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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|---|---|---|---|---|---|---|---|---|---|---|
ring.to_non_unital_ring :
non_unital_ring α | { zero_mul := λ a, add_left_cancel $ show 0 * a + 0 * a = 0 * a + 0,
by rw [← add_mul, zero_add, add_zero],
mul_zero := λ a, add_left_cancel $ show a * 0 + a * 0 = a * 0 + 0,
by rw [← mul_add, add_zero, add_zero],
..‹ring α› } | instance | ring.to_non_unital_ring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_zero",
"non_unital_ring",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring.to_non_assoc_ring :
non_assoc_ring α | { zero_mul := λ a, add_left_cancel $ show 0 * a + 0 * a = 0 * a + 0,
by rw [← add_mul, zero_add, add_zero],
mul_zero := λ a, add_left_cancel $ show a * 0 + a * 0 = a * 0 + 0,
by rw [← mul_add, add_zero, add_zero],
..‹ring α› } | instance | ring.to_non_assoc_ring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_zero",
"non_assoc_ring",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring.to_semiring : semiring α | { ..‹ring α›, .. ring.to_non_unital_ring } | instance | ring.to_semiring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"ring.to_non_unital_ring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_comm_ring (α : Type u) extends non_unital_ring α, comm_semigroup α | class | non_unital_comm_ring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"comm_semigroup",
"non_unital_ring"
] | A non-unital commutative ring is a `non_unital_ring` with commutative multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_comm_ring.to_non_unital_comm_semiring [s : non_unital_comm_ring α] :
non_unital_comm_semiring α | { ..s } | instance | non_unital_comm_ring.to_non_unital_comm_semiring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"non_unital_comm_ring",
"non_unital_comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_ring (α : Type u) extends ring α, comm_monoid α | class | comm_ring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"comm_monoid",
"ring"
] | A commutative ring is a `ring` with commutative multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_ring.to_comm_semiring [s : comm_ring α] : comm_semiring α | { mul_zero := mul_zero, zero_mul := zero_mul, ..s } | instance | comm_ring.to_comm_semiring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"comm_ring",
"comm_semiring",
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_ring.to_non_unital_comm_ring [s : comm_ring α] : non_unital_comm_ring α | { mul_zero := mul_zero, zero_mul := zero_mul, ..s } | instance | comm_ring.to_non_unital_comm_ring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"comm_ring",
"mul_zero",
"non_unital_comm_ring",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_domain (α : Type u) [semiring α] extends is_cancel_mul_zero α, nontrivial α : Prop | class | is_domain | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"is_cancel_mul_zero",
"nontrivial",
"semiring"
] | A domain is a nontrivial semiring such multiplication by a non zero element is cancellative,
on both sides. In other words, a nontrivial semiring `R` satisfying
`∀ {a b c : R}, a ≠ 0 → a * b = a * c → b = c` and
`∀ {a b c : R}, b ≠ 0 → a * b = c * b → a = c`.
This is implemented as a mixin for `semiring α`.
