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
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|---|---|---|---|---|---|---|---|---|---|---|
to_subsemiring_mono : monotone (to_subsemiring : subring R → subsemiring R) | to_subsemiring_strict_mono.monotone | lemma | subring.to_subsemiring_mono | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"monotone",
"subring",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_subgroup_injective : function.injective (to_add_subgroup : subring R → add_subgroup R) | | r s h := ext (set_like.ext_iff.mp h : _) | lemma | subring.to_add_subgroup_injective | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_subgroup_strict_mono : strict_mono (to_add_subgroup : subring R → add_subgroup R) | λ _ _, id | lemma | subring.to_add_subgroup_strict_mono | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup",
"strict_mono",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_subgroup_mono : monotone (to_add_subgroup : subring R → add_subgroup R) | to_add_subgroup_strict_mono.monotone | lemma | subring.to_add_subgroup_mono | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup",
"monotone",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submonoid_injective : function.injective (to_submonoid : subring R → submonoid R) | | r s h := ext (set_like.ext_iff.mp h : _) | lemma | subring.to_submonoid_injective | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"submonoid",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submonoid_strict_mono : strict_mono (to_submonoid : subring R → submonoid R) | λ _ _, id | lemma | subring.to_submonoid_strict_mono | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"strict_mono",
"submonoid",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submonoid_mono : monotone (to_submonoid : subring R → submonoid R) | to_submonoid_strict_mono.monotone | lemma | subring.to_submonoid_mono | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"monotone",
"submonoid",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (s : set R) (sm : submonoid R) (sa : add_subgroup R)
(hm : ↑sm = s) (ha : ↑sa = s) :
subring R | { carrier := s,
zero_mem' := ha ▸ sa.zero_mem,
one_mem' := hm ▸ sm.one_mem,
add_mem' := λ x y, by simpa only [← ha] using sa.add_mem,
mul_mem' := λ x y, by simpa only [← hm] using sm.mul_mem,
neg_mem' := λ x, by simpa only [← ha] using sa.neg_mem, } | def | subring.mk' | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup",
"mk'",
"submonoid",
"subring"
] | Construct a `subring R` from a set `s`, a submonoid `sm`, and an additive
subgroup `sa` such that `x ∈ s ↔ x ∈ sm ↔ x ∈ sa`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk' {s : set R} {sm : submonoid R} (hm : ↑sm = s)
{sa : add_subgroup R} (ha : ↑sa = s) :
(subring.mk' s sm sa hm ha : set R) = s | rfl | lemma | subring.coe_mk' | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup",
"submonoid",
"subring.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk' {s : set R} {sm : submonoid R} (hm : ↑sm = s)
{sa : add_subgroup R} (ha : ↑sa = s) {x : R} :
x ∈ subring.mk' s sm sa hm ha ↔ x ∈ s | iff.rfl | lemma | subring.mem_mk' | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup",
"submonoid",
"subring.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_to_submonoid {s : set R} {sm : submonoid R} (hm : ↑sm = s)
{sa : add_subgroup R} (ha : ↑sa = s) :
(subring.mk' s sm sa hm ha).to_submonoid = sm | set_like.coe_injective hm.symm | lemma | subring.mk'_to_submonoid | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup",
"set_like.coe_injective",
"submonoid",
"subring.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_to_add_subgroup {s : set R} {sm : submonoid R} (hm : ↑sm = s)
{sa : add_subgroup R} (ha : ↑sa =s) :
(subring.mk' s sm sa hm ha).to_add_subgroup = sa | set_like.coe_injective ha.symm | lemma | subring.mk'_to_add_subgroup | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup",
"set_like.coe_injective",
"submonoid",
"subring.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring.to_subring (s : subsemiring R) (hneg : (-1 : R) ∈ s) : subring R | { neg_mem' := by { rintros x, rw <-neg_one_mul, apply subsemiring.mul_mem, exact hneg, }
..s.to_submonoid, ..s.to_add_submonoid } | def | subsemiring.to_subring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"neg_one_mul",
"subring",
"subsemiring",
"subsemiring.mul_mem"
] | A `subsemiring` containing -1 is a `subring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_mem : (1 : R) ∈ s | one_mem _ | theorem | subring.one_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | A subring contains the ring's 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_mem : (0 : R) ∈ s | zero_mem _ | theorem | subring.zero_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | A subring contains the ring's 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_mem {x y : R} : x ∈ s → y ∈ s → x * y ∈ s | mul_mem | theorem | subring.mul_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | A subring is closed under multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_mem {x y : R} : x ∈ s → y ∈ s → x + y ∈ s | add_mem | theorem | subring.add_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | A subring is closed under addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_mem {x : R} : x ∈ s → -x ∈ s | neg_mem | theorem | subring.neg_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | A subring is closed under negation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_mem {x y : R} (hx : x ∈ s) (hy : y ∈ s) : x - y ∈ s | sub_mem hx hy | theorem | subring.sub_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | A subring is closed under subtraction | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
