one_div",
"one_mul"
] | See `inv_le_inv_of_le` for the implication from right-to-left with one fewer assumption. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_le (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a | by rw [← inv_le_inv hb (inv_pos.2 ha), inv_inv] | lemma | inv_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_inv",
"inv_le_inv"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_le_of_inv_le (ha : 0 < a) (h : a⁻¹ ≤ b) : b⁻¹ ≤ a | (inv_le ha ((inv_pos.2 ha).trans_le h)).1 h | lemma | inv_le_of_inv_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_inv (ha : 0 < a) (hb : 0 < b) : a ≤ b⁻¹ ↔ b ≤ a⁻¹ | by rw [← inv_le_inv (inv_pos.2 hb) ha, inv_inv] | lemma | le_inv | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_inv",
"inv_le_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b⁻¹ ↔ b < a | lt_iff_lt_of_le_iff_le (inv_le_inv hb ha) | lemma | inv_lt_inv | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_le_inv",
"lt_iff_lt_of_le_iff_le"
] | See `inv_lt_inv_of_lt` for the implication from right-to-left with one fewer assumption. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_lt_inv_of_lt (hb : 0 < b) (h : b < a) : a⁻¹ < b⁻¹ | (inv_lt_inv (hb.trans h) hb).2 h | lemma | inv_lt_inv_of_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_lt_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b ↔ b⁻¹ < a | lt_iff_lt_of_le_iff_le (le_inv hb ha) | lemma | inv_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"le_inv",
"lt_iff_lt_of_le_iff_le"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_lt_of_inv_lt (ha : 0 < a) (h : a⁻¹ < b) : b⁻¹ < a | (inv_lt ha ((inv_pos.2 ha).trans h)).1 h | lemma | inv_lt_of_inv_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_inv (ha : 0 < a) (hb : 0 < b) : a < b⁻¹ ↔ b < a⁻¹ | lt_iff_lt_of_le_iff_le (inv_le hb ha) | lemma | lt_inv | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_le",
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_one (ha : 1 < a) : a⁻¹ < 1 | by rwa [inv_lt (zero_lt_one.trans ha) zero_lt_one, inv_one] | lemma | inv_lt_one | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_lt",
"inv_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_inv (h₁ : 0 < a) (h₂ : a < 1) : 1 < a⁻¹ | by rwa [lt_inv zero_lt_one h₁, inv_one] | lemma | one_lt_inv | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_one",
"lt_inv",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_one (ha : 1 ≤ a) : a⁻¹ ≤ 1 | by rwa [inv_le (zero_lt_one.trans_le ha) zero_lt_one, inv_one] | lemma | inv_le_one | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_le",
"inv_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_inv (h₁ : 0 < a) (h₂ : a ≤ 1) : 1 ≤ a⁻¹ | by rwa [le_inv zero_lt_one h₁, inv_one] | lemma | one_le_inv | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_one",
"le_inv",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_one_iff_of_pos (h₀ : 0 < a) : a⁻¹ < 1 ↔ 1 < a | ⟨λ h₁, inv_inv a ▸ one_lt_inv (inv_pos.2 h₀) h₁, inv_lt_one⟩ | lemma | inv_lt_one_iff_of_pos | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_inv",
"one_lt_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_one_iff : a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a | begin
cases le_or_lt a 0 with ha ha,
{ simp [ha, (inv_nonpos.2 ha).trans_lt zero_lt_one] },
{ simp only [ha.not_le, false_or, inv_lt_one_iff_of_pos ha] }
end | lemma | inv_lt_one_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_lt_one_iff_of_pos",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_inv_iff : 1 < a⁻¹ ↔ 0 < a ∧ a < 1 | ⟨λ h, ⟨inv_pos.1 (zero_lt_one.trans h), inv_inv a ▸ inv_lt_one h⟩, and_imp.2 one_lt_inv⟩ | lemma | one_lt_inv_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_inv",
"inv_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_one_iff : a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a | begin
rcases em (a = 1) with (rfl|ha),
{ simp [le_rfl] },
{ simp only [ne.le_iff_lt (ne.symm ha), ne.le_iff_lt (mt inv_eq_one.1 ha), inv_lt_one_iff] }
end | lemma | inv_le_one_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"em",
"inv_lt_one_iff",
"le_rfl",
"ne.le_iff_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_inv_iff : 1 ≤ a⁻¹ ↔ 0 < a ∧ a ≤ 1 | ⟨λ h, ⟨inv_pos.1 (zero_lt_one.trans_le h), inv_inv a ▸ inv_le_one h⟩, and_imp.2 one_le_inv⟩ | lemma | one_le_inv_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_inv",
"inv_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_of_le (hc : 0 ≤ c) (h : a ≤ b) : a / c ≤ b / c | begin
