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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|---|---|---|---|---|---|---|---|---|---|---|
is_noetherian.disjoint_partial_sups_eventually_bot [I : is_noetherian R M]
(f : ℕ → submodule R M) (h : ∀ n, disjoint (partial_sups f n) (f (n+1))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊥ | begin
-- A little off-by-one cleanup first:
suffices t : ∃ n : ℕ, ∀ m, n ≤ m → f (m+1) = ⊥,
{ obtain ⟨n, w⟩ := t,
use n+1,
rintros (_|m) p,
{ cases p, },
{ apply w,
exact nat.succ_le_succ_iff.mp p }, },
obtain ⟨n, w⟩ := monotone_stabilizes_iff_noetherian.mpr I (partial_sups f),
exact ⟨n... | lemma | is_noetherian.disjoint_partial_sups_eventually_bot | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"disjoint",
"is_noetherian",
"partial_sups",
"submodule"
] | A sequence `f` of submodules of a noetherian module,
with `f (n+1)` disjoint from the supremum of `f 0`, ..., `f n`,
is eventually zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian.equiv_punit_of_prod_injective [is_noetherian R M]
(f : M × N →ₗ[R] M) (i : injective f) : N ≃ₗ[R] punit.{w+1} | begin
apply nonempty.some,
obtain ⟨n, w⟩ := is_noetherian.disjoint_partial_sups_eventually_bot (f.tailing i)
(f.tailings_disjoint_tailing i),
specialize w n (le_refl n),
apply nonempty.intro,
refine (f.tailing_linear_equiv i n).symm ≪≫ₗ _,
rw w,
exact submodule.bot_equiv_punit,
end | def | is_noetherian.equiv_punit_of_prod_injective | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"is_noetherian.disjoint_partial_sups_eventually_bot",
"nonempty.some",
"submodule.bot_equiv_punit"
] | If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian_ring (R) [semiring R] | is_noetherian R R | def | is_noetherian_ring | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"semiring"
] | A (semi)ring is Noetherian if it is Noetherian as a module over itself,
i.e. all its ideals are finitely generated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian_ring_iff {R} [semiring R] : is_noetherian_ring R ↔ is_noetherian R R | iff.rfl | theorem | is_noetherian_ring_iff | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian",
"is_noetherian_ring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring_iff_ideal_fg (R : Type*) [semiring R] :
is_noetherian_ring R ↔ ∀ I : ideal R, I.fg | is_noetherian_ring_iff.trans is_noetherian_def | lemma | is_noetherian_ring_iff_ideal_fg | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"ideal",
"is_noetherian_def",
"is_noetherian_ring",
"semiring"
] | A ring is Noetherian if and only if all its ideals are finitely-generated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian_of_finite (R M) [finite M] [semiring R] [add_comm_monoid M] [module R M] :
is_noetherian R M | ⟨λ s, ⟨(s : set M).to_finite.to_finset, by rw [set.finite.coe_to_finset, submodule.span_eq]⟩⟩ | instance | is_noetherian_of_finite | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_monoid",
"finite",
"is_noetherian",
"module",
"semiring",
"set.finite.coe_to_finset",
"submodule.span_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_of_subsingleton (R M) [subsingleton R] [semiring R] [add_comm_monoid M]
[module R M] : is_noetherian R M | by { haveI := module.subsingleton R M, exact is_noetherian_of_finite R M } | instance | is_noetherian_of_subsingleton | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_monoid",
"is_noetherian",
"is_noetherian_of_finite",
"module",
"module.subsingleton",
"semiring"
] | Modules over the trivial ring are Noetherian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian_of_submodule_of_noetherian (R M) [semiring R] [add_comm_monoid M] [module R M]
(N : submodule R M) (h : is_noetherian R M) : is_noetherian R N | begin
rw is_noetherian_iff_well_founded at h ⊢,
exact order_embedding.well_founded (submodule.map_subtype.order_embedding N).dual h,
end | theorem | is_noetherian_of_submodule_of_noetherian | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_monoid",
"is_noetherian",
"is_noetherian_iff_well_founded",
"module",
"order_embedding.well_founded",
"semiring",
"submodule",
"submodule.map_subtype.order_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.quotient.is_noetherian {R} [ring R] {M} [add_comm_group M] [module R M]
(N : submodule R M) [h : is_noetherian R M] : is_noetherian R (M ⧸ N) | begin
rw is_noetherian_iff_well_founded at h ⊢,
exact order_embedding.well_founded (submodule.comap_mkq.order_embedding N).dual h,
end | instance | submodule.quotient.is_noetherian | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_group",
"is_noetherian",
"is_noetherian_iff_well_founded",
"module",
"order_embedding.well_founded",
"ring",
"submodule",
"submodule.comap_mkq.order_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_of_tower (R) {S M} [semiring R] [semiring S]
[add_comm_monoid M] [has_smul R S] [module S M] [module R M] [is_scalar_tower R S M]
(h : is_noetherian R M) : is_noetherian S M | begin
rw is_noetherian_iff_well_founded at h ⊢,
refine (submodule.restrict_scalars_embedding R S M).dual.well_founded h
end | theorem | is_noetherian_of_tower | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_monoid",
"has_smul",
"is_noetherian",
"is_noetherian_iff_well_founded",
"is_scalar_tower",
"module",
"semiring",
"submodule.restrict_scalars_embedding"
] | If `M / S / R` is a scalar tower, and `M / R` is Noetherian, then `M / S` is
also noetherian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian_of_fg_of_noetherian {R M} [ring R] [add_comm_group M] [module R M]
(N : submodule R M) [is_noetherian_ring R] (hN : N.fg) : is_noetherian R N | let ⟨s, hs⟩ := hN in
begin
haveI := classical.dec_eq M,
haveI := classical.dec_eq R,
letI : is_noetherian R R := by apply_instance,
have : ∀ x ∈ s, x ∈ N, from λ x hx, hs ▸ submodule.subset_span hx,
refine @@is_noetherian_of_surjective ((↑s : set M) → R) _ _ _ (pi.module _ _ _)
_ _ _ is_noetherian_pi,
{... | theorem | is_noetherian_of_fg_of_noetherian | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_group",
"add_smul",
"classical.dec_eq",
"finsupp.mem_span_image_iff_total",
"finsupp.total_apply",
"is_noetherian",
"is_noetherian_of_surjective",
"is_noetherian_pi",