list_prod_mem {l : list R} : (∀x ∈ l, x ∈ s) → l.prod ∈ s | list_prod_mem | lemma | subring.list_prod_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"list_prod_mem"
] | Product of a list of elements in a subring is in the subring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
list_sum_mem {l : list R} : (∀x ∈ l, x ∈ s) → l.sum ∈ s | list_sum_mem | lemma | subring.list_sum_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | Sum of a list of elements in a subring is in the subring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiset_prod_mem {R} [comm_ring R] (s : subring R) (m : multiset R) :
(∀a ∈ m, a ∈ s) → m.prod ∈ s | multiset_prod_mem _ | lemma | subring.multiset_prod_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"comm_ring",
"multiset",
"multiset_prod_mem",
"subring"
] | Product of a multiset of elements in a subring of a `comm_ring` is in the subring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiset_sum_mem {R} [ring R] (s : subring R) (m : multiset R) :
(∀a ∈ m, a ∈ s) → m.sum ∈ s | multiset_sum_mem _ | lemma | subring.multiset_sum_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"multiset",
"ring",
"subring"
] | Sum of a multiset of elements in an `subring` of a `ring` is
in the `subring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mem {R : Type*} [comm_ring R] (s : subring R)
{ι : Type*} {t : finset ι} {f : ι → R} (h : ∀c ∈ t, f c ∈ s) :
∏ i in t, f i ∈ s | prod_mem h | lemma | subring.prod_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"comm_ring",
"finset",
"prod_mem",
"subring"
] | Product of elements of a subring of a `comm_ring` indexed by a `finset` is in the
subring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_mem {R : Type*} [ring R] (s : subring R)
{ι : Type*} {t : finset ι} {f : ι → R} (h : ∀c ∈ t, f c ∈ s) :
∑ i in t, f i ∈ s | sum_mem h | lemma | subring.sum_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"finset",
"ring",
"subring"
] | Sum of elements in a `subring` of a `ring` indexed by a `finset`
is in the `subring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zsmul_mem {x : R} (hx : x ∈ s) (n : ℤ) : n • x ∈ s | zsmul_mem hx n | lemma | subring.zsmul_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mem {x : R} (hx : x ∈ s) (n : ℕ) : x^n ∈ s | pow_mem hx n | lemma | subring.pow_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"pow_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (x y : s) : (↑(x + y) : R) = ↑x + ↑y | rfl | lemma | subring.coe_add | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (x : s) : (↑(-x) : R) = -↑x | rfl | lemma | subring.coe_neg | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (x y : s) : (↑(x * y) : R) = ↑x * ↑y | rfl | lemma | subring.coe_mul | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : s) : R) = 0 | rfl | lemma | subring.coe_zero | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : s) : R) = 1 | rfl | lemma | subring.coe_one | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (x : s) (n : ℕ) : (↑(x ^ n) : R) = x ^ n | submonoid_class.coe_pow x n | lemma | subring.coe_pow | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"submonoid_class.coe_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_zero_iff {x : s} : (x : R) = 0 ↔ x = 0 | ⟨λ h, subtype.ext (trans h s.coe_zero.symm),
λ h, h.symm ▸ s.coe_zero⟩ | lemma | subring.coe_eq_zero_iff | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_comm_ring {R} [comm_ring R] (s : subring R) : comm_ring s | subtype.coe_injective.comm_ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) | instance | subring.to_comm_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"comm_ring",
"subring"
] | A subring of a `comm_ring` is a `comm_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_ring {R} [ordered_ring R] (s : subring R) : ordered_ring s | subtype.coe_injective.ordered_ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) | instance | subring.to_ordered_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"ordered_ring",
"subring"
] | A subring of an `ordered_ring` is an `ordered_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_comm_ring {R} [ordered_comm_ring R] (s : subring R) : ordered_comm_ring s | subtype.coe_injective.ordered_comm_ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) | instance | subring.to_ordered_comm_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"ordered_comm_ring",
"subring"