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_add [left_distrib_class α] {a b c : α} (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b + c | dvd.elim h₁ (λ d hd, dvd.elim h₂ (λ e he, dvd.intro (d + e) (by simp [left_distrib, hd, he]))) | theorem | dvd_add | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd.elim",
"dvd.intro",
"left_distrib",
"left_distrib_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_dvd_bit0 [semiring α] {a : α} : 2 ∣ bit0 a | ⟨a, bit0_eq_two_mul _⟩ | theorem | two_dvd_bit0 | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"bit0_eq_two_mul",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_dvd.dvd.linear_comb {d x y : α} (hdx : d ∣ x) (hdy : d ∣ y) (a b : α) :
d ∣ (a * x + b * y) | dvd_add (hdx.mul_left a) (hdy.mul_left b) | lemma | has_dvd.dvd.linear_comb | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_neg : a ∣ -b ↔ a ∣ b | (equiv.neg _).exists_congr_left.trans $ by simpa | lemma | dvd_neg | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [] | An element `a` of a semigroup with a distributive negation divides the negation of an element
`b` iff `a` divides `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_dvd : -a ∣ b ↔ a ∣ b | (equiv.neg _).exists_congr_left.trans $ by simpa | lemma | neg_dvd | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [] | The negation of an element `a` of a semigroup with a distributive negation divides another
element `b` iff `a` divides `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_sub (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b - c | by simpa only [←sub_eq_add_neg] using h₁.add h₂.neg_right | theorem | dvd_sub | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_add_left (h : a ∣ c) : a ∣ b + c ↔ a ∣ b | ⟨λ H, by simpa only [add_sub_cancel] using dvd_sub H h, λ h₂, dvd_add h₂ h⟩ | theorem | dvd_add_left | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_add",
"dvd_sub"
] | If an element `a` divides another element `c` in a ring, `a` divides the sum of another element
`b` with `c` iff `a` divides `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_add_right (h : a ∣ b) : a ∣ b + c ↔ a ∣ c | by rw add_comm; exact dvd_add_left h | theorem | dvd_add_right | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_add_left"
] | If an element `a` divides another element `b` in a ring, `a` divides the sum of `b` and another
element `c` iff `a` divides `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_sub_left (h : a ∣ c) : a ∣ b - c ↔ a ∣ b | by simpa only [←sub_eq_add_neg] using dvd_add_left (dvd_neg.2 h) | theorem | dvd_sub_left | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_add_left"
] | If an element `a` divides another element `c` in a ring, `a` divides the difference of another
element `b` with `c` iff `a` divides `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_sub_right (h : a ∣ b) : a ∣ b - c ↔ a ∣ c | by rw [sub_eq_add_neg, dvd_add_right h, dvd_neg] | theorem | dvd_sub_right | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_add_right",
"dvd_neg"
] | If an element `a` divides another element `b` in a ring, `a` divides the difference of `b` and
another element `c` iff `a` divides `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_iff_dvd_of_dvd_sub (h : a ∣ b - c) : a ∣ b ↔ a ∣ c | by rw [←sub_add_cancel b c, dvd_add_right h] | lemma | dvd_iff_dvd_of_dvd_sub | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_sub_comm : a ∣ b - c ↔ a ∣ c - b | by rw [←dvd_neg, neg_sub] | lemma | dvd_sub_comm | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_dvd_bit1 : 2 ∣ bit1 a ↔ (2 : α) ∣ 1 | dvd_add_right two_dvd_bit0 | theorem | two_dvd_bit1 | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_add_right",
"two_dvd_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_add_self_left {a b : α} : a ∣ a + b ↔ a ∣ b | dvd_add_right (dvd_refl a) | lemma | dvd_add_self_left | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_add_right",
"dvd_refl"
] | An element a divides the sum a + b if and only if a divides b. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_add_self_right {a b : α} : a ∣ b + a ↔ a ∣ b | dvd_add_left (dvd_refl a) | lemma | dvd_add_self_right | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_add_left",
"dvd_refl"
] | An element a divides the sum b + a if and only if a divides b. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_sub_self_left : a ∣ a - b ↔ a ∣ b | dvd_sub_right dvd_rfl | lemma | dvd_sub_self_left | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_rfl",
"dvd_sub_right"
] | An element `a` divides the difference `a - b` if and only if `a` divides `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_sub_self_right : a ∣ b - a ↔ a ∣ b | dvd_sub_left dvd_rfl | lemma | dvd_sub_self_right | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_rfl",
"dvd_sub_left"
] | An element `a` divides the difference `b - a` if and only if `a` divides `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_mul_sub_mul {k a b x y : α} (hab : k ∣ a - b) (hxy : k ∣ x - y) :
k ∣ a * x - b * y | begin
convert dvd_add (hxy.mul_left a) (hab.mul_right y),
rw [mul_sub_left_distrib, mul_sub_right_distrib],
simp only [sub_eq_add_neg, add_assoc, neg_add_cancel_left],
end | lemma | dvd_mul_sub_mul | algebra.ring | src/algebra/ring/divisibility.lean | [
"algebra.divisibility.basic",
"algebra.hom.equiv.basic",
"algebra.ring.defs"
] | [
"dvd_add",
"mul_sub_left_distrib",
"mul_sub_right_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv (R S : Type*) [has_mul R] [has_add R] [has_mul S] [has_add S]
extends R ≃ S, R ≃* S, R ≃+ S | structure | ring_equiv | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | An equivalence between two (non-unital non-associative semi)rings that preserves the
algebraic structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv_class (F : Type*) (R S : out_param Type*)
[has_mul R] [has_add R] [has_mul S] [has_add S]
extends mul_equiv_class F R S | (map_add : ∀ (f : F) a b, f (a + b) = f a + f b) | class | ring_equiv_class | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"mul_equiv_class"
] | `ring_equiv_class F R S` states that `F` is a type of ring structure preserving equivalences.
