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
value |
|---|---|---|---|---|---|---|---|---|---|---|
is_upper_set.smul_subset (hs : is_upper_set s) (hx : 1 ≤ x) : x • s ⊆ s | smul_set_subset_iff.2 $ λ y, hs $ le_mul_of_one_le_left' hx | lemma | is_upper_set.smul_subset | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"is_upper_set",
"le_mul_of_one_le_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.smul_subset (hs : is_lower_set s) (hx : x ≤ 1) : x • s ⊆ s | smul_set_subset_iff.2 $ λ y, hs $ mul_le_of_le_one_left' hx | lemma | is_lower_set.smul_subset | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"is_lower_set",
"mul_le_of_le_one_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.smul (hs : is_upper_set s) : is_upper_set (a • s) | hs.image $ order_iso.mul_left _ | lemma | is_upper_set.smul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"is_upper_set",
"order_iso.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.smul (hs : is_lower_set s) : is_lower_set (a • s) | hs.image $ order_iso.mul_left _ | lemma | is_lower_set.smul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"is_lower_set",
"order_iso.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.ord_connected.smul (hs : s.ord_connected) : (a • s).ord_connected | begin
rw [←hs.upper_closure_inter_lower_closure, smul_set_inter],
exact (upper_closure _).upper.smul.ord_connected.inter (lower_closure _).lower.smul.ord_connected,
end | lemma | set.ord_connected.smul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"lower_closure",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.mul_left (ht : is_upper_set t) : is_upper_set (s * t) | by { rw [←smul_eq_mul, ←bUnion_smul_set], exact is_upper_set_Union₂ (λ x hx, ht.smul) } | lemma | is_upper_set.mul_left | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"is_upper_set",
"is_upper_set_Union₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.mul_right (hs : is_upper_set s) : is_upper_set (s * t) | by { rw mul_comm, exact hs.mul_left } | lemma | is_upper_set.mul_right | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"is_upper_set",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.mul_left (ht : is_lower_set t) : is_lower_set (s * t) | ht.to_dual.mul_left | lemma | is_lower_set.mul_left | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.mul_right (hs : is_lower_set s) : is_lower_set (s * t) | hs.to_dual.mul_right | lemma | is_lower_set.mul_right | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.inv (hs : is_upper_set s) : is_lower_set s⁻¹ | λ x y h, hs $ inv_le_inv' h | lemma | is_upper_set.inv | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"inv_le_inv'",
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.inv (hs : is_lower_set s) : is_upper_set s⁻¹ | λ x y h, hs $ inv_le_inv' h | lemma | is_lower_set.inv | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"inv_le_inv'",
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.div_left (ht : is_upper_set t) : is_lower_set (s / t) | by { rw div_eq_mul_inv, exact ht.inv.mul_left } | lemma | is_upper_set.div_left | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"div_eq_mul_inv",
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.div_right (hs : is_upper_set s) : is_upper_set (s / t) | by { rw div_eq_mul_inv, exact hs.mul_right } | lemma | is_upper_set.div_right | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"div_eq_mul_inv",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.div_left (ht : is_lower_set t) : is_upper_set (s / t) | ht.to_dual.div_left | lemma | is_lower_set.div_left | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.div_right (hs : is_lower_set s) : is_lower_set (s / t) | hs.to_dual.div_right | lemma | is_lower_set.div_right | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : upper_set α) : set α) = set.Ici 1 | rfl | lemma | upper_set.coe_one | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"set.Ici",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (a : α) (s : upper_set α) : (↑(a • s) : set α) = a • s | rfl | lemma | upper_set.coe_smul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (s t : upper_set α) : (↑(s * t) : set α) = s * t | rfl | lemma | upper_set.coe_mul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div (s t : upper_set α) : (↑(s / t) : set α) = s / t | rfl | lemma | upper_set.coe_div | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_one : Ici (1 : α) = 1 | rfl | lemma | upper_set.Ici_one | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mul (s : upper_set α) : 1 * s = s | set_like.coe_injective $ (subset_mul_right _ left_mem_Ici).antisymm' $
by { rw [←smul_eq_mul, ←bUnion_smul_set], exact Union₂_subset (λ _, s.upper.smul_subset) } | lemma | upper_set.one_mul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"antisymm'",
"one_mul",
"set_like.coe_injective",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (a : α) (s : lower_set α) : (↑(a • s) : set α) = a • s | rfl | lemma | lower_set.coe_smul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (s t : lower_set α) : (↑(s * t) : set α) = s * t | rfl | lemma | lower_set.coe_mul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div (s t : lower_set α) : (↑(s / t) : set α) = s / t | rfl | lemma | lower_set.coe_div | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_one : Iic (1 : α) = 1 | rfl | lemma | lower_set.Iic_one | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mul (s : lower_set α) : 1 * s = s | set_like.coe_injective $ (subset_mul_right _ right_mem_Iic).antisymm' $
by { rw [←smul_eq_mul, ←bUnion_smul_set], exact Union₂_subset (λ _, s.lower.smul_subset) } | lemma | lower_set.one_mul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"antisymm'",
"lower_set",
"one_mul",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_one : upper_closure (1 : set α) = 1 | upper_closure_singleton _ | lemma | upper_closure_one | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"upper_closure",
"upper_closure_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_one : lower_closure (1 : set α) = 1 | lower_closure_singleton _ | lemma | lower_closure_one | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"lower_closure",
"lower_closure_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_smul : upper_closure (a • s) = a • upper_closure s | upper_closure_image $ order_iso.mul_left a | lemma | upper_closure_smul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"order_iso.mul_left",
"upper_closure",
"upper_closure_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_smul : lower_closure (a • s) = a • lower_closure s | lower_closure_image $ order_iso.mul_left a | lemma | lower_closure_smul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"lower_closure",
"lower_closure_image",
"order_iso.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_upper_closure : s * upper_closure t = upper_closure (s * t) | by simp_rw [←smul_eq_mul, ←bUnion_smul_set, upper_closure_Union, upper_closure_smul,
upper_set.coe_infi₂, upper_set.coe_smul] | lemma | mul_upper_closure | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"upper_closure",
"upper_closure_Union",
"upper_closure_smul",
"upper_set.coe_infi₂",
"upper_set.coe_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lower_closure : s * lower_closure t = lower_closure (s * t) | by simp_rw [←smul_eq_mul, ←bUnion_smul_set, lower_closure_Union, lower_closure_smul,
lower_set.coe_supr₂, lower_set.coe_smul] | lemma | mul_lower_closure | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"lower_closure",
"lower_closure_Union",
"lower_closure_smul",
"lower_set.coe_smul",
"lower_set.coe_supr₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_mul : ↑(upper_closure s) * t = upper_closure (s * t) | by { simp_rw mul_comm _ t, exact mul_upper_closure _ _ } | lemma | upper_closure_mul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"mul_comm",
