chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
23,955 | lean_file_chunk | 22e343d8728d3b9ccdf7b0eab5fb806f86bbd00ddc2d5631f7345fad23cc6a01 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.Algebra.Ring.CharZero
public import Mathlib.Algebra.Ring.Int.Units
public import Mathlib.GroupTheory.Coprod.Basic
public import Mathlib.Grou... |
5,329 | lean_file_chunk | 121f09a8ca9b25de6d952fdc113cf21c0a80c36939afcdc609a5060efa6dba7a | lean_source_file | pkg_mathlib_source_file_chunks | rw [dif_neg (not_cancels_of_cons_hyp u w h2)]
-- Before https://github.com/leanprover/lean4/pull/2644, this was just
-- simp [unitsSMulGroup, (d.compl _).equiv_fst_eq_one_of_mem_of_one_mem (one_mem _) h1,
-- -SetLike.coe_sort_coe]
-- ext <;> simp [-SetLike.coe_sort_coe]
simp only [unitsSMulGroup, ... |
11,191 | lean_file_chunk | 4b62ca0a3084065880f995980966e5635a6ff1083b16448600eb02397ed26a30 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.Algebra.Group.Pointwise.Finset.Basic
public import Mathlib.GroupTheory.Abelianization.Defs
public import Mathlib.GroupTheory.Commutato... |
6,292 | lean_file_chunk | c575275b1e5dc0fdc40a1e115925d74f510224932013b5d7b8686e0120c8584f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
module
public import Mathlib.Algebra.Group.Commute.Hom
public import Mathlib.Algebra.Group.Prod
public import Mathlib.Algebra.Group.Subgroup.Ker
public... |
9,070 | lean_file_chunk | fbf00a851ad99494c78d42846ba452d692d4db3b4d29299df8cb0d6b08959367 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Francisco Silva. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Francisco Silva
-/
module
public import Mathlib.Algebra.Group.PUnit
public import Mathlib.Data.Finite.Perm
public import Mathlib.Data.Nat.Multiplicity
public import Mathlib.GroupTheo... |
23,989 | lean_file_chunk | 52f0303fa295550083d3c709b3a328fc4320dd079bf704ed9875d9870ca94dc3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
module
public import Mathlib.Algebra.GCDMonoid.Finset
public import Mathlib.Algebra.GCDMonoid.Nat
public import Mathlib.Algebra.Order.BigOperators.GroupWithZ... |
3,200 | lean_file_chunk | 08e2837c50a21f3ec7f4d70e377fb74d43a5fd4b34edf5dd4ed59ced39848c42 | lean_source_file | pkg_mathlib_source_file_chunks | · exact MonoidHom.exponent_dvd (f := MonoidHom.fst M₁ M₂) Prod.fst_surjective
· exact MonoidHom.exponent_dvd (f := MonoidHom.snd M₁ M₂) Prod.snd_surjective
end PiProd
/-! ### Properties of monoids with exponent two -/
section ExponentTwo
section Monoid
variable [Monoid G]
@[to_additive]
lemma orderOf_eq_two_iff... |
4,967 | lean_file_chunk | 207ba21f7a756aadae8c9268e43dddceee67f8a922bc2f0bd1956e27286cab63 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
module
public import Mathlib.GroupTheory.Index
/-!
# Commensurability for subgroups
Two subgroups `H` and `K` of a group `G` are commensurable if `H ∩ K` has finit... |
15,268 | lean_file_chunk | 63d3a2c946e6d176fc9bd0246098656fcd1e4c8dd9a40a4beff62ae053a1a1ca | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.GroupTheory.Complement
public import Mathlib.GroupTheory.Sylow
public import Mathlib.Data.ZMod.QuotientGroup
/-!
# The Transfer Homom... |
4,443 | lean_file_chunk | bd03fd8bc218d7845fde1b46e3481cb0796469e580fc7677d7015a3cf5be8862 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
module
public import Mathlib.GroupTheory.Index
public import Mathlib.LinearAlgebra.Dimension.Finrank
public import Mathlib.LinearAlgebra.FreeModule.Basic
public import... |
8,032 | lean_file_chunk | f036a5229b8fd89792d997d5dee4eb98f493d579716db6e6f4b5b2f03ca68b0c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.Group.Graph
public import Mathlib.Algebra.Group.Subgroup.Basic
public import Mathlib.GroupTheory.QuotientGroup.Defs
/-!