You should extend this class when you extend `ring_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_add_equiv_class (F R S : Type*)
[has_mul R] [has_add R] [has_mul S] [has_add S] [h : ring_equiv_class F R S] :
add_equiv_class F R S | { coe := coe_fn,
.. h } | instance | ring_equiv_class.to_add_equiv_class | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"add_equiv_class",
"ring_equiv_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_hom_class (F R S : Type*)
[non_assoc_semiring R] [non_assoc_semiring S] [h : ring_equiv_class F R S] :
ring_hom_class F R S | { coe := coe_fn,
coe_injective' := fun_like.coe_injective,
map_zero := map_zero,
map_one := map_one,
.. h } | instance | ring_equiv_class.to_ring_hom_class | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"fun_like.coe_injective",
"map_one",
"non_assoc_semiring",
"ring_equiv_class",
"ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_ring_hom_class (F R S : Type*)
[non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] [h : ring_equiv_class F R S] :
non_unital_ring_hom_class F R S | { coe := coe_fn,
coe_injective' := fun_like.coe_injective,
map_zero := map_zero,
.. h } | instance | ring_equiv_class.to_non_unital_ring_hom_class | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"fun_like.coe_injective",
"non_unital_non_assoc_semiring",
"non_unital_ring_hom_class",
"ring_equiv_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_eq_coe (f : R ≃+* S) : f.to_equiv = f | rfl | lemma | ring_equiv.to_equiv_eq_coe | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (f : R ≃+* S) : f.to_fun = f | rfl | lemma | ring_equiv.to_fun_eq_coe | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_equiv (f : R ≃+* S) : ⇑(f : R ≃ S) = f | rfl | lemma | ring_equiv.coe_to_equiv | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul (e : R ≃+* S) (x y : R) : e (x * y) = e x * e y | map_mul e x y | lemma | ring_equiv.map_mul | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"map_mul"
] | A ring isomorphism preserves multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_add (e : R ≃+* S) (x y : R) : e (x + y) = e x + e y | map_add e x y | lemma | ring_equiv.map_add | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | A ring isomorphism preserves addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {f g : R ≃+* S} (h : ∀ x, f x = g x) : f = g | fun_like.ext f g h | lemma | ring_equiv.ext | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"fun_like.ext"
] | Two ring isomorphisms agree if they are defined by the
same underlying function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk (e e' h₁ h₂ h₃ h₄) :
⇑(⟨e, e', h₁, h₂, h₃, h₄⟩ : R ≃+* S) = e | rfl | theorem | ring_equiv.coe_mk | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (e : R ≃+* S) (e' h₁ h₂ h₃ h₄) :
(⟨e, e', h₁, h₂, h₃, h₄⟩ : R ≃+* S) = e | ext $ λ _, rfl | theorem | ring_equiv.mk_coe | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg {f : R ≃+* S} {x x' : R} : x = x' → f x = f x' | fun_like.congr_arg f | lemma | ring_equiv.congr_arg | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"fun_like.congr_arg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : R ≃+* S} (h : f = g) (x : R) : f x = g x | fun_like.congr_fun h x | lemma | ring_equiv.congr_fun | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : R ≃+* S} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | lemma | ring_equiv.ext_iff | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_equiv_eq_coe (f : R ≃+* S) : f.to_add_equiv = ↑f | rfl | lemma | ring_equiv.to_add_equiv_eq_coe | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_equiv_eq_coe (f : R ≃+* S) : f.to_mul_equiv = ↑f | rfl | lemma | ring_equiv.to_mul_equiv_eq_coe | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_mul_equiv (f : R ≃+* S) : ⇑(f : R ≃* S) = f | rfl | lemma | ring_equiv.coe_to_mul_equiv | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_add_equiv (f : R ≃+* S) : ⇑(f : R ≃+ S) = f | rfl | lemma | ring_equiv.coe_to_add_equiv | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv_of_unique {M N}
[unique M] [unique N] [has_add M] [has_mul M] [has_add N] [has_mul N] : M ≃+* N | { ..add_equiv.add_equiv_of_unique,