"is_noetherian_ring",
"linear_map.range_eq_top",
"module",
"pi.module",
"ring",
"set.image_id",
"smul_eq_mul",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_of_fg_of_noetherian' {R M} [ring R] [add_comm_group M] [module R M]
[is_noetherian_ring R] (h : (⊤ : submodule R M).fg) : is_noetherian R M | have is_noetherian R (⊤ : submodule R M), from is_noetherian_of_fg_of_noetherian _ h,
by exactI is_noetherian_of_linear_equiv (linear_equiv.of_top (⊤ : submodule R M) rfl) | lemma | is_noetherian_of_fg_of_noetherian' | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_group",
"is_noetherian",
"is_noetherian_of_fg_of_noetherian",
"is_noetherian_of_linear_equiv",
"is_noetherian_ring",
"linear_equiv.of_top",
"module",
"ring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_span_of_finite (R) {M} [ring R] [add_comm_group M] [module R M]
[is_noetherian_ring R] {A : set M} (hA : A.finite) : is_noetherian R (submodule.span R A) | is_noetherian_of_fg_of_noetherian _ (submodule.fg_def.mpr ⟨A, hA, rfl⟩) | theorem | is_noetherian_span_of_finite | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"add_comm_group",
"is_noetherian",
"is_noetherian_of_fg_of_noetherian",
"is_noetherian_ring",
"module",
"ring",
"submodule.span"
] | In a module over a noetherian ring, the submodule generated by finitely many vectors is
noetherian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_noetherian_ring_of_surjective (R) [ring R] (S) [ring S]
(f : R →+* S) (hf : function.surjective f)
[H : is_noetherian_ring R] : is_noetherian_ring S | begin
rw [is_noetherian_ring_iff, is_noetherian_iff_well_founded] at H ⊢,
exact order_embedding.well_founded (ideal.order_embedding_of_surjective f hf).dual H,
end | theorem | is_noetherian_ring_of_surjective | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"ideal.order_embedding_of_surjective",
"is_noetherian_iff_well_founded",
"is_noetherian_ring",
"is_noetherian_ring_iff",
"order_embedding.well_founded",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring_range {R} [ring R] {S} [ring S] (f : R →+* S)
[is_noetherian_ring R] : is_noetherian_ring f.range | is_noetherian_ring_of_surjective R f.range f.range_restrict
f.range_restrict_surjective | instance | is_noetherian_ring_range | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian_ring",
"is_noetherian_ring_of_surjective",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring_of_ring_equiv (R) [ring R] {S} [ring S]
(f : R ≃+* S) [is_noetherian_ring R] : is_noetherian_ring S | is_noetherian_ring_of_surjective R S f.to_ring_hom f.to_equiv.surjective | theorem | is_noetherian_ring_of_ring_equiv | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"is_noetherian_ring",
"is_noetherian_ring_of_surjective",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring.is_nilpotent_nilradical (R : Type*) [comm_ring R] [is_noetherian_ring R] :
is_nilpotent (nilradical R) | begin
obtain ⟨n, hn⟩ := ideal.exists_radical_pow_le_of_fg (⊥ : ideal R) (is_noetherian.noetherian _),
exact ⟨n, eq_bot_iff.mpr hn⟩
end | lemma | is_noetherian_ring.is_nilpotent_nilradical | ring_theory | src/ring_theory/noetherian.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.algebra.tower",
"algebra.ring.idempotents",
"group_theory.finiteness",
"linear_algebra.linear_independent",
"order.compactly_generated",
"order.order_iso_nat",
"ring_theory.finiteness",
"ring_theory.nilpotent"
] | [
"comm_ring",
"ideal",
"ideal.exists_radical_pow_le_of_fg",
"is_nilpotent",
"is_noetherian_ring",
"nilradical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_zero_divisors (R : Type*) [monoid_with_zero R] : submonoid R | { carrier := {x | ∀ z, z * x = 0 → z = 0},
one_mem' := λ z hz, by rwa mul_one at hz,
mul_mem' := λ x₁ x₂ hx₁ hx₂ z hz,
have z * x₁ * x₂ = 0, by rwa mul_assoc,
hx₁ z $ hx₂ (z * x₁) this } | def | non_zero_divisors | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"monoid_with_zero",
"mul_assoc",
"mul_one",
"submonoid"
] | The submonoid of non-zero-divisors of a `monoid_with_zero` `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_non_zero_divisors_iff {r : M} : r ∈ M⁰ ↔ ∀ x, x * r = 0 → x = 0 | iff.rfl | lemma | mem_non_zero_divisors_iff | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_mem_non_zero_divisors_eq_zero_iff {x r : M} (hr : r ∈ M⁰) :
x * r = 0 ↔ x = 0 | ⟨hr _, by simp {contextual := tt}⟩ | lemma | mul_right_mem_non_zero_divisors_eq_zero_iff | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_coe_non_zero_divisors_eq_zero_iff {x : M} {c : M⁰} :
x * c = 0 ↔ x = 0 | mul_right_mem_non_zero_divisors_eq_zero_iff c.prop | lemma | mul_right_coe_non_zero_divisors_eq_zero_iff | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"mul_right_mem_non_zero_divisors_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_mem_non_zero_divisors_eq_zero_iff {r x : M₁} (hr : r ∈ M₁⁰) :
r * x = 0 ↔ x = 0 | by rw [mul_comm, mul_right_mem_non_zero_divisors_eq_zero_iff hr] | lemma | mul_left_mem_non_zero_divisors_eq_zero_iff | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"mul_comm",
"mul_right_mem_non_zero_divisors_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_coe_non_zero_divisors_eq_zero_iff {c : M₁⁰} {x : M₁} :
(c : M₁) * x = 0 ↔ x = 0 | mul_left_mem_non_zero_divisors_eq_zero_iff c.prop | lemma | mul_left_coe_non_zero_divisors_eq_zero_iff | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"mul_left_mem_non_zero_divisors_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_cancel_right_mem_non_zero_divisor {x y r : R} (hr : r ∈ R⁰) :
x * r = y * r ↔ x = y | begin
refine ⟨λ h, _, congr_arg _⟩,
rw [←sub_eq_zero, ←mul_right_mem_non_zero_divisors_eq_zero_iff hr, sub_mul, h, sub_self]
end | lemma | mul_cancel_right_mem_non_zero_divisor | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_cancel_right_coe_non_zero_divisor {x y : R} {c : R⁰} :
x * c = y * c ↔ x = y | mul_cancel_right_mem_non_zero_divisor c.prop | lemma | mul_cancel_right_coe_non_zero_divisor | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"mul_cancel_right_mem_non_zero_divisor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_cancel_left_mem_non_zero_divisor {x y r : R'} (hr : r ∈ R'⁰) :