# Goursat's... |
2,898 | lean_file_chunk | bc58dd97b8aa1586490c1c2f73ec3786523714ecfc1e4b0e8252e6deafb48b98 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Colva Roney-Dougal. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Colva Roney-Dougal, Inna Capdeboscq, Susanna Fishel, Kim Morrison
-/
module
public import Mathlib.GroupTheory.Nilpotent
public import Mathlib.Order.Radical
/-!
# The Frattini sub... |
3,600 | lean_file_chunk | ef20764cea7be19d5fefa6a88ab811075ebcd88a27607be557b37b79063e5e94 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Inna Capdeboscq, Damiano Testa
-/
module
public import Mathlib.GroupTheory.Nilpotent
/-!
# Perfect groups
A group `G` is perfect if it equals its commutator subgroup, that is `⁅G, G⁆... |
2,754 | lean_file_chunk | 4c8a0e5a68e295be4ba9ccab43c7e065459af299f05f70ffa8536566afa478bc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
module
public import Mathlib.Data.Finite.Perm
public import Mathlib.Data.Nat.Prime.Factorial
public import Mathlib.GroupTheory.Index
/-! # Subgroups o... |
23,981 | lean_file_chunk | 3f9868cf41495888e3f41b8000c3f5aa381cde580b5bade46f7b4e9213c8acca | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.GroupTheory.CoprodI
public import Mathlib.GroupTheory.Coprod.Basic
public import Mathlib.GroupTheory.Complement
/-!
## Pushouts of Monoids... |
3,804 | lean_file_chunk | b9c7d6880d19ea8dff0a23cb32639a2787fe42698955a8330770f444d95dd796 | lean_source_file | pkg_mathlib_source_file_chunks | theorem base_injective (hφ : ∀ i, Function.Injective (φ i)) :
Function.Injective (base φ) := by
rcases transversal_nonempty φ hφ with ⟨d⟩
let _ := Classical.decEq ι
let _ := fun i => Classical.decEq (G i)
refine Function.Injective.of_comp
(f := ((· • ·) : PushoutI φ → NormalWord d → NormalWord d)) ?_
... |
5,258 | lean_file_chunk | 334c3c3445b1f281c992332938a948bf194a23bc5ff226d1adaf9f4413bcb718 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
module
public import Mathlib.GroupTheory.Index
/-!
# Finite-index normal subgroups
This file builds the lattice `FiniteIndexNormalSubgroup G` of finite-index normal subgro... |
23,997 | lean_file_chunk | 49f18d4768d93432f7f4388d5e40ba2192de124f8943a2f2cc88b580d89cb400 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module
public import Mathlib.Algebra.Group.Pointwise.Set.Finite
public import Mathlib.Algebra.Group.Subgroup.Pointwise
public import Mathlib.Algebra.Group.Subgroup... |
1,604 | lean_file_chunk | 5c0abde16f9a4f35c047464462a97b7930546c3069cb27c103f0389543a3839c | lean_source_file | pkg_mathlib_source_file_chunks | by_cases hx₂ : x = 1
· simp [hx₂]
refine hpwo.wellFoundedOn.induction ⟨hx₁, hx₂⟩ fun y ⟨hy₁, hy₂⟩ ih => ?_
simp only [Set.mem_setOf_eq, and_imp] at ih
by_cases hy₃ : Minimal (· ∈ { x | x ∈ P ∧ x ≠ 1 }) y
· exact mem_closure_of_mem hy₃
rcases exists_lt_of_not_minimal ⟨hy₁, hy₂⟩ hy₃ with ⟨z, hz₁, ... |
10,418 | lean_file_chunk | 724c570235a8c93c2196c28e7b722508304721bddc8a6ac2d89528c9a4786e50 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.GroupTheory.Complement
/-!