] | A subring of an `ordered_comm_ring` is an `ordered_comm_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_ring {R} [linear_ordered_ring R] (s : subring R) :
linear_ordered_ring s | subtype.coe_injective.linear_ordered_ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) | instance | subring.to_linear_ordered_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"linear_ordered_ring",
"subring"
] | A subring of a `linear_ordered_ring` is a `linear_ordered_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_comm_ring {R} [linear_ordered_comm_ring R] (s : subring R) :
linear_ordered_comm_ring s | subtype.coe_injective.linear_ordered_comm_ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) | instance | subring.to_linear_ordered_comm_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"linear_ordered_comm_ring",
"subring"
] | A subring of a `linear_ordered_comm_ring` is a `linear_ordered_comm_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype (s : subring R) : s →+* R | { to_fun := coe,
.. s.to_submonoid.subtype, .. s.to_add_subgroup.subtype } | def | subring.subtype | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | The natural ring hom from a subring of ring `R` to `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_nat_cast : ∀ n : ℕ, ((n : s) : R) = n | map_nat_cast s.subtype | lemma | subring.coe_nat_cast | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"map_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_int_cast : ∀ n : ℤ, ((n : s) : R) = n | map_int_cast s.subtype | lemma | subring.coe_int_cast | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"map_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_submonoid {s : subring R} {x : R} : x ∈ s.to_submonoid ↔ x ∈ s | iff.rfl | lemma | subring.mem_to_submonoid | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submonoid (s : subring R) : (s.to_submonoid : set R) = s | rfl | lemma | subring.coe_to_submonoid | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_add_subgroup {s : subring R} {x : R} :
x ∈ s.to_add_subgroup ↔ x ∈ s | iff.rfl | lemma | subring.mem_to_add_subgroup | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_add_subgroup (s : subring R) : (s.to_add_subgroup : set R) = s | rfl | lemma | subring.coe_to_add_subgroup | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top (x : R) : x ∈ (⊤ : subring R) | set.mem_univ x | lemma | subring.mem_top | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.mem_univ",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ((⊤ : subring R) : set R) = set.univ | rfl | lemma | subring.coe_top | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_equiv : (⊤ : subring R) ≃+* R | subsemiring.top_equiv | def | subring.top_equiv | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring",
"subsemiring.top_equiv"
] | The ring equiv between the top element of `subring R` and `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap {R : Type u} {S : Type v} [ring R] [ring S]
(f : R →+* S) (s : subring S) : subring R | { carrier := f ⁻¹' s.carrier,
.. s.to_submonoid.comap (f : R →* S),
.. s.to_add_subgroup.comap (f : R →+ S) } | def | subring.comap | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"ring",
"subring"
] | The preimage of a subring along a ring homomorphism is a subring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comap (s : subring S) (f : R →+* S) : (s.comap f : set R) = f ⁻¹' s | rfl | lemma | subring.coe_comap | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap {s : subring S} {f : R →+* S} {x : R} : x ∈ s.comap f ↔ f x ∈ s | iff.rfl | lemma | subring.mem_comap | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap (s : subring T) (g : S →+* T) (f : R →+* S) :
(s.comap g).comap f = s.comap (g.comp f) | rfl | lemma | subring.comap_comap | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map {R : Type u} {S : Type v} [ring R] [ring S]
(f : R →+* S) (s : subring R) : subring S | { carrier := f '' s.carrier,
.. s.to_submonoid.map (f : R →* S),
.. s.to_add_subgroup.map (f : R →+ S) } | def | subring.map | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"ring",
"subring"
] | The image of a subring along a ring homomorphism is a subring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map (f : R →+* S) (s : subring R) : (s.map f : set S) = f '' s | rfl | lemma | subring.coe_map | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map {f : R →+* S} {s : subring R} {y : S} :
y ∈ s.map f ↔ ∃ x ∈ s, f x = y | set.mem_image_iff_bex | lemma | subring.mem_map | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"mem_map",
"set.mem_image_iff_bex",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : s.map (ring_hom.id R) = s | set_like.coe_injective $ set.image_id _ | lemma | subring.map_id | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"map_id",
"ring_hom.id",
"set.image_id",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (g : S →+* T) (f : R →+* S) : (s.map f).map g = s.map (g.comp f) | set_like.coe_injective $ set.image_image _ _ _ | lemma | subring.map_map | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.image_image",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_iff_le_comap {f : R →+* S} {s : subring R} {t : subring S} :