r * x = r * y ↔ x = y | by simp_rw [mul_comm r, mul_cancel_right_mem_non_zero_divisor hr] | lemma | mul_cancel_left_mem_non_zero_divisor | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"mul_cancel_right_mem_non_zero_divisor",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_cancel_left_coe_non_zero_divisor {x y : R'} {c : R'⁰} :
(c : R') * x = c * y ↔ x = y | mul_cancel_left_mem_non_zero_divisor c.prop | lemma | mul_cancel_left_coe_non_zero_divisor | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"mul_cancel_left_mem_non_zero_divisor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_zero_divisors.ne_zero [nontrivial M] {x} (hx : x ∈ M⁰) : x ≠ 0 | λ h, one_ne_zero (hx _ $ (one_mul _).trans h) | lemma | non_zero_divisors.ne_zero | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"nontrivial",
"one_mul",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_zero_divisors.coe_ne_zero [nontrivial M] (x : M⁰) : (x : M) ≠ 0 | non_zero_divisors.ne_zero x.2 | lemma | non_zero_divisors.coe_ne_zero | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"non_zero_divisors.ne_zero",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_non_zero_divisors {a b : M₁} :
a * b ∈ M₁⁰ ↔ a ∈ M₁⁰ ∧ b ∈ M₁⁰ | begin
split,
{ intro h,
split; intros x h'; apply h,
{ rw [←mul_assoc, h', zero_mul] },
{ rw [mul_comm a b, ←mul_assoc, h', zero_mul] } },
{ rintros ⟨ha, hb⟩ x hx,
apply ha,
apply hb,
rw [mul_assoc, hx] },
end | lemma | mul_mem_non_zero_divisors | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"mul_assoc",
"mul_comm",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_mem_non_zero_divisors {G₀ : Type*} [group_with_zero G₀]
{x : G₀} (hx : x ∈ non_zero_divisors G₀) : is_unit x | ⟨⟨x, x⁻¹, mul_inv_cancel (non_zero_divisors.ne_zero hx),
inv_mul_cancel (non_zero_divisors.ne_zero hx)⟩, rfl⟩ | lemma | is_unit_of_mem_non_zero_divisors | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"group_with_zero",
"inv_mul_cancel",
"is_unit",
"mul_inv_cancel",
"non_zero_divisors",
"non_zero_divisors.ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_ne_zero_of_mul_right_eq_zero [no_zero_divisors M]
{x y : M} (hnx : x ≠ 0) (hxy : y * x = 0) : y = 0 | or.resolve_right (eq_zero_or_eq_zero_of_mul_eq_zero hxy) hnx | lemma | eq_zero_of_ne_zero_of_mul_right_eq_zero | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_ne_zero_of_mul_left_eq_zero [no_zero_divisors M]
{x y : M} (hnx : x ≠ 0) (hxy : x * y = 0) : y = 0 | or.resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero hxy) hnx | lemma | eq_zero_of_ne_zero_of_mul_left_eq_zero | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_non_zero_divisors_of_ne_zero [no_zero_divisors M] {x : M} (hx : x ≠ 0) : x ∈ M⁰ | λ _, eq_zero_of_ne_zero_of_mul_right_eq_zero hx | lemma | mem_non_zero_divisors_of_ne_zero | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"eq_zero_of_ne_zero_of_mul_right_eq_zero",
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_non_zero_divisors_iff_ne_zero [no_zero_divisors M] [nontrivial M] {x : M} :
x ∈ M⁰ ↔ x ≠ 0 | ⟨non_zero_divisors.ne_zero, mem_non_zero_divisors_of_ne_zero⟩ | lemma | mem_non_zero_divisors_iff_ne_zero | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"no_zero_divisors",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne_zero_of_mem_non_zero_divisors [nontrivial M] [zero_hom_class F M M']
(g : F) (hg : function.injective (g : M → M')) {x : M} (h : x ∈ M⁰) : g x ≠ 0 | λ h0, one_ne_zero (h 1 ((one_mul x).symm ▸ (hg (trans h0 (map_zero g).symm)))) | lemma | map_ne_zero_of_mem_non_zero_divisors | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"nontrivial",
"one_mul",
"one_ne_zero",
"zero_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mem_non_zero_divisors [nontrivial M] [no_zero_divisors M'] [zero_hom_class F M M']
(g : F) (hg : function.injective g) {x : M} (h : x ∈ M⁰) : g x ∈ M'⁰ | λ z hz, eq_zero_of_ne_zero_of_mul_right_eq_zero
(map_ne_zero_of_mem_non_zero_divisors g hg h) hz | lemma | map_mem_non_zero_divisors | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"eq_zero_of_ne_zero_of_mul_right_eq_zero",
"map_ne_zero_of_mem_non_zero_divisors",
"no_zero_divisors",
"nontrivial",
"zero_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_non_zero_divisors_of_no_zero_divisors [no_zero_divisors M] {S : submonoid M}
(hS : (0 : M) ∉ S) : S ≤ M⁰ | λ x hx y hy, or.rec_on (eq_zero_or_eq_zero_of_mul_eq_zero hy)
(λ h, h) (λ h, absurd (h ▸ hx : (0 : M) ∈ S) hS) | lemma | le_non_zero_divisors_of_no_zero_divisors | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"no_zero_divisors",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powers_le_non_zero_divisors_of_no_zero_divisors [no_zero_divisors M]
{a : M} (ha : a ≠ 0) : submonoid.powers a ≤ M⁰ | le_non_zero_divisors_of_no_zero_divisors (λ h, absurd (h.rec_on (λ _ hn, pow_eq_zero hn)) ha) | lemma | powers_le_non_zero_divisors_of_no_zero_divisors | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"le_non_zero_divisors_of_no_zero_divisors",
"no_zero_divisors",
"pow_eq_zero",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_non_zero_divisors_of_injective [no_zero_divisors M']
[monoid_with_zero_hom_class F M M'] (f : F) (hf : function.injective f) {S : submonoid M}
(hS : S ≤ M⁰) : S.map f ≤ M'⁰ | begin
casesI subsingleton_or_nontrivial M,
{ simp [subsingleton.elim S ⊥] },
{ exact le_non_zero_divisors_of_no_zero_divisors (λ h, let ⟨x, hx, hx0⟩ := h in
zero_ne_one (hS (hf (trans hx0 ((map_zero f).symm)) ▸ hx : 0 ∈ S) 1 (mul_zero 1)).symm) }
end | lemma | map_le_non_zero_divisors_of_injective | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"le_non_zero_divisors_of_no_zero_divisors",
"monoid_with_zero_hom_class",
"mul_zero",
"no_zero_divisors",
"submonoid",
"subsingleton_or_nontrivial",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_zero_divisors_le_comap_non_zero_divisors_of_injective [no_zero_divisors M']
[monoid_with_zero_hom_class F M M'] (f : F) (hf : function.injective f) : M⁰ ≤ M'⁰.comap f | submonoid.le_comap_of_map_le _ (map_le_non_zero_divisors_of_injective _ hf le_rfl) | lemma | non_zero_divisors_le_comap_non_zero_divisors_of_injective | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"le_rfl",