# Semidirect product
This file defines semidirect products of groups, and the canonical maps in and out of the
... |
23,995 | lean_file_chunk | cab6afb3ada0dc515c56c5ac9bf81985d92dc49894aac5749c775c396d85830d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
module
public import Mathlib.Algebra.CharP.Two
public import Mathlib.Algebra.Group.Commute.Basic
public import Mathlib.Algebra.Group.Pointwise.Se... |
23,986 | lean_file_chunk | b58865d8f32135bcddcbb964f1390f075be00b81611d7c6112b3f199a6dac692 | lean_source_file | pkg_mathlib_source_file_chunks | `AddSubmonoid.multiples a`, sending `i` to `i • a` -/]
noncomputable def finEquivPowers {x : G} (hx : IsOfFinOrder x) : Fin (orderOf x) ≃ powers x :=
Equiv.ofBijective (fun n ↦ ⟨x ^ (n : ℕ), ⟨n, rfl⟩⟩) ⟨fun ⟨_, h₁⟩ ⟨_, h₂⟩ ij ↦
Fin.ext (pow_injOn_Iio_orderOf h₁ h₂ (Subtype.mk_eq_mk.1 ij)), fun ⟨_, i, rfl⟩ ↦
... |
12,972 | lean_file_chunk | 3534ed6fee75b51d50b5f04a3919b2c90e668221be3e951ed9c37e8879074213 | lean_source_file | pkg_mathlib_source_file_chunks | @[to_additive]
theorem card_zpowers_le (a : G) {k : ℕ} (k_pos : k ≠ 0)
(ha : a ^ k = 1) : Fintype.card (Subgroup.zpowers a) ≤ k := by
rw [Fintype.card_zpowers]
apply orderOf_le_of_pow_eq_one k_pos.bot_lt ha
open QuotientGroup
@[to_additive]
theorem orderOf_dvd_card : orderOf x ∣ Fintype.card G := by
use Fin... |
5,713 | lean_file_chunk | fc9c8643d1172541a2ec62db8c13778f975ce332a32a65c49cbe6be1eaea375b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Algebra.Homology.HomologicalComplex
public import Mathlib.AlgebraicTopology.SimplicialObject.Basic
public import Mathlib.CategoryTheory.Abel... |
1,922 | lean_file_chunk | eb1cccd04192ba99a06f3936177cb758c74e8701b73595d066f6000660c23184 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Adam Topaz, Joël Riou
-/
module
public import Mathlib.AlgebraicTopology.SimplicialObject.Basic
public import Mathlib.Analysis.Convex.StdSimplex
public import Mathlib.Topol... |
12,331 | lean_file_chunk | 268f42d038785413eaea232e4cdd27c2e5623e23b273ad94120c6867387718fb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Adam Topaz, Johan Commelin
-/
module
public import Mathlib.Algebra.Homology.Additive
public import Mathlib.AlgebraicTopology.MooreComplex
public import Mathlib.Algebra.BigOperat... |
12,668 | lean_file_chunk | 18785dc982ea76e72eb33832e59020973c6088f6b20039f11e61574b711aa642 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex
public import Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex
public import Mathlib.AlgebraicTopolo... |
7,625 | lean_file_chunk | 7f1520c3916a7440f64939e6b95309c73c57715d4aab81993d7fe7c2bdd99e47 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.AlgebraicTopology.SimplicialCategory.Basic
public import Mathlib.AlgebraicTopology.SimplicialSet.Nerve
/-!
# The simplicial nerve o... |
15,244 | lean_file_chunk | 7a38b8562ac21efe4c8641db9b6016d90c8b7f3d9dbc34d9fb0132d7a65694b2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
module
public import Mathlib.AlgebraicTopology.SimplicialObject.Basic
public import Mathlib.CategoryTheory.Comma.Arrow
public import Mathlib.CategoryTheory.Limits.Shapes.Wid... |
5,033 | lean_file_chunk | 702565b79b72206d8156b9f20c7d646b8ac1dd97b1f61a1223fbc7bcbf95c316 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison, Adam Topaz
-/
module
public import Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex
public import Mathlib.AlgebraicTopology.TopologicalSimplex
public i... |
2,775 | lean_file_chunk | 1d3901f18f977d957da67b35ca49ae97f66dae8442314818116a63c025638b0d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.Algebra.Category.ModuleCat.Abelian
public import Mathlib.Algebra.Category.ModuleCat.Colimits
public import Mathlib.Algebra.Category.... |
1,113 | lean_file_chunk | a8e5bcb43104daec84024f4c8947329cc11fb16e07d60f27aaee3b54972e6658 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jonas van der Schaaf. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jonas van der Schaaf, Dagur Asgeirsson
-/
module
public import Mathlib.CategoryTheory.Sites.RegularEpi
public import Mathlib.Condensed.Epi
public import Mathlib.Condensed.Functo... |
2,834 | lean_file_chunk | 0eaacd691788049483b5ea929580c80f9905ac2cc74b004cca7eda8a6441f8aa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.CategoryTheory.Sites.PreservesLimits
public import Mathlib.Condensed.Explicit
/-!