s.map f ≤ t ↔ s ≤ t.comap f | set.image_subset_iff | lemma | subring.map_le_iff_le_comap | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.image_subset_iff",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_map_comap (f : R →+* S) : galois_connection (map f) (comap f) | λ S T, map_le_iff_le_comap | lemma | subring.gc_map_comap | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_map_of_injective
(f : R →+* S) (hf : function.injective f) : s ≃+* s.map f | { map_mul' := λ _ _, subtype.ext (f.map_mul _ _),
map_add' := λ _ _, subtype.ext (f.map_add _ _),
..equiv.set.image f s hf } | def | subring.equiv_map_of_injective | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"equiv.set.image",
"subtype.ext"
] | A subring is isomorphic to its image under an injective function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_equiv_map_of_injective_apply
(f : R →+* S) (hf : function.injective f) (x : s) :
(equiv_map_of_injective s f hf x : S) = f x | rfl | lemma | subring.coe_equiv_map_of_injective_apply | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) : subring S | ((⊤ : subring R).map f).copy (set.range f) set.image_univ.symm | def | ring_hom.range | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"ring",
"set.range",
"subring"
] | The range of a ring homomorphism, as a subring of the target. See Note [range copy pattern]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_range : (f.range : set S) = set.range f | rfl | lemma | ring_hom.coe_range | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_range {f : R →+* S} {y : S} : y ∈ f.range ↔ ∃ x, f x = y | iff.rfl | lemma | ring_hom.mem_range | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq_map (f : R →+* S) : f.range = subring.map f ⊤ | by { ext, simp } | lemma | ring_hom.range_eq_map | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_range_self (f : R →+* S) (x : R) : f x ∈ f.range | mem_range.mpr ⟨x, rfl⟩ | lemma | ring_hom.mem_range_self | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_range : f.range.map g = (g.comp f).range | by simpa only [range_eq_map] using (⊤ : subring R).map_map g f | lemma | ring_hom.map_range | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype_range [fintype R] [decidable_eq S] (f : R →+* S) : fintype (range f) | set.fintype_range f | instance | ring_hom.fintype_range | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"fintype",
"set.fintype_range"
] | The range of a ring homomorphism is a fintype, if the domain is a fintype.
Note: this instance can form a diamond with `subtype.fintype` in the
presence of `fintype S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_bot : ((⊥ : subring R) : set R) = set.range (coe : ℤ → R) | ring_hom.coe_range (int.cast_ring_hom R) | lemma | subring.coe_bot | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"int.cast_ring_hom",
"ring_hom.coe_range",
"set.range",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot {x : R} : x ∈ (⊥ : subring R) ↔ ∃ (n : ℤ), ↑n = x | ring_hom.mem_range | lemma | subring.mem_bot | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"ring_hom.mem_range",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (p p' : subring R) : ((p ⊓ p' : subring R) : set R) = p ∩ p' | rfl | lemma | subring.coe_inf | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf {p p' : subring R} {x : R} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p' | iff.rfl | lemma | subring.mem_inf | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Inf (S : set (subring R)) :
((Inf S : subring R) : set R) = ⋂ s ∈ S, ↑s | rfl | lemma | subring.coe_Inf | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Inf {S : set (subring R)} {x : R} : x ∈ Inf S ↔ ∀ p ∈ S, x ∈ p | set.mem_Inter₂ | lemma | subring.mem_Inf | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.mem_Inter₂",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_infi {ι : Sort*} {S : ι → subring R} :
(↑(⨅ i, S i) : set R) = ⋂ i, S i | by simp only [infi, coe_Inf, set.bInter_range] | lemma | subring.coe_infi | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"infi",
"set.bInter_range",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi {ι : Sort*} {S : ι → subring R} {x : R} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i | by simp only [infi, mem_Inf, set.forall_range_iff] | lemma | subring.mem_infi | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"infi",
"set.forall_range_iff",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_to_submonoid (s : set (subring R)) :
(Inf s).to_submonoid = ⨅ t ∈ s, subring.to_submonoid t | mk'_to_submonoid _ _ | lemma | subring.Inf_to_submonoid | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring",
"subring.to_submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_to_add_subgroup (s : set (subring R)) :