"map_le_non_zero_divisors_of_injective",
"monoid_with_zero_hom_class",
"no_zero_divisors",
"submonoid.le_comap_of_map_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_zero_iff_exists_zero [no_zero_divisors M₁] [nontrivial M₁]
{s : multiset M₁} : s.prod = 0 ↔ ∃ (r : M₁) (hr : r ∈ s), r = 0 | begin
split, swap,
{ rintros ⟨r, hrs, rfl⟩,
exact multiset.prod_eq_zero hrs, },
refine multiset.induction _ (λ a s ih, _) s,
{ intro habs,
simpa using habs, },
{ rw multiset.prod_cons,
intro hprod,
replace hprod := eq_zero_or_eq_zero_of_mul_eq_zero hprod,
cases hprod with ha,
{ exact ⟨... | lemma | prod_zero_iff_exists_zero | ring_theory | src/ring_theory/non_zero_divisors.lean | [
"group_theory.submonoid.operations",
"group_theory.submonoid.membership"
] | [
"ih",
"multiset",
"multiset.induction",
"multiset.mem_cons_self",
"multiset.prod_cons",
"multiset.prod_eq_zero",
"no_zero_divisors",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm : S →* R | linear_map.det.comp (lmul R S).to_ring_hom.to_monoid_hom | def | algebra.norm | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [] | The norm of an element `s` of an `R`-algebra is the determinant of `(*) s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_apply (x : S) : norm R x = linear_map.det (lmul R S x) | rfl | lemma | algebra.norm_apply | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"linear_map.det"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_one_of_not_exists_basis
(h : ¬ ∃ (s : finset S), nonempty (basis s R S)) (x : S) : norm R x = 1 | by { rw [norm_apply, linear_map.det], split_ifs with h, refl } | lemma | algebra.norm_eq_one_of_not_exists_basis | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"basis",
"finset",
"linear_map.det"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_one_of_not_module_finite (h : ¬ module.finite R S) (x : S) :
norm R x = 1 | begin
refine norm_eq_one_of_not_exists_basis _ (mt _ h) _,
rintro ⟨s, ⟨b⟩⟩,
exact module.finite.of_basis b,
end | lemma | algebra.norm_eq_one_of_not_module_finite | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"module.finite",
"module.finite.of_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_matrix_det [fintype ι] [decidable_eq ι] (b : basis ι R S) (s : S) :
norm R s = matrix.det (algebra.left_mul_matrix b s) | by { rwa [norm_apply, ← linear_map.det_to_matrix b, ← to_matrix_lmul_eq], refl } | lemma | algebra.norm_eq_matrix_det | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra.left_mul_matrix",
"basis",
"fintype",
"linear_map.det_to_matrix",
"matrix.det"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_algebra_map_of_basis [fintype ι] (b : basis ι R S) (x : R) :
norm R (algebra_map R S x) = x ^ fintype.card ι | begin
haveI := classical.dec_eq ι,
rw [norm_apply, ← det_to_matrix b, lmul_algebra_map],
convert @det_diagonal _ _ _ _ _ (λ (i : ι), x),
{ ext i j, rw [to_matrix_lsmul, matrix.diagonal] },
{ rw [finset.prod_const, finset.card_univ] }
end | lemma | algebra.norm_algebra_map_of_basis | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra_map",
"basis",
"classical.dec_eq",
"finset.card_univ",
"finset.prod_const",
"fintype",
"fintype.card",
"matrix.diagonal"
] | If `x` is in the base ring `K`, then the norm is `x ^ [L : K]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_algebra_map {L : Type*} [ring L] [algebra K L] (x : K) :
norm K (algebra_map K L x) = x ^ finrank K L | begin
by_cases H : ∃ (s : finset L), nonempty (basis s K L),
{ rw [norm_algebra_map_of_basis H.some_spec.some, finrank_eq_card_basis H.some_spec.some] },
{ rw [norm_eq_one_of_not_exists_basis K H, finrank_eq_zero_of_not_exists_basis, pow_zero],
rintros ⟨s, ⟨b⟩⟩,
exact H ⟨s, ⟨b⟩⟩ },
end | lemma | algebra.norm_algebra_map | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra",
"algebra_map",
"basis",
"finrank_eq_zero_of_not_exists_basis",
"finset",
"norm_algebra_map",
"pow_zero",
"ring"
] | If `x` is in the base field `K`, then the norm is `x ^ [L : K]`.
(If `L` is not finite-dimensional over `K`, then `norm = 1 = x ^ 0 = x ^ (finrank L K)`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
power_basis.norm_gen_eq_coeff_zero_minpoly (pb : power_basis R S) :
norm R pb.gen = (-1) ^ pb.dim * coeff (minpoly R pb.gen) 0 | by rw [norm_eq_matrix_det pb.basis, det_eq_sign_charpoly_coeff,
charpoly_left_mul_matrix, fintype.card_fin] | lemma | algebra.power_basis.norm_gen_eq_coeff_zero_minpoly | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"charpoly_left_mul_matrix",
"fintype.card_fin",
"minpoly",
"power_basis"
] | Given `pb : power_basis K S`, then the norm of `pb.gen` is
`(-1) ^ pb.dim * coeff (minpoly K pb.gen) 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
power_basis.norm_gen_eq_prod_roots [algebra R F] (pb : power_basis R S)
(hf : (minpoly R pb.gen).splits (algebra_map R F)) :
algebra_map R F (norm R pb.gen) =
((minpoly R pb.gen).map (algebra_map R F)).roots.prod | begin
haveI := module.nontrivial R F,
have := minpoly.monic pb.is_integral_gen,
rw [power_basis.norm_gen_eq_coeff_zero_minpoly, ← pb.nat_degree_minpoly, ring_hom.map_mul,
← coeff_map, prod_roots_eq_coeff_zero_of_monic_of_split (this.map _)
((splits_id_iff_splits _).2 hf), this.nat_degree_map, map_pow, ←... | lemma | algebra.power_basis.norm_gen_eq_prod_roots | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra",
"algebra_map",
"map_pow",
"minpoly",
"minpoly.monic",
"module.nontrivial",
"mul_assoc",
"mul_pow",
"neg_mul",
"one_mul",
"one_pow",
"power_basis",
"ring_hom.map_mul"
] | Given `pb : power_basis R S`, then the norm of `pb.gen` is
`((minpoly R pb.gen).map (algebra_map R F)).roots.prod`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_zero [nontrivial S] [module.free R S] [module.finite R S] :
norm R (0 : S) = 0 | begin
nontriviality,
rw [norm_apply, coe_lmul_eq_mul, map_zero, linear_map.det_zero' (module.free.choose_basis R S)]
end | lemma | algebra.norm_zero | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"linear_map.det_zero'",
"module.finite",
"module.free",
"module.free.choose_basis",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_zero_iff [is_domain R] [is_domain S] [module.free R S] [module.finite R S]