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],
exact mul_le_mul_of_nonneg_right h (one_div_nonneg.2 hc)
end | lemma | div_le_div_of_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_one_div",
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_of_le_left (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) : a / b ≤ a / c | begin
rw [div_eq_mul_inv, div_eq_mul_inv],
exact mul_le_mul_of_nonneg_left ((inv_le_inv (hc.trans_le h) hc).mpr h) ha
end | lemma | div_le_div_of_le_left | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_inv",
"inv_le_inv",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_of_le_of_nonneg (hab : a ≤ b) (hc : 0 ≤ c) : a / c ≤ b / c | div_le_div_of_le hc hab | lemma | div_le_div_of_le_of_nonneg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_of_lt (hc : 0 < c) (h : a < b) : a / c < b / c | begin
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],
exact mul_lt_mul_of_pos_right h (one_div_pos.2 hc)
end | lemma | div_lt_div_of_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_one_div",
"mul_lt_mul_of_pos_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_right (hc : 0 < c) : a / c ≤ b / c ↔ a ≤ b | ⟨le_imp_le_of_lt_imp_lt $ div_lt_div_of_lt hc, div_le_div_of_le $ hc.le⟩ | lemma | div_le_div_right | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_of_le",
"div_lt_div_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_right (hc : 0 < c) : a / c < b / c ↔ a < b | lt_iff_lt_of_le_iff_le $ div_le_div_right hc | lemma | div_lt_div_right | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_right",
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b < a / c ↔ c < b | by simp only [div_eq_mul_inv, mul_lt_mul_left ha, inv_lt_inv hb hc] | lemma | div_lt_div_left | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_inv",
"inv_lt_inv",
"mul_lt_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b ≤ a / c ↔ c ≤ b | le_iff_le_iff_lt_iff_lt.2 (div_lt_div_left ha hc hb) | lemma | div_le_div_left | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_div_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_iff (b0 : 0 < b) (d0 : 0 < d) :
a / b < c / d ↔ a * d < c * b | by rw [lt_div_iff d0, div_mul_eq_mul_div, div_lt_iff b0] | lemma | div_lt_div_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_iff",
"div_mul_eq_mul_div",
"lt_div_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_iff (b0 : 0 < b) (d0 : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b | by rw [le_div_iff d0, div_mul_eq_mul_div, div_le_iff b0] | lemma | div_le_div_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff",
"div_mul_eq_mul_div",
"le_div_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hbd : d ≤ b) : a / b ≤ c / d | by { rw div_le_div_iff (hd.trans_le hbd) hd, exact mul_le_mul hac hbd hd.le hc } | lemma | div_le_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_iff",
"mul_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div (hac : a < c) (hbd : d ≤ b) (c0 : 0 ≤ c) (d0 : 0 < d) :
a / b < c / d | (div_lt_div_iff (d0.trans_le hbd) d0).2 (mul_lt_mul hac hbd d0 c0) | lemma | div_lt_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_div_iff",
"mul_lt_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div' (hac : a ≤ c) (hbd : d < b) (c0 : 0 < c) (d0 : 0 < d) :
a / b < c / d | (div_lt_div_iff (d0.trans hbd) d0).2 (mul_lt_mul' hac hbd d0.le c0) | lemma | div_lt_div' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_div_iff",
"mul_lt_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_of_lt_left (hc : 0 < c) (hb : 0 < b) (h : b < a) : c / a < c / b | (div_lt_div_left hc (hb.trans h) hb).mpr h | lemma | div_lt_div_of_lt_left | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_div_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_self (ha : 0 ≤ a) (hb : 1 ≤ b) : a / b ≤ a | by simpa only [div_one] using div_le_div_of_le_left ha zero_lt_one hb | lemma | div_le_self | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_of_le_left",
"div_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_self (ha : 0 < a) (hb : 1 < b) : a / b < a | by simpa only [div_one] using div_lt_div_of_lt_left ha zero_lt_one hb | lemma | div_lt_self | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_div_of_lt_left",
"div_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_self (ha : 0 ≤ a) (hb₀ : 0 < b) (hb₁ : b ≤ 1) : a ≤ a / b | by simpa only [div_one] using div_le_div_of_le_left ha hb₀ hb₁ | lemma | le_div_self | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_of_le_left",