"mul_upper_closure",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_mul : ↑(lower_closure s) * t = lower_closure (s * t) | by { simp_rw mul_comm _ t, exact mul_lower_closure _ _ } | lemma | lower_closure_mul | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"lower_closure",
"mul_comm",
"mul_lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_mul_distrib : upper_closure (s * t) = upper_closure s * upper_closure t | set_like.coe_injective $
by rw [upper_set.coe_mul, mul_upper_closure, upper_closure_mul, upper_set.upper_closure] | lemma | upper_closure_mul_distrib | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"mul_upper_closure",
"set_like.coe_injective",
"upper_closure",
"upper_closure_mul",
"upper_set.coe_mul",
"upper_set.upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_mul_distrib : lower_closure (s * t) = lower_closure s * lower_closure t | set_like.coe_injective $
by rw [lower_set.coe_mul, mul_lower_closure, lower_closure_mul, lower_set.lower_closure] | lemma | lower_closure_mul_distrib | algebra.order | src/algebra/order/upper_lower.lean | [
"algebra.order.group.defs",
"data.set.pointwise.smul",
"order.upper_lower.basic"
] | [
"lower_closure",
"lower_closure_mul",
"lower_set.coe_mul",
"lower_set.lower_closure",
"mul_lower_closure",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_group_with_zero (α : Type*)
extends linear_ordered_comm_monoid_with_zero α, comm_group_with_zero α | class | linear_ordered_comm_group_with_zero | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"comm_group_with_zero",
"linear_ordered_comm_monoid_with_zero"
] | A linearly ordered commutative group with a zero element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.linear_ordered_comm_monoid_with_zero {β : Type*}
[has_zero β] [has_one β] [has_mul β] [has_pow β ℕ] [has_sup β] [has_inf β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(hsup : ∀ x y, f ... | { zero_le_one := show f 0 ≤ f 1, by simp only [zero, one,
linear_ordered_comm_monoid_with_zero.zero_le_one],
..linear_order.lift f hf hsup hinf,
..hf.ordered_comm_monoid f one mul npow,
..hf.comm_monoid_with_zero f zero one mul npow } | def | function.injective.linear_ordered_comm_monoid_with_zero | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"has_inf",
"has_sup",
"linear_order.lift",
"linear_ordered_comm_monoid_with_zero",
"zero_le_one"
] | Pullback a `linear_ordered_comm_monoid_with_zero` under an injective map.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_le' : 0 ≤ a | by simpa only [mul_zero, mul_one] using mul_le_mul_left' zero_le_one a | lemma | zero_le' | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"mul_le_mul_left'",
"mul_one",
"mul_zero",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_lt_zero' : ¬a < 0 | not_lt_of_le zero_le' | lemma | not_lt_zero' | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"not_lt_of_le",
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_zero_iff : a ≤ 0 ↔ a = 0 | ⟨λ h, le_antisymm h zero_le', λ h, h ▸ le_rfl⟩ | lemma | le_zero_iff | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_iff : 0 < a ↔ a ≠ 0 | ⟨ne_of_gt, λ h, lt_of_le_of_ne zero_le' h.symm⟩ | lemma | zero_lt_iff | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_lt (h : b < a) : a ≠ 0 | λ h1, not_lt_zero' $ show b < 0, from h1 ▸ h | lemma | ne_zero_of_lt | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"not_lt_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_one₀ (ha : a ≤ 1) (hb : b ≤ 1) : a * b ≤ 1 | mul_le_one' ha hb | lemma | mul_le_one₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [] | Alias of `mul_le_one'` for unification. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_le_mul₀ (ha : 1 ≤ a) (hb : 1 ≤ b) : 1 ≤ a * b | one_le_mul ha hb | lemma | one_le_mul₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [] | Alias of `one_le_mul'` for unification. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_le_mul_right (h : c ≠ 0) (hab : a * c ≤ b * c) : a ≤ b | by simpa only [mul_inv_cancel_right₀ h] using (mul_le_mul_right' hab c⁻¹) | lemma | le_of_le_mul_right | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"mul_inv_cancel_right₀",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_inv_of_mul_le (h : c ≠ 0) (hab : a * c ≤ b) : a ≤ b * c⁻¹ | le_of_le_mul_right h (by simpa [h] using hab) | lemma | le_mul_inv_of_mul_le | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"le_of_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_le_of_le_mul (hab : a ≤ b * c) : a * c⁻¹ ≤ b | begin