{x : S} :
norm R x = 0 ↔ x = 0 | begin
split,
let b := module.free.choose_basis R S,
swap, { rintro rfl, exact norm_zero },
{ letI := classical.dec_eq (module.free.choose_basis_index R S),
rw [norm_eq_matrix_det b,
← matrix.exists_mul_vec_eq_zero_iff],
rintros ⟨v, v_ne, hv⟩,
rw [← b.equiv_fun.apply_symm_apply v, b.equiv_fun... | lemma | algebra.norm_eq_zero_iff | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"classical.dec_eq",
"is_domain",
"linear_equiv.map_zero",
"matrix.exists_mul_vec_eq_zero_iff",
"module.finite",
"module.free",
"module.free.choose_basis",
"module.free.choose_basis_index"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_ne_zero_iff [is_domain R] [is_domain S] [module.free R S] [module.finite R S]
{x : S} :
norm R x ≠ 0 ↔ x ≠ 0 | not_iff_not.mpr norm_eq_zero_iff | lemma | algebra.norm_ne_zero_iff | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"is_domain",
"module.finite",
"module.free"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_zero_iff' [is_domain R] [is_domain S] [module.free R S] [module.finite R S]
{x : S} :
linear_map.det (linear_map.mul R S x) = 0 ↔ x = 0 | norm_eq_zero_iff | lemma | algebra.norm_eq_zero_iff' | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"is_domain",
"linear_map.det",
"linear_map.mul",
"module.finite",
"module.free"
] | This is `algebra.norm_eq_zero_iff` composed with `algebra.norm_apply`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eq_zero_iff_of_basis [is_domain R] [is_domain S] (b : basis ι R S) {x : S} :
algebra.norm R x = 0 ↔ x = 0 | begin
haveI : module.free R S := module.free.of_basis b,
haveI : module.finite R S := module.finite.of_basis b,
exact norm_eq_zero_iff
end | lemma | algebra.norm_eq_zero_iff_of_basis | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra.norm",
"basis",
"is_domain",
"module.finite",
"module.finite.of_basis",
"module.free",
"module.free.of_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_ne_zero_iff_of_basis [is_domain R] [is_domain S] (b : basis ι R S) {x : S} :
algebra.norm R x ≠ 0 ↔ x ≠ 0 | not_iff_not.mpr (norm_eq_zero_iff_of_basis b) | lemma | algebra.norm_ne_zero_iff_of_basis | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra.norm",
"basis",
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_norm_adjoin [finite_dimensional K L] [is_separable K L] (x : L) :
norm K x = norm K (adjoin_simple.gen K x) ^ finrank K⟮x⟯ L | begin
letI := is_separable_tower_top_of_is_separable K K⟮x⟯ L,
let pbL := field.power_basis_of_finite_of_separable K⟮x⟯ L,
let pbx := intermediate_field.adjoin.power_basis (is_separable.is_integral K x),
rw [← adjoin_simple.algebra_map_gen K x, norm_eq_matrix_det (pbx.basis.smul pbL.basis) _,
smul_left_mul_... | lemma | algebra.norm_eq_norm_adjoin | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"field.power_basis_of_finite_of_separable",
"finite_dimensional",
"finset.card_fin",
"finset.prod_const",
"intermediate_field.adjoin.power_basis",
"is_separable",
"is_separable.is_integral",
"is_separable_tower_top_of_is_separable",
"power_basis.finrank"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.intermediate_field.adjoin_simple.norm_gen_eq_one {x : L}
(hx : ¬is_integral K x) : norm K (adjoin_simple.gen K x) = 1 | begin
rw [norm_eq_one_of_not_exists_basis],
contrapose! hx,
obtain ⟨s, ⟨b⟩⟩ := hx,
refine is_integral_of_mem_of_fg (K⟮x⟯).to_subalgebra _ x _,
{ exact (submodule.fg_iff_finite_dimensional _).mpr (of_fintype_basis b) },
{ exact intermediate_field.subset_adjoin K _ (set.mem_singleton x) }
end | lemma | intermediate_field.adjoin_simple.norm_gen_eq_one | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"intermediate_field.subset_adjoin",
"is_integral",
"is_integral_of_mem_of_fg",
"set.mem_singleton",
"submodule.fg_iff_finite_dimensional"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.intermediate_field.adjoin_simple.norm_gen_eq_prod_roots (x : L)
(hf : (minpoly K x).splits (algebra_map K F)) :
(algebra_map K F) (norm K (adjoin_simple.gen K x)) =
((minpoly K x).map (algebra_map K F)).roots.prod | begin
have injKxL := (algebra_map K⟮x⟯ L).injective,
by_cases hx : is_integral K x, swap,
{ simp [minpoly.eq_zero hx, intermediate_field.adjoin_simple.norm_gen_eq_one hx] },
have hx' : is_integral K (adjoin_simple.gen K x),
{ rwa [← is_integral_algebra_map_iff injKxL, adjoin_simple.algebra_map_gen],
apply... | lemma | intermediate_field.adjoin_simple.norm_gen_eq_prod_roots | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra_map",
"intermediate_field.adjoin_simple.norm_gen_eq_one",
"is_integral",
"is_integral_algebra_map_iff",
"minpoly",
"minpoly.eq_of_algebra_map_eq",
"minpoly.eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_prod_embeddings_gen [algebra R F] (pb : power_basis R S)
(hE : (minpoly R pb.gen).splits (algebra_map R F)) (hfx : (minpoly R pb.gen).separable) :
algebra_map R F (norm R pb.gen) = (@@finset.univ pb^.alg_hom.fintype).prod (λ σ, σ pb.gen) | begin
letI := classical.dec_eq F,
rw [pb.norm_gen_eq_prod_roots hE, fintype.prod_equiv pb.lift_equiv',
finset.prod_mem_multiset, finset.prod_eq_multiset_prod, multiset.to_finset_val,
multiset.dedup_eq_self.mpr, multiset.map_id],
{ exact nodup_roots hfx.map },
{ intro x, refl },
{ intro σ, rw [pb.lift_... | lemma | algebra.norm_eq_prod_embeddings_gen | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra",
"algebra_map",
"classical.dec_eq",
"finset.prod_eq_multiset_prod",
"finset.prod_mem_multiset",
"finset.univ",
"fintype.prod_equiv",
"minpoly",
"multiset.map_id",
"multiset.to_finset_val",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_prod_roots [is_separable K L] [finite_dimensional K L]
{x : L} (hF : (minpoly K x).splits (algebra_map K F)) :
algebra_map K F (norm K x) = ((minpoly K x).map (algebra_map K F)).roots.prod ^ finrank K⟮x⟯ L | by rw [norm_eq_norm_adjoin K x, map_pow,
intermediate_field.adjoin_simple.norm_gen_eq_prod_roots _ hF] | lemma | algebra.norm_eq_prod_roots | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra_map",
"finite_dimensional",
"intermediate_field.adjoin_simple.norm_gen_eq_prod_roots",
"is_separable",
"map_pow",
"minpoly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_embeddings_eq_finrank_pow [algebra L F] [is_scalar_tower K L F] [is_alg_closed E]