"div_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_div (hb : 0 < b) : 1 ≤ a / b ↔ b ≤ a | by rw [le_div_iff hb, one_mul] | lemma | one_le_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"le_div_iff",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_one (hb : 0 < b) : a / b ≤ 1 ↔ a ≤ b | by rw [div_le_iff hb, one_mul] | lemma | div_le_one | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_div (hb : 0 < b) : 1 < a / b ↔ b < a | by rw [lt_div_iff hb, one_mul] | lemma | one_lt_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"lt_div_iff",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_one (hb : 0 < b) : a / b < 1 ↔ a < b | by rw [div_lt_iff hb, one_mul] | lemma | div_lt_one | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_iff",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_le (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ b ↔ 1 / b ≤ a | by simpa using inv_le ha hb | lemma | one_div_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_lt (ha : 0 < a) (hb : 0 < b) : 1 / a < b ↔ 1 / b < a | by simpa using inv_lt ha hb | lemma | one_div_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_div (ha : 0 < a) (hb : 0 < b) : a ≤ 1 / b ↔ b ≤ 1 / a | by simpa using le_inv ha hb | lemma | le_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"le_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_one_div (ha : 0 < a) (hb : 0 < b) : a < 1 / b ↔ b < 1 / a | by simpa using lt_inv ha hb | lemma | lt_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"lt_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_le_one_div_of_le (ha : 0 < a) (h : a ≤ b) : 1 / b ≤ 1 / a | by simpa using inv_le_inv_of_le ha h | lemma | one_div_le_one_div_of_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_le_inv_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_lt_one_div_of_lt (ha : 0 < a) (h : a < b) : 1 / b < 1 / a | by rwa [lt_div_iff' ha, ← div_eq_mul_one_div, div_lt_one (ha.trans h)] | lemma | one_div_lt_one_div_of_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_one_div",
"div_lt_one",
"lt_div_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_one_div_le_one_div (ha : 0 < a) (h : 1 / a ≤ 1 / b) : b ≤ a | le_imp_le_of_lt_imp_lt (one_div_lt_one_div_of_lt ha) h | lemma | le_of_one_div_le_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div_lt_one_div_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_one_div_lt_one_div (ha : 0 < a) (h : 1 / a < 1 / b) : b < a | lt_imp_lt_of_le_imp_le (one_div_le_one_div_of_le ha) h | lemma | lt_of_one_div_lt_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"lt_imp_lt_of_le_imp_le",
"one_div_le_one_div_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_le_one_div (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ 1 / b ↔ b ≤ a | div_le_div_left zero_lt_one ha hb | lemma | one_div_le_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_left",
"zero_lt_one"
] | For the single implications with fewer assumptions, see `one_div_le_one_div_of_le` and
`le_of_one_div_le_one_div` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_div_lt_one_div (ha : 0 < a) (hb : 0 < b) : 1 / a < 1 / b ↔ b < a | div_lt_div_left zero_lt_one ha hb | lemma | one_div_lt_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_div_left",
"zero_lt_one"
] | For the single implications with fewer assumptions, see `one_div_lt_one_div_of_lt` and
`lt_of_one_div_lt_one_div` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_lt_one_div (h1 : 0 < a) (h2 : a < 1) : 1 < 1 / a | by rwa [lt_one_div zero_lt_one h1, one_div_one] | lemma | one_lt_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"lt_one_div",
"one_div_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_one_div (h1 : 0 < a) (h2 : a ≤ 1) : 1 ≤ 1 / a | by rwa [le_one_div zero_lt_one h1, one_div_one] | lemma | one_le_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"le_one_div",
"one_div_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_halves (a : α) : a / 2 + a / 2 = a | by rw [div_add_div_same, ← two_mul, mul_div_cancel_left a two_ne_zero] | lemma | add_halves | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_add_div_same",
"mul_div_cancel_left",
"two_mul",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_self_div_two (a : α) : (a + a) / 2 = a | by rw [← mul_two, mul_div_cancel a two_ne_zero] | lemma | add_self_div_two | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"mul_div_cancel",
"mul_two",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