by_cases h : c = 0,
{ simp [h], },
{ exact le_of_le_mul_right h (by simpa [h] using hab), },
end | lemma | mul_inv_le_of_le_mul | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"le_of_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_one₀ (ha : a ≠ 0) : a⁻¹ ≤ 1 ↔ 1 ≤ a | @inv_le_one' _ _ _ _ $ units.mk0 a ha | lemma | inv_le_one₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_inv₀ (ha : a ≠ 0) : 1 ≤ a⁻¹ ↔ a ≤ 1 | @one_le_inv' _ _ _ _ $ units.mk0 a ha | lemma | one_le_inv₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_inv_iff₀ (hc : c ≠ 0) : a ≤ b * c⁻¹ ↔ a * c ≤ b | ⟨λ h, inv_inv c ▸ mul_inv_le_of_le_mul h, le_mul_inv_of_mul_le hc⟩ | lemma | le_mul_inv_iff₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"inv_inv",
"le_mul_inv_of_mul_le",
"mul_inv_le_of_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_le_iff₀ (hc : c ≠ 0) : a * c⁻¹ ≤ b ↔ a ≤ b * c | ⟨λ h, inv_inv c ▸ le_mul_inv_of_mul_le (inv_ne_zero hc) h, mul_inv_le_of_le_mul⟩ | lemma | mul_inv_le_iff₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"inv_inv",
"inv_ne_zero",
"le_mul_inv_of_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div₀ (a b c d : α) (hb : b ≠ 0) (hd : d ≠ 0) :
a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b | if ha : a = 0 then by simp [ha] else
if hc : c = 0 then by simp [inv_ne_zero hb, hc, hd] else
show (units.mk0 a ha) * (units.mk0 b hb)⁻¹ ≤ (units.mk0 c hc) * (units.mk0 d hd)⁻¹ ↔
(units.mk0 a ha) * (units.mk0 d hd) ≤ (units.mk0 c hc) * (units.mk0 b hb),
from mul_inv_le_mul_inv_iff' | lemma | div_le_div₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"inv_ne_zero",
"mul_inv_le_mul_inv_iff'",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.zero_lt (u : αˣ) : (0 : α) < u | zero_lt_iff.2 $ u.ne_zero | lemma | units.zero_lt | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_lt_of_le₀ (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d | have hd : d ≠ 0 := ne_zero_of_lt hcd,
if ha : a = 0 then by { rw [ha, zero_mul, zero_lt_iff], exact mul_ne_zero hb hd } else
if hc : c = 0 then by { rw [hc, mul_zero, zero_lt_iff], exact mul_ne_zero hb hd } else
show (units.mk0 a ha) * (units.mk0 c hc) < (units.mk0 b hb) * (units.mk0 d hd),
from mul_lt_mul_of_le_of_lt ... | lemma | mul_lt_mul_of_lt_of_le₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"mul_lt_mul_of_le_of_lt",
"mul_ne_zero",
"mul_zero",
"ne_zero_of_lt",
"units.mk0",
"zero_lt_iff",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul₀ (hab : a < b) (hcd : c < d) : a * c < b * d | mul_lt_mul_of_lt_of_le₀ hab.le (ne_zero_of_lt hab) hcd | lemma | mul_lt_mul₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"mul_lt_mul_of_lt_of_le₀",
"ne_zero_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_lt_of_lt_mul₀ (h : x < y * z) : x * z⁻¹ < y | by { contrapose! h, simpa only [inv_inv] using mul_inv_le_of_le_mul h } | lemma | mul_inv_lt_of_lt_mul₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"inv_inv",
"mul_inv_le_of_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_lt_of_lt_mul₀ (h : x < y * z) : y⁻¹ * x < z | by { rw mul_comm at *, exact mul_inv_lt_of_lt_mul₀ h } | lemma | inv_mul_lt_of_lt_mul₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"mul_comm",