[is_separable K F] [finite_dimensional K F] (pb : power_basis K L) :
∏ σ : F →ₐ[K] E, σ (algebra_map L F pb.gen) =
((@@finset.univ pb^.alg_hom.fintype).prod (λ σ : L →ₐ[K] E, σ pb.gen)) ^ finrank L F | begin
haveI : finite_dimensional L F := finite_dimensional.right K L F,
haveI : is_separable L F := is_separable_tower_top_of_is_separable K L F,
letI : fintype (L →ₐ[K] E) := power_basis.alg_hom.fintype pb,
letI : ∀ (f : L →ₐ[K] E), fintype (@@alg_hom L F E _ _ _ _ f.to_ring_hom.to_algebra) := _,
rw [fintype... | lemma | algebra.prod_embeddings_eq_finrank_pow | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"alg_hom",
"alg_hom.card",
"alg_hom.comp_apply",
"alg_hom.restrict_domain",
"alg_hom_equiv_sigma",
"algebra",
"algebra_map",
"equiv.coe_fn_mk",
"finite_dimensional",
"finite_dimensional.right",
"finset.card_univ",
"finset.prod_congr",
"finset.prod_const",
"finset.prod_pow",
"finset.prod_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_prod_embeddings [finite_dimensional K L] [is_separable K L] [is_alg_closed E]
(x : L) : algebra_map K E (norm K x) = ∏ σ : L →ₐ[K] E, σ x | begin
have hx := is_separable.is_integral K x,
rw [norm_eq_norm_adjoin K x, ring_hom.map_pow, ← adjoin.power_basis_gen hx,
norm_eq_prod_embeddings_gen E (adjoin.power_basis hx) (is_alg_closed.splits_codomain _)],
{ exact (prod_embeddings_eq_finrank_pow L E (adjoin.power_basis hx)).symm },
{ haveI := is_sepa... | lemma | algebra.norm_eq_prod_embeddings | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra_map",
"finite_dimensional",
"is_alg_closed",
"is_alg_closed.splits_codomain",
"is_separable",
"is_separable.is_integral",
"is_separable.separable",
"is_separable_tower_bot_of_is_separable",
"ring_hom.map_pow"
] | For `L/K` a finite separable extension of fields and `E` an algebraically closed extension
of `K`, the norm (down to `K`) of an element `x` of `L` is equal to the product of the images
of `x` over all the `K`-embeddings `σ` of `L` into `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eq_prod_automorphisms [finite_dimensional K L] [is_galois K L] (x : L) :
algebra_map K L (norm K x) = ∏ (σ : L ≃ₐ[K] L), σ x | begin
apply no_zero_smul_divisors.algebra_map_injective L (algebraic_closure L),
rw map_prod (algebra_map L (algebraic_closure L)),
rw ← fintype.prod_equiv (normal.alg_hom_equiv_aut K (algebraic_closure L) L),
{ rw ← norm_eq_prod_embeddings,
simp only [algebra_map_eq_smul_one, smul_one_smul] },
{ intro σ,... | lemma | algebra.norm_eq_prod_automorphisms | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"alg_equiv.coe_of_bijective",
"alg_hom.restrict_normal'",
"alg_hom.restrict_normal_commutes",
"algebra_map",
"algebraic_closure",
"equiv.coe_fn_mk",
"finite_dimensional",
"fintype.prod_equiv",
"is_galois",
"map_prod",
"no_zero_smul_divisors.algebra_map_injective",
"normal.alg_hom_equiv_aut",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_norm [algebra R L] [algebra R K] [is_scalar_tower R K L]
[is_separable K L] [finite_dimensional K L] {x : L} (hx : is_integral R x) :
is_integral R (norm K x) | begin
have hx' : is_integral K x := is_integral_of_is_scalar_tower hx,
rw [← is_integral_algebra_map_iff (algebra_map K (algebraic_closure L)).injective,
norm_eq_prod_roots],
{ refine (is_integral.multiset_prod (λ y hy, _)).pow _,
rw mem_roots_map (minpoly.ne_zero hx') at hy,
use [minpoly R x, minpo... | lemma | algebra.is_integral_norm | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"algebra",
"algebra_map",
"algebraic_closure",
"finite_dimensional",
"is_alg_closed.splits_codomain",
"is_integral",
"is_integral.multiset_prod",
"is_integral_algebra_map_iff",
"is_integral_of_is_scalar_tower",
"is_scalar_tower",
"is_separable",
"minpoly",
"minpoly.aeval_of_is_scalar_tower",... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_norm [algebra L F] [is_scalar_tower K L F] [is_separable K F] (x : F) :
norm K (norm L x) = norm K x | begin
by_cases hKF : finite_dimensional K F,
{ haveI := hKF,
let A := algebraic_closure K,
apply (algebra_map K A).injective,
haveI : finite_dimensional L F := finite_dimensional.right K L F,
haveI : finite_dimensional K L := finite_dimensional.left K L F,
haveI : is_separable K L := is_separabl... | lemma | algebra.norm_norm | ring_theory | src/ring_theory/norm.lean | [
"field_theory.primitive_element",
"linear_algebra.determinant",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.minpoly",
"linear_algebra.matrix.to_linear_equiv",
"field_theory.is_alg_closed.algebraic_closure",
"field_theory.galois"
] | [
"alg_hom_equiv_sigma",
"algebra",
"algebra_map",
"algebraic_closure",
"finite_dimensional",
"finite_dimensional.left",
"finite_dimensional.right",
"finite_dimensional.trans",
"finset.prod_sigma",
"finset.univ_sigma_univ",
"fintype",
"fintype.prod_equiv",
"is_scalar_tower",
"is_separable",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus (I : ideal (mv_polynomial σ k)) : set (σ → k) | {x : σ → k | ∀ p ∈ I, eval x p = 0} | def | mv_polynomial.zero_locus | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"mv_polynomial"
] | Set of points that are zeroes of all polynomials in an ideal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_zero_locus_iff {I : ideal (mv_polynomial σ k)} {x : σ → k} :
x ∈ zero_locus I ↔ ∀ p ∈ I, eval x p = 0 | iff.rfl | lemma | mv_polynomial.mem_zero_locus_iff | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_anti_mono {I J : ideal (mv_polynomial σ k)} (h : I ≤ J) :
zero_locus J ≤ zero_locus I | λ x hx p hp, hx p $ h hp | lemma | mv_polynomial.zero_locus_anti_mono | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_bot : zero_locus (⊥ : ideal (mv_polynomial σ k)) = ⊤ | eq_top_iff.2 (λ x hx p hp, trans (congr_arg (eval x) (mem_bot.1 hp)) (eval x).map_zero) | lemma | mv_polynomial.zero_locus_bot | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_top : zero_locus (⊤ : ideal (mv_polynomial σ k)) = ⊥ | eq_bot_iff.2 $ λ x hx, one_ne_zero ((eval x).map_one ▸ (hx 1 submodule.mem_top) : (1 : k) = 0) | lemma | mv_polynomial.zero_locus_top | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"map_one",
"mv_polynomial",