half_pos (h : 0 < a) : 0 < a / 2 | div_pos h zero_lt_two | lemma | half_pos | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_pos",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_half_pos : (0:α) < 1 / 2 | half_pos zero_lt_one | lemma | one_half_pos | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"half_pos",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
half_le_self_iff : a / 2 ≤ a ↔ 0 ≤ a | by rw [div_le_iff (zero_lt_two' α), mul_two, le_add_iff_nonneg_left] | lemma | half_le_self_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff",
"mul_two",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
half_lt_self_iff : a / 2 < a ↔ 0 < a | by rw [div_lt_iff (zero_lt_two' α), mul_two, lt_add_iff_pos_left] | lemma | half_lt_self_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_iff",
"mul_two",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_half_lt_one : (1 / 2 : α) < 1 | half_lt_self zero_lt_one | lemma | one_half_lt_one | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_inv_lt_one : (2⁻¹ : α) < 1 | (one_div _).symm.trans_lt one_half_lt_one | lemma | two_inv_lt_one | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div",
"one_half_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lt_add_div_two : a < (a + b) / 2 ↔ a < b | by simp [lt_div_iff, mul_two] | lemma | left_lt_add_div_two | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"lt_div_iff",
"mul_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_div_two_lt_right : (a + b) / 2 < b ↔ a < b | by simp [div_lt_iff, mul_two] | lemma | add_div_two_lt_right | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_iff",
"mul_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_mul_div_le (h : a * (b / c) ≤ d) (hc : 0 < c) : b * a ≤ d * c | begin
rw [← mul_div_assoc] at h,
rwa [mul_comm b, ← div_le_iff hc],
end | lemma | mul_le_mul_of_mul_div_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff",
"mul_comm",
"mul_div_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_le_div_mul_of_div_le_div (h : a / b ≤ c / d) (he : 0 ≤ e) :
a / (b * e) ≤ c / (d * e) | begin
rw [div_mul_eq_div_mul_one_div, div_mul_eq_div_mul_one_div],
exact mul_le_mul_of_nonneg_right h (one_div_nonneg.2 he)
end | lemma | div_mul_le_div_mul_of_div_le_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_mul_eq_div_mul_one_div",
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pos_mul_lt {a : α} (h : 0 < a) (b : α) : ∃ c : α, 0 < c ∧ b * c < a | begin
have : 0 < a / max (b + 1) 1, from div_pos h (lt_max_iff.2 (or.inr zero_lt_one)),
refine ⟨a / max (b + 1) 1, this, _⟩,
rw [← lt_div_iff this, div_div_cancel' h.ne'],
exact lt_max_iff.2 (or.inl $ lt_add_one _)
end | lemma | exists_pos_mul_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_div_cancel'",
"div_pos",
"lt_add_one",
"lt_div_iff",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pos_lt_mul {a : α} (h : 0 < a) (b : α) : ∃ c : α, 0 < c ∧ b < c * a | let ⟨c, hc₀, hc⟩ := exists_pos_mul_lt h b
in ⟨c⁻¹, inv_pos.2 hc₀, by rwa [← div_eq_inv_mul, lt_div_iff hc₀]⟩ | lemma | exists_pos_lt_mul | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_inv_mul",
"exists_pos_mul_lt",
"lt_div_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.div_const {β : Type*} [preorder β] {f : β → α} (hf : monotone f)
{c : α} (hc : 0 ≤ c) : monotone (λ x, (f x) / c) | begin
haveI := @linear_order.decidable_le α _,
simpa only [div_eq_mul_inv] using (monotone_mul_right_of_nonneg (inv_nonneg.2 hc)).comp hf
end | lemma | monotone.div_const | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_inv",
"monotone",
"monotone_mul_right_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.div_const {β : Type*} [preorder β] {f : β → α} (hf : strict_mono f)
{c : α} (hc : 0 < c) :
strict_mono (λ x, (f x) / c) | by simpa only [div_eq_mul_inv] using hf.mul_const (inv_pos.2 hc) | lemma | strict_mono.div_const | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_inv",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_semifield.to_densely_ordered : densely_ordered α | { dense := λ a₁ a₂ h, ⟨(a₁ + a₂) / 2,
calc a₁ = (a₁ + a₁) / 2 : (add_self_div_two a₁).symm
... < (a₁ + a₂) / 2 : div_lt_div_of_lt zero_lt_two (add_lt_add_left h _),
calc (a₁ + a₂) / 2 < (a₂ + a₂) / 2 : div_lt_div_of_lt zero_lt_two (add_lt_add_right h _)
... = a₂ : add_self_div_two ... | instance | linear_ordered_semifield.to_densely_ordered | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"add_self_div_two",
"dense",