..mul_equiv.mul_equiv_of_unique} | def | ring_equiv.ring_equiv_of_unique | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"mul_equiv.mul_equiv_of_unique",
"unique"
] | The `ring_equiv` between two semirings with a unique element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl : R ≃+* R | { .. mul_equiv.refl R, .. add_equiv.refl R } | def | ring_equiv.refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"mul_equiv.refl"
] | The identity map is a ring isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_apply (x : R) : ring_equiv.refl R x = x | rfl | lemma | ring_equiv.refl_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_equiv_refl : (ring_equiv.refl R : R ≃+ R) = add_equiv.refl R | rfl | lemma | ring_equiv.coe_add_equiv_refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_equiv_refl : (ring_equiv.refl R : R ≃* R) = mul_equiv.refl R | rfl | lemma | ring_equiv.coe_mul_equiv_refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"mul_equiv.refl",
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (e : R ≃+* S) : S ≃+* R | { .. e.to_mul_equiv.symm, .. e.to_add_equiv.symm } | def | ring_equiv.symm | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | The inverse of a ring isomorphism is a ring isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.symm_apply (e : R ≃+* S) : S → R | e.symm
initialize_simps_projections ring_equiv (to_fun → apply, inv_fun → symm_apply) | def | ring_equiv.simps.symm_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"inv_fun",
"ring_equiv"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_fun_eq_symm (f : R ≃+* S) : f.inv_fun = f.symm | rfl | lemma | ring_equiv.inv_fun_eq_symm | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm (e : R ≃+* S) : e.symm.symm = e | ext $ λ x, rfl | lemma | ring_equiv.symm_symm | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_equiv_symm (e : R ≃+* S) : (e.symm : S ≃ R) = (e : R ≃ S).symm | rfl | lemma | ring_equiv.coe_to_equiv_symm | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_bijective : function.bijective (ring_equiv.symm : (R ≃+* S) → (S ≃+* R)) | equiv.bijective ⟨ring_equiv.symm, ring_equiv.symm, symm_symm, symm_symm⟩ | lemma | ring_equiv.symm_bijective | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"equiv.bijective",
"ring_equiv.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe' (e : R ≃+* S) (f h₁ h₂ h₃ h₄) :
(ring_equiv.mk f ⇑e h₁ h₂ h₃ h₄ : S ≃+* R) = e.symm | symm_bijective.injective $ ext $ λ x, rfl | lemma | ring_equiv.mk_coe' | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_mk (f : R → S) (g h₁ h₂ h₃ h₄) :
(mk f g h₁ h₂ h₃ h₄).symm =
{ to_fun := g, inv_fun := f, ..(mk f g h₁ h₂ h₃ h₄).symm} | rfl | lemma | ring_equiv.symm_mk | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') : R ≃+* S' | { .. (e₁.to_mul_equiv.trans e₂.to_mul_equiv), .. (e₁.to_add_equiv.trans e₂.to_add_equiv) } | def | ring_equiv.trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | Transitivity of `ring_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : R) :
e₁.trans e₂ a = e₂ (e₁ a) | rfl | lemma | ring_equiv.trans_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂ : R → S') = e₂ ∘ e₁ | rfl | lemma | ring_equiv.coe_trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_apply (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : S') :
(e₁.trans e₂).symm a = e₁.symm (e₂.symm a) | rfl | lemma | ring_equiv.symm_trans_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂).symm = e₂.symm.trans (e₁.symm) | rfl | lemma | ring_equiv.symm_trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bijective (e : R ≃+* S) : function.bijective e | equiv_like.bijective e | lemma | ring_equiv.bijective | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"equiv_like.bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (e : R ≃+* S) : function.injective e | equiv_like.injective e | lemma | ring_equiv.injective | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"equiv_like.injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective (e : R ≃+* S) : function.surjective e | equiv_like.surjective e | lemma | ring_equiv.surjective | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"equiv_like.surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_symm_apply (e : R ≃+* S) : ∀ x, e (e.symm x) = x | e.to_equiv.apply_symm_apply | lemma | ring_equiv.apply_symm_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (e : R ≃+* S) : ∀ x, e.symm (e x) = x | e.to_equiv.symm_apply_apply | lemma | ring_equiv.symm_apply_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_eq_preimage (e : R ≃+* S) (s : set R) : e '' s = e.symm ⁻¹' s | e.to_equiv.image_eq_preimage s | lemma | ring_equiv.image_eq_preimage | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_equiv_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂ : R ≃* S') = (e₁ : R ≃* S).trans ↑e₂ | rfl | lemma | ring_equiv.coe_mul_equiv_trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_equiv_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂ : R ≃+ S') = (e₁ : R ≃+ S).trans ↑e₂ | rfl | lemma | ring_equiv.coe_add_equiv_trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op {α β} [has_add α] [has_mul α] [has_add β] [has_mul β] :
(α ≃+* β) ≃ (αᵐᵒᵖ ≃+* βᵐᵒᵖ) | { to_fun := λ f, { ..f.to_add_equiv.mul_op, ..f.to_mul_equiv.op},
inv_fun := λ f, { ..add_equiv.mul_op.symm f.to_add_equiv, ..mul_equiv.op.symm f.to_mul_equiv },
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, refl } } | def | ring_equiv.op | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"inv_fun"
] | A ring iso `α ≃+* β` can equivalently be viewed as a ring iso `αᵐᵒᵖ ≃+* βᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop {α β} [has_add α] [has_mul α] [has_add β] [has_mul β] :
(αᵐᵒᵖ ≃+* βᵐᵒᵖ) ≃ (α ≃+* β) | ring_equiv.op.symm | def | ring_equiv.unop | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | The 'unopposite' of a ring iso `αᵐᵒᵖ ≃+* βᵐᵒᵖ`. Inverse to `ring_equiv.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_opposite : R ≃+* Rᵐᵒᵖ | { map_add' := λ x y, rfl,
map_mul' := λ x y, mul_comm (op y) (op x),
.. mul_opposite.op_equiv } | def | ring_equiv.to_opposite | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"mul_comm",
"mul_opposite.op_equiv"
] | A non-unital commutative ring is isomorphic to its opposite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_opposite_apply (r : R) : to_opposite R r = op r | rfl | lemma | ring_equiv.to_opposite_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_opposite_symm_apply (r : Rᵐᵒᵖ) : (to_opposite R).symm r = unop r | rfl | lemma | ring_equiv.to_opposite_symm_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero_iff : f x = 0 ↔ x = 0 | add_equiv_class.map_eq_zero_iff f | lemma | ring_equiv.map_eq_zero_iff | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne_zero_iff : f x ≠ 0 ↔ x ≠ 0 | add_equiv_class.map_ne_zero_iff f | lemma | ring_equiv.map_ne_zero_iff | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bijective [non_unital_ring_hom_class F R S] (f : F)
(hf : function.bijective f) : R ≃+* S | { map_mul' := map_mul f,
map_add' := map_add f,
.. equiv.of_bijective f hf,} | def | ring_equiv.of_bijective | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"equiv.of_bijective",
"map_mul",
"non_unital_ring_hom_class"
] | Produce a ring isomorphism from a bijective ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_bijective [non_unital_ring_hom_class F R S] (f : F)
(hf : function.bijective f) : (of_bijective f hf : R → S) = f | rfl | lemma | ring_equiv.coe_of_bijective | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_unital_ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bijective_apply [non_unital_ring_hom_class F R S] (f : F)
(hf : function.bijective f) (x : R) : of_bijective f hf x = f x | rfl | lemma | ring_equiv.of_bijective_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_unital_ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pi_congr_right {ι : Type*} {R S : ι → Type*}
[Π i, non_unital_non_assoc_semiring (R i)] [Π i, non_unital_non_assoc_semiring (S i)]
(e : Π i, R i ≃+* S i) : (Π i, R i) ≃+* Π i, S i | { to_fun := λ x j, e j (x j),
inv_fun := λ x j, (e j).symm (x j),
.. @mul_equiv.Pi_congr_right ι R S _ _ (λ i, (e i).to_mul_equiv),
.. @add_equiv.Pi_congr_right ι R S _ _ (λ i, (e i).to_add_equiv) } | def | ring_equiv.Pi_congr_right | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"inv_fun",
"mul_equiv.Pi_congr_right",
"non_unital_non_assoc_semiring"
] | A family of ring isomorphisms `Π j, (R j ≃+* S j)` generates a
ring isomorphisms between `Π j, R j` and `Π j, S j`.