# Functors from categories of topological spaces ... |
2,441 | lean_file_chunk | 0a2e6bd775e45294f0d233780617813016cc06908cdbab68b3deb997ece000db | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
module
public import Mathlib.CategoryTheory.Sites.Sheaf
public import Mathlib.Topology.Category.CompHaus.EffectiveEpi
/-!
# Condensed Objects
This file defines the catego... |
6,146 | lean_file_chunk | b792a996dc5b6e57b4e84582f9413c7672505eb256538b4e8b5453dcaa2740d0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.CategoryTheory.Limits.Preserves.Opposites
public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
public import Mathlib.... |
475 | lean_file_chunk | 5d75d93d78fee9c436e7aa6f2493142f35de38208f99309403a77d988f5f4ae3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.CategoryTheory.Monoidal.Closed.Types
public import Mathlib.CategoryTheory.Sites.CartesianClosed
public import Mathlib.Condensed.Basi... |
3,399 | lean_file_chunk | f9aea435c8dbaf24bd643069a59e19cbe8067549138bf03162b46efbc1d869ef | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
public import Mathlib.Condensed.Functors
public import Mathlib.Condensed.Limits
/-!
... |
8,441 | lean_file_chunk | 6d035fb429cf6a0e227206378bcabd933de3587fb5d0d0722d90f0e4ebd18e91 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Riccardo Brasca, Filippo A. E. Nuccio
-/
module
public import Mathlib.Condensed.Module
public import Mathlib.Condensed.Equivalence
/-!
# The explicit sheaf condit... |
8,766 | lean_file_chunk | 8b9ab94c889aa327246a7b035c31077a0e8c9ed7d55e82b336d60c60c51d974c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.Condensed.TopComparison
public import Mathlib.Topology.Category.CompactlyGenerated
/-!
# The adjunction between condensed sets and ... |
4,003 | lean_file_chunk | 46937627bb42e2e23bd436221d69c43ffd350c7a1591bb725abda9bad6ee92ca | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.CategoryTheory.ConcreteCategory.EpiMono
public import Mathlib.CategoryTheory.Sites.Coherent.LocallySurjective
public import Mathlib.... |
2,612 | lean_file_chunk | f3a66f517ea29e50a25ee6a97005ae15eee6ce2da785da0ac06e1b828fe7d63a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.Algebra.Category.ModuleCat.AB
public import Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Sheaf
public import Mathlib.CategoryTh... |
1,687 | lean_file_chunk | ac56cb7af0eba170e0782a03ef67485d8fe05cb2c5a88b6cc937dfc005375c8d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.Condensed.Module
/-!
# Limits in categories of condensed objects
This file adds some instances for limits in condensed sets and c... |
4,457 | lean_file_chunk | 41207b130002bdd56768bfcc9a16a555aa822ba20eadf483eafd7f94b1bee5ac | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Nikolas Kuhn, Dagur Asgeirsson
-/
module
public import Mathlib.Topology.Category.Profinite.EffectiveEpi
public import Mathlib.Topology.Category.Stonean.EffectiveEpi
public imp... |
2,571 | lean_file_chunk | 1c6cb8e572fdcb2a868a8a6ce5aaba117d85ba6dbab382ca671f9de3e5cb3dc6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.RightExactness
public import Mathlib.RingTheory.FinitePresentation
public import Mathlib.RingTheory.Tens... |
5,618 | lean_file_chunk | 234e258bef83004fa0342ea29adb3f838de2eccd8d6e314bde63cb8c0efd2650 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.RingTheory.Artinian.Module
/-!