"mul_inv_lt_of_lt_mul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_right₀ (c : α) (h : a < b) (hc : c ≠ 0) : a * c < b * c | by { contrapose! h, exact le_of_le_mul_right hc h } | lemma | mul_lt_right₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"le_of_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_inv₀ (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ < b⁻¹ ↔ b < a | show (units.mk0 a ha)⁻¹ < (units.mk0 b hb)⁻¹ ↔ (units.mk0 b hb) < (units.mk0 a ha),
from inv_lt_inv_iff | lemma | inv_lt_inv₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"inv_lt_inv_iff",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_inv₀ (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a | show (units.mk0 a ha)⁻¹ ≤ (units.mk0 b hb)⁻¹ ↔ (units.mk0 b hb) ≤ (units.mk0 a ha),
from inv_le_inv_iff | lemma | inv_le_inv₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"inv_le_inv_iff",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d | begin
have ha : a ≠ 0 := ne_of_gt (lt_of_lt_of_le hc hh),
simp_rw ← inv_le_inv₀ ha (ne_of_gt hc) at hh,
have := mul_lt_mul_of_lt_of_le₀ hh (inv_ne_zero (ne_of_gt hc)) h,
simpa [inv_mul_cancel_left₀ ha, inv_mul_cancel_left₀ (ne_of_gt hc)] using this,
end | lemma | lt_of_mul_lt_mul_of_le₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"inv_le_inv₀",
"inv_mul_cancel_left₀",
"inv_ne_zero",
"mul_lt_mul_of_lt_of_le₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_right₀ (hc : c ≠ 0) : a * c ≤ b * c ↔ a ≤ b | ⟨le_of_le_mul_right hc, λ hab, mul_le_mul_right' hab _⟩ | lemma | mul_le_mul_right₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_left₀ (ha : a ≠ 0) : a * b ≤ a * c ↔ b ≤ c | by {simp only [mul_comm a], exact mul_le_mul_right₀ ha } | lemma | mul_le_mul_left₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"mul_comm",
"mul_le_mul_right₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_right₀ (hc : c ≠ 0) : a / c ≤ b / c ↔ a ≤ b | by rw [div_eq_mul_inv, div_eq_mul_inv, mul_le_mul_right₀ (inv_ne_zero hc)] | lemma | div_le_div_right₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"div_eq_mul_inv",
"inv_ne_zero",
"mul_le_mul_right₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_left₀ (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) : a / b ≤ a / c ↔ c ≤ b | by simp only [div_eq_mul_inv, mul_le_mul_left₀ ha, inv_le_inv₀ hb hc] | lemma | div_le_div_left₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"div_eq_mul_inv",
"inv_le_inv₀",
"mul_le_mul_left₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff₀ (hc : c ≠ 0) : a ≤ b / c ↔ a*c ≤ b | by rw [div_eq_mul_inv, le_mul_inv_iff₀ hc] | lemma | le_div_iff₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"div_eq_mul_inv",
"le_mul_inv_iff₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_iff₀ (hc : c ≠ 0) : a / c ≤ b ↔ a ≤ b*c | by rw [div_eq_mul_inv, mul_inv_le_iff₀ hc] | lemma | div_le_iff₀ | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"div_eq_mul_inv",
"mul_inv_le_iff₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.mul_left₀' {a : α} (ha : a ≠ 0) : α ≃o α | { map_rel_iff' := λ x y, mul_le_mul_left₀ ha, ..equiv.mul_left₀ a ha } | def | order_iso.mul_left₀' | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"equiv.mul_left₀",
"mul_le_mul_left₀"
] | `equiv.mul_left₀` as an order_iso on a `linear_ordered_comm_group_with_zero.`.
Note that `order_iso.mul_left₀` refers to the `linear_ordered_field` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso.mul_left₀'_symm {a : α} (ha : a ≠ 0) :
(order_iso.mul_left₀' ha).symm = order_iso.mul_left₀' (inv_ne_zero ha) | by { ext, refl } | lemma | order_iso.mul_left₀'_symm | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"inv_ne_zero",
"order_iso.mul_left₀'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.mul_right₀' {a : α} (ha : a ≠ 0) : α ≃o α | { map_rel_iff' := λ _ _, mul_le_mul_right₀ ha, ..equiv.mul_right₀ a ha } | def | order_iso.mul_right₀' | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"equiv.mul_right₀",
"mul_le_mul_right₀"
] | `equiv.mul_right₀` as an order_iso on a `linear_ordered_comm_group_with_zero.`.