This is the `ring_equiv` version of `equiv.Pi_congr_right`, and the dependent version of
`ring_equiv.arrow_congr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi_congr_right_refl {ι : Type*} {R : ι → Type*} [Π i, non_unital_non_assoc_semiring (R i)] :
Pi_congr_right (λ i, ring_equiv.refl (R i)) = ring_equiv.refl _ | rfl | lemma | ring_equiv.Pi_congr_right_refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_unital_non_assoc_semiring",
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pi_congr_right_symm {ι : Type*} {R S : ι → Type*}
[Π i, non_unital_non_assoc_semiring (R i)] [Π i, non_unital_non_assoc_semiring (S i)]
(e : Π i, R i ≃+* S i) : (Pi_congr_right e).symm = (Pi_congr_right $ λ i, (e i).symm) | rfl | lemma | ring_equiv.Pi_congr_right_symm | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pi_congr_right_trans {ι : Type*} {R S T : ι → Type*}
[Π i, non_unital_non_assoc_semiring (R i)] [Π i, non_unital_non_assoc_semiring (S i)]
[Π i, non_unital_non_assoc_semiring (T i)]
(e : Π i, R i ≃+* S i) (f : Π i, S i ≃+* T i) :
(Pi_congr_right e).trans (Pi_congr_right f) = (Pi_congr_right $ λ i, (e i).trans (... | rfl | lemma | ring_equiv.Pi_congr_right_trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one : f 1 = 1 | map_one f | lemma | ring_equiv.map_one | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"map_one"
] | A ring isomorphism sends one to one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_eq_one_iff : f x = 1 ↔ x = 1 | mul_equiv_class.map_eq_one_iff f | lemma | ring_equiv.map_eq_one_iff | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"map_eq_one_iff",
"mul_equiv_class.map_eq_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne_one_iff : f x ≠ 1 ↔ x ≠ 1 | mul_equiv_class.map_ne_one_iff f | lemma | ring_equiv.map_ne_one_iff | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"map_ne_one_iff",
"mul_equiv_class.map_ne_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monoid_hom_refl : (ring_equiv.refl R : R →* R) = monoid_hom.id R | rfl | lemma | ring_equiv.coe_monoid_hom_refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"monoid_hom.id",
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom_refl : (ring_equiv.refl R : R →+ R) = add_monoid_hom.id R | rfl | lemma | ring_equiv.coe_add_monoid_hom_refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_hom_refl : (ring_equiv.refl R : R →* R) = ring_hom.id R | rfl | lemma | ring_equiv.coe_ring_hom_refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.refl",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monoid_hom_trans [non_assoc_semiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂ : R →* S') = (e₂ : S →* S').comp ↑e₁ | rfl | lemma | ring_equiv.coe_monoid_hom_trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom_trans [non_assoc_semiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂ : R →+ S') = (e₂ : S →+ S').comp ↑e₁ | rfl | lemma | ring_equiv.coe_add_monoid_hom_trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_hom_trans [non_assoc_semiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂ : R →+* S') = (e₂ : S →+* S').comp ↑e₁ | rfl | lemma | ring_equiv.coe_ring_hom_trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_symm (e : R ≃+* S) :
(e : R →+* S).comp (e.symm : S →+* R) = ring_hom.id S | ring_hom.ext e.apply_symm_apply | lemma | ring_equiv.comp_symm | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_hom.ext",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_comp (e : R ≃+* S) :
(e.symm : S →+* R).comp (e : R →+* S) = ring_hom.id R | ring_hom.ext e.symm_apply_apply | lemma | ring_equiv.symm_comp | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_hom.ext",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg : f (-x) = -f x | map_neg f x | lemma | ring_equiv.map_neg | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub : f (x - y) = f x - f y | map_sub f x y | lemma | ring_equiv.map_sub | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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