"densely_ordered",
"div_lt_div_of_lt",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_div_div_right {c : α} (hc : 0 ≤ c) (a b : α) : min (a / c) (b / c) = (min a b) / c | eq.symm $ monotone.map_min (λ x y, div_le_div_of_le hc) | lemma | min_div_div_right | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_of_le",
"monotone.map_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_div_div_right {c : α} (hc : 0 ≤ c) (a b : α) : max (a / c) (b / c) = (max a b) / c | eq.symm $ monotone.map_max (λ x y, div_le_div_of_le hc) | lemma | max_div_div_right | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_of_le",
"monotone.map_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_strict_anti_on : strict_anti_on (λ x : α, 1 / x) (set.Ioi 0) | λ x x1 y y1 xy, (one_div_lt_one_div (set.mem_Ioi.mp y1) (set.mem_Ioi.mp x1)).mpr xy | lemma | one_div_strict_anti_on | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div_lt_one_div",
"set.Ioi",
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_pow_le_one_div_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) :
1 / a ^ n ≤ 1 / a ^ m | by refine (one_div_le_one_div _ _).mpr (pow_le_pow a1 mn);
exact pow_pos (zero_lt_one.trans_le a1) _ | lemma | one_div_pow_le_one_div_pow_of_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div_le_one_div",
"pow_le_pow",
"pow_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_pow_lt_one_div_pow_of_lt (a1 : 1 < a) {m n : ℕ} (mn : m < n) :
1 / a ^ n < 1 / a ^ m | by refine (one_div_lt_one_div _ _).mpr (pow_lt_pow a1 mn);
exact pow_pos (trans zero_lt_one a1) _ | lemma | one_div_pow_lt_one_div_pow_of_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div_lt_one_div",
"pow_lt_pow",
"pow_pos",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_pow_anti (a1 : 1 ≤ a) : antitone (λ n : ℕ, 1 / a ^ n) | λ m n, one_div_pow_le_one_div_pow_of_le a1 | lemma | one_div_pow_anti | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"antitone",
"one_div_pow_le_one_div_pow_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_pow_strict_anti (a1 : 1 < a) : strict_anti (λ n : ℕ, 1 / a ^ n) | λ m n, one_div_pow_lt_one_div_pow_of_lt a1 | lemma | one_div_pow_strict_anti | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div_pow_lt_one_div_pow_of_lt",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_strict_anti_on : strict_anti_on (λ x : α, x⁻¹) (set.Ioi 0) | λ x hx y hy xy, (inv_lt_inv hy hx).2 xy | lemma | inv_strict_anti_on | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_lt_inv",
"set.Ioi",
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pow_le_inv_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) :
(a ^ n)⁻¹ ≤ (a ^ m)⁻¹ | by convert one_div_pow_le_one_div_pow_of_le a1 mn; simp | lemma | inv_pow_le_inv_pow_of_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div_pow_le_one_div_pow_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pow_lt_inv_pow_of_lt (a1 : 1 < a) {m n : ℕ} (mn : m < n) :
(a ^ n)⁻¹ < (a ^ m)⁻¹ | by convert one_div_pow_lt_one_div_pow_of_lt a1 mn; simp | lemma | inv_pow_lt_inv_pow_of_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div_pow_lt_one_div_pow_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pow_anti (a1 : 1 ≤ a) : antitone (λ n : ℕ, (a ^ n)⁻¹) | λ m n, inv_pow_le_inv_pow_of_le a1 | lemma | inv_pow_anti | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"antitone",
"inv_pow_le_inv_pow_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pow_strict_anti (a1 : 1 < a) : strict_anti (λ n : ℕ, (a ^ n)⁻¹) | λ m n, inv_pow_lt_inv_pow_of_lt a1 | lemma | inv_pow_strict_anti | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_pow_lt_inv_pow_of_lt",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.mul_left {s : set α} (ha : 0 ≤ a) (hs : is_glb s b) :
is_glb ((λ b, a * b) '' s) (a * b) | begin
rcases lt_or_eq_of_le ha with ha | rfl,
{ exact (order_iso.mul_left₀ _ ha).is_glb_image'.2 hs, },
{ simp_rw zero_mul,
rw hs.nonempty.image_const,
exact is_glb_singleton },
end | lemma | is_glb.mul_left | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"is_glb",
"is_glb_singleton",
"order_iso.mul_left₀",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.mul_right {s : set α} (ha : 0 ≤ a) (hs : is_glb s b) :
is_glb ((λ b, b * a) '' s) (b * a) | by simpa [mul_comm] using hs.mul_left ha | lemma | is_glb.mul_right | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"is_glb",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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