"one_ne_zero",
"submodule.mem_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal (V : set (σ → k)) : ideal (mv_polynomial σ k) | { carrier := {p | ∀ x ∈ V, eval x p = 0},
zero_mem' := λ x hx, ring_hom.map_zero _,
add_mem' := λ p q hp hq x hx, by simp only [hq x hx, hp x hx, add_zero, ring_hom.map_add],
smul_mem' := λ p q hq x hx,
by simp only [hq x hx, algebra.id.smul_eq_mul, mul_zero, ring_hom.map_mul] } | def | mv_polynomial.vanishing_ideal | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"algebra.id.smul_eq_mul",
"ideal",
"mul_zero",
"mv_polynomial",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_zero"
] | Ideal of polynomials with common zeroes at all elements of a set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_vanishing_ideal_iff {V : set (σ → k)} {p : mv_polynomial σ k} :
p ∈ vanishing_ideal V ↔ ∀ x ∈ V, eval x p = 0 | iff.rfl | lemma | mv_polynomial.mem_vanishing_ideal_iff | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_anti_mono {A B : set (σ → k)} (h : A ≤ B) :
vanishing_ideal B ≤ vanishing_ideal A | λ p hp x hx, hp x $ h hx | lemma | mv_polynomial.vanishing_ideal_anti_mono | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_empty : vanishing_ideal (∅ : set (σ → k)) = ⊤ | le_antisymm le_top (λ p hp x hx, absurd hx (set.not_mem_empty x)) | lemma | mv_polynomial.vanishing_ideal_empty | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"le_top",
"set.not_mem_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_vanishing_ideal_zero_locus (I : ideal (mv_polynomial σ k)) :
I ≤ vanishing_ideal (zero_locus I) | λ p hp x hx, hx p hp | lemma | mv_polynomial.le_vanishing_ideal_zero_locus | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_vanishing_ideal_le (V : set (σ → k)) :
V ≤ zero_locus (vanishing_ideal V) | λ V hV p hp, hp V hV | lemma | mv_polynomial.zero_locus_vanishing_ideal_le | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_vanishing_ideal_galois_connection :
@galois_connection (ideal (mv_polynomial σ k)) (set (σ → k))ᵒᵈ _ _
zero_locus vanishing_ideal | λ I V, ⟨λ h, le_trans (le_vanishing_ideal_zero_locus I) (vanishing_ideal_anti_mono h),
λ h, le_trans (zero_locus_anti_mono h) (zero_locus_vanishing_ideal_le V)⟩ | theorem | mv_polynomial.zero_locus_vanishing_ideal_galois_connection | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"galois_connection",
"ideal",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_vanishing_ideal_singleton_iff (x : σ → k) (p : mv_polynomial σ k) :
p ∈ (vanishing_ideal {x} : ideal (mv_polynomial σ k)) ↔ (eval x p = 0) | ⟨λ h, h x rfl, λ hpx y hy, hy.symm ▸ hpx⟩ | lemma | mv_polynomial.mem_vanishing_ideal_singleton_iff | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_singleton_is_maximal {x : σ → k} :
(vanishing_ideal {x} : ideal (mv_polynomial σ k)).is_maximal | begin
have : mv_polynomial σ k ⧸ vanishing_ideal {x} ≃+* k := ring_equiv.of_bijective
(ideal.quotient.lift _ (eval x) (λ p h, (mem_vanishing_ideal_singleton_iff x p).mp h))
begin
refine ⟨(injective_iff_map_eq_zero _).mpr (λ p hp, _), λ z,
⟨(ideal.quotient.mk (vanishing_ideal {x} : ideal (mv_poly... | instance | mv_polynomial.vanishing_ideal_singleton_is_maximal | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"ideal.quotient.lift",
"ideal.quotient.lift_mk",
"ideal.quotient.mk",
"mv_polynomial",
"ring_equiv.of_bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_le_vanishing_ideal_zero_locus (I : ideal (mv_polynomial σ k)) :
I.radical ≤ vanishing_ideal (zero_locus I) | begin
intros p hp x hx,
rw ← mem_vanishing_ideal_singleton_iff,
rw radical_eq_Inf at hp,
refine (mem_Inf.mp hp) ⟨le_trans (le_vanishing_ideal_zero_locus I)
(vanishing_ideal_anti_mono (λ y hy, hy.symm ▸ hx)), is_maximal.is_prime' _⟩,
end | lemma | mv_polynomial.radical_le_vanishing_ideal_zero_locus | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point_to_point (x : σ → k) : prime_spectrum (mv_polynomial σ k) | ⟨(vanishing_ideal {x} : ideal (mv_polynomial σ k)), by apply_instance⟩ | def | mv_polynomial.point_to_point | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"mv_polynomial",
"prime_spectrum"
] | The point in the prime spectrum assosiated to a given point | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vanishing_ideal_point_to_point (V : set (σ → k)) :
prime_spectrum.vanishing_ideal (point_to_point '' V) = mv_polynomial.vanishing_ideal V | le_antisymm
(λ p hp x hx, (((prime_spectrum.mem_vanishing_ideal _ _).1 hp)
⟨vanishing_ideal {x}, by apply_instance⟩ ⟨x, ⟨hx, rfl⟩⟩) x rfl)
(λ p hp, (prime_spectrum.mem_vanishing_ideal _ _).2
(λ I hI, let ⟨x, hx⟩ := hI in hx.2 ▸ λ x' hx', (set.mem_singleton_iff.1 hx').symm ▸ hp x hx.1)) | lemma | mv_polynomial.vanishing_ideal_point_to_point | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"mv_polynomial.vanishing_ideal",
"prime_spectrum.mem_vanishing_ideal",
"prime_spectrum.vanishing_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point_to_point_zero_locus_le (I : ideal (mv_polynomial σ k)) :
point_to_point '' (mv_polynomial.zero_locus I) ≤ prime_spectrum.zero_locus ↑I | λ J hJ, let ⟨x, hx⟩ := hJ in (le_trans (le_vanishing_ideal_zero_locus I)
(hx.2 ▸ vanishing_ideal_anti_mono (set.singleton_subset_iff.2 hx.1)) : I ≤ J.as_ideal) | lemma | mv_polynomial.point_to_point_zero_locus_le | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"mv_polynomial",
"mv_polynomial.zero_locus",
"prime_spectrum.zero_locus"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal_iff_eq_vanishing_ideal_singleton (I : ideal (mv_polynomial σ k)) :
I.is_maximal ↔ ∃ (x : σ → k), I = vanishing_ideal {x} | begin
casesI nonempty_fintype σ,
refine ⟨λ hI, _, λ h, let ⟨x, hx⟩ := h in
hx.symm ▸ (mv_polynomial.vanishing_ideal_singleton_is_maximal)⟩,
letI : I.is_maximal := hI,
letI : field (mv_polynomial σ k ⧸ I) := quotient.field I,
let ϕ : k →+* mv_polynomial σ k ⧸ I := (ideal.quotient.mk I).comp C,
have hϕ : ... | lemma | mv_polynomial.is_maximal_iff_eq_vanishing_ideal_singleton | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"field",
"ideal",
"ideal.quotient.mk",
"is_alg_closed.algebra_map_surjective_of_is_integral'",
"mv_polynomial",