Note that `order_iso.mul_right₀` refers to the `linear_ordered_field` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso.mul_right₀'_symm {a : α} (ha : a ≠ 0) :
(order_iso.mul_right₀' ha).symm = order_iso.mul_right₀' (inv_ne_zero ha) | by { ext, refl } | lemma | order_iso.mul_right₀'_symm | algebra.order | src/algebra/order/with_zero.lean | [
"algebra.hom.equiv.units.group_with_zero",
"algebra.group_with_zero.inj_surj",
"algebra.order.group.units",
"algebra.order.monoid.basic",
"algebra.order.monoid.with_zero.defs",
"algebra.order.group.instances",
"algebra.order.monoid.type_tags"
] | [
"inv_ne_zero",
"order_iso.mul_right₀'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_one_class (α : Type*) [has_zero α] [has_one α] [has_le α] | (zero_le_one : (0 : α) ≤ 1) | class | zero_le_one_class | algebra.order | src/algebra/order/zero_le_one.lean | [
"order.basic",
"algebra.ne_zero"
] | [
"zero_le_one"
] | Typeclass for expressing that the `0` of a type is less or equal to its `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_le_one [has_zero α] [has_one α] [has_le α] [zero_le_one_class α] : (0 : α) ≤ 1 | zero_le_one_class.zero_le_one | lemma | zero_le_one | algebra.order | src/algebra/order/zero_le_one.lean | [
"order.basic",
"algebra.ne_zero"
] | [
"zero_le_one_class"
] | `zero_le_one` with the type argument implicit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_le_one' (α) [has_zero α] [has_one α] [has_le α] [zero_le_one_class α] : (0 : α) ≤ 1 | zero_le_one | lemma | zero_le_one' | algebra.order | src/algebra/order/zero_le_one.lean | [
"order.basic",
"algebra.ne_zero"
] | [
"zero_le_one",
"zero_le_one_class"
] | `zero_le_one` with the type argument explicit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_lt_one : (0 : α) < 1 | zero_le_one.lt_of_ne (ne_zero.ne' 1) | lemma | zero_lt_one | algebra.order | src/algebra/order/zero_le_one.lean | [
"order.basic",
"algebra.ne_zero"
] | [
"ne_zero.ne'"
] | See `zero_lt_one'` for a version with the type explicit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_lt_one' : (0 : α) < 1 | zero_lt_one | lemma | zero_lt_one' | algebra.order | src/algebra/order/zero_le_one.lean | [
"order.basic",
"algebra.ne_zero"
] | [
"zero_lt_one"
] | See `zero_lt_one` for a version with the type implicit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso.mul_left₀ (a : α) (ha : 0 < a) : α ≃o α | { map_rel_iff' := λ _ _, mul_le_mul_left ha, ..equiv.mul_left₀ a ha.ne' } | def | order_iso.mul_left₀ | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"equiv.mul_left₀",
"mul_le_mul_left"
] | `equiv.mul_left₀` as an order_iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso.mul_right₀ (a : α) (ha : 0 < a) : α ≃o α | { map_rel_iff' := λ _ _, mul_le_mul_right ha, ..equiv.mul_right₀ a ha.ne' } | def | order_iso.mul_right₀ | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"equiv.mul_right₀",
"mul_le_mul_right"
] | `equiv.mul_right₀` as an order_iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_pos : 0 < a⁻¹ ↔ 0 < a | suffices ∀ a : α, 0 < a → 0 < a⁻¹,
from ⟨λ h, inv_inv a ▸ this _ h, this a⟩,
assume a ha, flip lt_of_mul_lt_mul_left ha.le $ by simp [ne_of_gt ha, zero_lt_one] | lemma | inv_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",
"lt_of_mul_lt_mul_left",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_nonneg : 0 ≤ a⁻¹ ↔ 0 ≤ a | by simp only [le_iff_eq_or_lt, inv_pos, zero_eq_inv] | lemma | inv_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"
] | [
"inv_pos",
"le_iff_eq_or_lt",
"zero_eq_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_zero : a⁻¹ < 0 ↔ a < 0 | by simp only [← not_le, inv_nonneg] | lemma | inv_lt_zero | 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_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_nonpos : a⁻¹ ≤ 0 ↔ a ≤ 0 | by simp only [← not_lt, inv_pos] | lemma | inv_nonpos | 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_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_pos : 0 < 1 / a ↔ 0 < a | inv_eq_one_div a ▸ inv_pos | lemma | one_div_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_eq_one_div",
"inv_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_neg : 1 / a < 0 ↔ a < 0 | inv_eq_one_div a ▸ inv_lt_zero | lemma | one_div_neg | 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_eq_one_div",