(Inf s).to_add_subgroup = ⨅ t ∈ s, subring.to_add_subgroup t | mk'_to_add_subgroup _ _ | lemma | subring.Inf_to_add_subgroup | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_iff' (A : subring R) : A = ⊤ ↔ ∀ x : R, x ∈ A | eq_top_iff.trans ⟨λ h m, h $ mem_top m, λ h m _, h m⟩ | lemma | subring.eq_top_iff' | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center : subring R | { carrier := set.center R,
neg_mem' := λ a, set.neg_mem_center,
.. subsemiring.center R } | def | subring.center | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.center",
"set.neg_mem_center",
"subring",
"subsemiring.center"
] | The center of a ring `R` is the set of elements that commute with everything in `R` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_center : ↑(center R) = set.center R | rfl | lemma | subring.coe_center | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.center"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_to_subsemiring : (center R).to_subsemiring = subsemiring.center R | rfl | lemma | subring.center_to_subsemiring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subsemiring.center"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_center_iff {z : R} : z ∈ center R ↔ ∀ g, g * z = z * g | iff.rfl | lemma | subring.mem_center_iff | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decidable_mem_center [decidable_eq R] [fintype R] : decidable_pred (∈ center R) | λ _, decidable_of_iff' _ mem_center_iff | instance | subring.decidable_mem_center | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"decidable_of_iff'",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_eq_top (R) [comm_ring R] : center R = ⊤ | set_like.coe_injective (set.center_eq_univ R) | lemma | subring.center_eq_top | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"comm_ring",
"set.center_eq_univ",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center.coe_inv (a : center K) : ((a⁻¹ : center K) : K) = (a : K)⁻¹ | rfl | lemma | subring.center.coe_inv | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center.coe_div (a b : center K) : ((a / b : center K) : K) = (a : K) / (b : K) | rfl | lemma | subring.center.coe_div | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer (s : set R) : subring R | { neg_mem' := λ x, set.neg_mem_centralizer,
..subsemiring.centralizer s } | def | subring.centralizer | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.neg_mem_centralizer",
"subring",
"subsemiring.centralizer"
] | The centralizer of a set inside a ring as a `subring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_centralizer (s : set R) : (centralizer s : set R) = s.centralizer | rfl | lemma | subring.coe_centralizer | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_to_submonoid (s : set R) :
(centralizer s).to_submonoid = submonoid.centralizer s | rfl | lemma | subring.centralizer_to_submonoid | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"submonoid.centralizer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_to_subsemiring (s : set R) :
(centralizer s).to_subsemiring = subsemiring.centralizer s | rfl | lemma | subring.centralizer_to_subsemiring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subsemiring.centralizer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_centralizer_iff {s : set R} {z : R} :
z ∈ centralizer s ↔ ∀ g ∈ s, g * z = z * g | iff.rfl | lemma | subring.mem_centralizer_iff | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_le_centralizer (s) : center R ≤ centralizer s | s.center_subset_centralizer | lemma | subring.center_le_centralizer | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_le (s t : set R) (h : s ⊆ t) : centralizer t ≤ centralizer s | set.centralizer_subset h | lemma | subring.centralizer_le | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.centralizer_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_eq_top_iff_subset {s : set R} : centralizer s = ⊤ ↔ s ⊆ center R | set_like.ext'_iff.trans set.centralizer_eq_top_iff_subset | lemma | subring.centralizer_eq_top_iff_subset | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.centralizer_eq_top_iff_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_univ : centralizer set.univ = center R | set_like.ext' (set.centralizer_univ R) | lemma | subring.centralizer_univ | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.centralizer_univ",
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure (s : set R) : subring R | Inf {S | s ⊆ S} | def | subring.closure | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"subring"
] | The `subring` generated by a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure {x : R} {s : set R} : x ∈ closure s ↔ ∀ S : subring R, s ⊆ S → x ∈ S | mem_Inf | lemma | subring.mem_closure | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_le {s : set R} {t : subring R} : closure s ≤ t ↔ s ⊆ t | ⟨set.subset.trans subset_closure, λ h, Inf_le h⟩ | lemma | subring.closure_le | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"Inf_le",
"closure",
"subring",
"subset_closure"
] | A subring `t` includes `closure s` if and only if it includes `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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