"mv_polynomial.vanishing_ideal_singleton_is_maximal",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_zero_locus_eq_radical (I : ideal (mv_polynomial σ k)) :
vanishing_ideal (zero_locus I) = I.radical | begin
rw I.radical_eq_jacobson,
refine le_antisymm (le_Inf _) (λ p hp x hx, _),
{ rintros J ⟨hJI, hJ⟩,
obtain ⟨x, hx⟩ := (is_maximal_iff_eq_vanishing_ideal_singleton J).1 hJ,
refine hx.symm ▸ vanishing_ideal_anti_mono (λ y hy p hp, _),
rw [← mem_vanishing_ideal_singleton_iff, set.mem_singleton_iff.1 h... | theorem | mv_polynomial.vanishing_ideal_zero_locus_eq_radical | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"le_Inf",
"mv_polynomial"
] | Main statement of the Nullstellensatz | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_prime.vanishing_ideal_zero_locus (P : ideal (mv_polynomial σ k)) [h : P.is_prime] :
vanishing_ideal (zero_locus P) = P | trans (vanishing_ideal_zero_locus_eq_radical P) h.radical | lemma | mv_polynomial.is_prime.vanishing_ideal_zero_locus | ring_theory | src/ring_theory/nullstellensatz.lean | [
"ring_theory.jacobson",
"field_theory.is_alg_closed.basic",
"field_theory.mv_polynomial",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid.perfection (M : Type u₁) [comm_monoid M] (p : ℕ) : submonoid (ℕ → M) | { carrier := { f | ∀ n, f (n + 1) ^ p = f n },
one_mem' := λ n, one_pow _,
mul_mem' := λ f g hf hg n, (mul_pow _ _ _).trans $ congr_arg2 _ (hf n) (hg n) } | def | monoid.perfection | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"comm_monoid",
"congr_arg2",
"mul_pow",
"one_pow",
"submonoid"
] | The perfection of a monoid `M`, defined to be the projective limit of `M`
using the `p`-th power maps `M → M` indexed by the natural numbers, implemented as
`{ f : ℕ → M | ∀ n, f (n + 1) ^ p = f n }`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring.perfection_subsemiring (R : Type u₁) [comm_semiring R]
(p : ℕ) [hp : fact p.prime] [char_p R p] :
subsemiring (ℕ → R) | { zero_mem' := λ n, zero_pow $ hp.1.pos,
add_mem' := λ f g hf hg n, (frobenius_add R p _ _).trans $ congr_arg2 _ (hf n) (hg n),
.. monoid.perfection R p } | def | ring.perfection_subsemiring | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"char_p",
"comm_semiring",
"congr_arg2",
"fact",
"frobenius_add",
"monoid.perfection",
"subsemiring",
"zero_pow"
] | The perfection of a ring `R` with characteristic `p`, as a subsemiring,
defined to be the projective limit of `R` using the Frobenius maps `R → R`
indexed by the natural numbers, implemented as `{ f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring.perfection_subring (R : Type u₁) [comm_ring R]
(p : ℕ) [hp : fact p.prime] [char_p R p] :
subring (ℕ → R) | (ring.perfection_subsemiring R p).to_subring $ λ n, by simp_rw [← frobenius_def, pi.neg_apply,
pi.one_apply, ring_hom.map_neg, ring_hom.map_one] | def | ring.perfection_subring | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"char_p",
"comm_ring",
"fact",
"frobenius_def",
"pi.one_apply",
"ring.perfection_subsemiring",
"ring_hom.map_neg",
"ring_hom.map_one",
"subring"
] | The perfection of a ring `R` with characteristic `p`, as a subring,
defined to be the projective limit of `R` using the Frobenius maps `R → R`
indexed by the natural numbers, implemented as `{ f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring.perfection (R : Type u₁) [comm_semiring R] (p : ℕ) : Type u₁ | {f // ∀ (n : ℕ), (f : ℕ → R) (n + 1) ^ p = f n} | def | ring.perfection | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"comm_semiring"
] | The perfection of a ring `R` with characteristic `p`,
defined to be the projective limit of `R` using the Frobenius maps `R → R`
indexed by the natural numbers, implemented as `{f : ℕ → R // ∀ n, f (n + 1) ^ p = f n}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring (R : Type u₁) [comm_ring R] [char_p R p] : ring (ring.perfection R p) | (ring.perfection_subring R p).to_ring | instance | perfection.ring | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"char_p",
"comm_ring",
"ring",
"ring.perfection",
"ring.perfection_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_ring (R : Type u₁) [comm_ring R] [char_p R p] : comm_ring (ring.perfection R p) | (ring.perfection_subring R p).to_comm_ring | instance | perfection.comm_ring | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"char_p",
"comm_ring",
"ring.perfection",
"ring.perfection_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff (n : ℕ) : ring.perfection R p →+* R | { to_fun := λ f, f.1 n,
map_one' := rfl,
map_mul' := λ f g, rfl,
map_zero' := rfl,
map_add' := λ f g, rfl } | def | perfection.coeff | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ring.perfection"
] | The `n`-th coefficient of an element of the perfection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {f g : ring.perfection R p} (h : ∀ n, coeff R p n f = coeff R p n g) : f = g | subtype.eq $ funext h | lemma | perfection.ext | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ring.perfection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pth_root : ring.perfection R p →+* ring.perfection R p | { to_fun := λ f, ⟨λ n, coeff R p (n + 1) f, λ n, f.2 _⟩,
map_one' := rfl,
map_mul' := λ f g, rfl,
map_zero' := rfl,
map_add' := λ f g, rfl } | def | perfection.pth_root | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"pth_root",
"ring.perfection"
] | The `p`-th root of an element of the perfection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_mk (f : ℕ → R) (hf) (n : ℕ) : coeff R p n ⟨f, hf⟩ = f n | rfl | lemma | perfection.coeff_mk | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_pth_root (f : ring.perfection R p) (n : ℕ) :
coeff R p n (pth_root R p f) = coeff R p (n + 1) f | rfl | lemma | perfection.coeff_pth_root | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"pth_root",
"ring.perfection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_pow_p (f : ring.perfection R p) (n : ℕ) :
coeff R p (n + 1) (f ^ p) = coeff R p n f | by { rw ring_hom.map_pow, exact f.2 n } | lemma | perfection.coeff_pow_p | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ring.perfection",
"ring_hom.map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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