"inv_lt_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_nonneg : 0 ≤ 1 / a ↔ 0 ≤ a | inv_eq_one_div a ▸ inv_nonneg | lemma | one_div_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"
] | [
"inv_eq_one_div",
"inv_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_nonpos : 1 / a ≤ 0 ↔ a ≤ 0 | inv_eq_one_div a ▸ inv_nonpos | lemma | one_div_nonpos | 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_eq_one_div",
"inv_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_pos (ha : 0 < a) (hb : 0 < b) : 0 < a / b | by { rw div_eq_mul_inv, exact mul_pos ha (inv_pos.2 hb) } | lemma | div_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_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_nonneg (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a / b | by { rw div_eq_mul_inv, exact mul_nonneg ha (inv_nonneg.2 hb) } | lemma | div_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_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_nonpos_of_nonpos_of_nonneg (ha : a ≤ 0) (hb : 0 ≤ b) : a / b ≤ 0 | by { rw div_eq_mul_inv, exact mul_nonpos_of_nonpos_of_nonneg ha (inv_nonneg.2 hb) } | lemma | div_nonpos_of_nonpos_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_eq_mul_inv",
"mul_nonpos_of_nonpos_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_nonpos_of_nonneg_of_nonpos (ha : 0 ≤ a) (hb : b ≤ 0) : a / b ≤ 0 | by { rw div_eq_mul_inv, exact mul_nonpos_of_nonneg_of_nonpos ha (inv_nonpos.2 hb) } | lemma | div_nonpos_of_nonneg_of_nonpos | 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",
"mul_nonpos_of_nonneg_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_nonneg (ha : 0 ≤ a) : ∀ n : ℤ, 0 ≤ a ^ n | | (n : ℕ) := by { rw zpow_coe_nat, exact pow_nonneg ha _ }
| -[1+n] := by { rw zpow_neg_succ_of_nat, exact inv_nonneg.2 (pow_nonneg ha _) } | lemma | zpow_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"
] | [
"pow_nonneg",
"zpow_coe_nat",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_pos_of_pos (ha : 0 < a) : ∀ n : ℤ, 0 < a ^ n | | (n : ℕ) := by { rw zpow_coe_nat, exact pow_pos ha _ }
| -[1+n] := by { rw zpow_neg_succ_of_nat, exact inv_pos.2 (pow_pos ha _) } | lemma | zpow_pos_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"
] | [
"pow_pos",
"zpow_coe_nat",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff (hc : 0 < c) : a ≤ b / c ↔ a * c ≤ b | ⟨λ h, div_mul_cancel b (ne_of_lt hc).symm ▸ mul_le_mul_of_nonneg_right h hc.le,
λ h, calc
a = a * c * (1 / c) : mul_mul_div a (ne_of_lt hc).symm
... ≤ b * (1 / c) : mul_le_mul_of_nonneg_right h (one_div_pos.2 hc).le
... = b / c : (div_eq_mul_one_div b c).symm⟩ | lemma | 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_eq_mul_one_div",
"div_mul_cancel",
"mul_le_mul_of_nonneg_right",
"mul_mul_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff' (hc : 0 < c) : a ≤ b / c ↔ c * a ≤ b | by rw [mul_comm, le_div_iff hc] | lemma | 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"
] | [
"le_div_iff",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_iff (hb : 0 < b) : a / b ≤ c ↔ a ≤ c * b | ⟨λ h, calc
a = a / b * b : by rw (div_mul_cancel _ (ne_of_lt hb).symm)
... ≤ c * b : mul_le_mul_of_nonneg_right h hb.le,
λ h, calc
a / b = a * (1 / b) : div_eq_mul_one_div a b
... ≤ (c * b) * (1 / b) : mul_le_mul_of_nonneg_right h (one_div_pos.2 hb).le
... = (c * b) / b : (div_eq_mul_one_div (... | lemma | div_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_iff",
"div_eq_mul_one_div",
"div_mul_cancel",
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_iff' (hb : 0 < b) : a / b ≤ c ↔ a ≤ b * c | by rw [mul_comm, div_le_iff hb] | lemma | div_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_le_iff",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_div_iff (hc : 0 < c) : a < b / c ↔ a * c < b | lt_iff_lt_of_le_iff_le $ div_le_iff hc | lemma | 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_le_iff",
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_div_iff' (hc : 0 < c) : a < b / c ↔ c * a < b | by rw [mul_comm, lt_div_iff hc] | lemma | 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"
] | [
"lt_div_iff",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_iff (hc : 0 < c) : b / c < a ↔ b < a * c | lt_iff_lt_of_le_iff_le (le_div_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"
] | [
"le_div_iff",
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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