# Modules of finite length
We define modules of finite length (`IsFiniteLength`) to be finite iterated extensions... |
2,515 | lean_file_chunk | fca0771ad2c221cda6e6786a341d868c1680a845ff9e2ec3a0b79271cfb9f43c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Data.ZMod.Basic
public import Mathlib.Tactic.NormNum
/-!
# Some facts about finite rings
-/
public section
open Finset ZMod
section Rin... |
3,372 | lean_file_chunk | ad3b71bb76f563f41823760f4d308d98b237ff925c4fd359aedc8fd441be7645 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kenny Lau
-/
module
public import Mathlib.Algebra.Ring.Equiv
public import Mathlib.Algebra.Ring.Opposite
/-!
# Ring involutions
This file defines a ring involuti... |
13,779 | lean_file_chunk | 04a74eda888af996dec9ebdb66aa3fcc14e88b12a43564375bc39fd0d83114cc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Algebra.Exact
public import Mathlib.LinearAlgebra.Basis.VectorSpace
public import Mathlib.Order.KrullDimension
public import Mathlib.RingTheor... |
10,817 | lean_file_chunk | 1007e68c74168467227f94a0dbfeebadbc0bce485add58586b30cca01c9bc523 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Localization.AtPrime.Basic
public import Mathlib.RingTheory.TensorProduct.Maps
/-!
# Ring Homomorphisms surjective on stalks
In t... |
23,992 | lean_file_chunk | 6675afbfb90e2192a78c6cbe85465a6d392131b05723f6a77a2ffdcbda3bd833 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.TensorProduct.Maps
/-!
# The characteristic predicate of tensor product
## Main definitions
- `IsTensorProduct`: A predicate on ... |
9,088 | lean_file_chunk | 2068bb8396157f5c9928ec08d5a80f07aebee985eea87a95d6ceaeb0751d9995 | lean_source_file | pkg_mathlib_source_file_chunks | variable (R S R' S')
theorem Algebra.IsPushout.comm : Algebra.IsPushout R S R' S' ↔ Algebra.IsPushout R R' S S' :=
⟨Algebra.IsPushout.symm, Algebra.IsPushout.symm⟩
instance : Algebra.IsPushout R R S S where
out := .of_equiv (TensorProduct.lid R S) fun _ ↦ by simp
instance : Algebra.IsPushout R S R S := .symm inf... |
11,067 | lean_file_chunk | cb89b607197302fd709f497f12a3e3d8cfb7282005d7438920bcd75f23afc5d5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
module
public import Mathlib.LinearAlgebra.PiTensorProduct
public import Mathlib.Algebra.Algebra.Bilinear
public import Mathlib.Algebra.Algebra.Equiv
public import Mathl... |
23,984 | lean_file_chunk | bc3f844892837f50fa793712f204dcadcddc0cb4d0fa7b6d08003af81eaa58a0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.Algebra.CharP.Frobenius
public import Mathlib.Algebra.CharP.Pi
public import Mathlib.Algebra.CharP.Quotient
public import Mathlib.Algebra.CharP.Su... |
9,476 | lean_file_chunk | 748ecde30f5c261104c405a86c38002908443eb4a1a0a373059100687aea62ec | lean_source_file | pkg_mathlib_source_file_chunks | rw [← v_p_lt_preVal (hv := hv), hvx] at hx0
exact not_lt_zero' hx0,
fun hx => hx.symm ▸ preVal_zero⟩
theorem v_p_lt_val {x : O} :
v p < v (algebraMap O K x) ↔ (Ideal.Quotient.mk _ x : ModP O p) ≠ 0 := by
rw [lt_iff_not_ge, not_iff_not, ← map_natCast (algebraMap O K) p, hv.le_iff_dvd,
Ideal.Quotient... |
9,346 | lean_file_chunk | 01b7d265defd0131add4483c69fab9c9dc4c8a3eda78905a70088b49c5309937 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.Algebra.Module.Torsion.Basic
public import Mathlib.RingTheory.FiniteLength
public import Mathlib.RingTheory.Noetherian.Nilpotent
public import Mat... |
12,155 | lean_file_chunk | 449c2672992567ac74a71b2b45feb3d90fdf231327218a1a6eb9f6b3e7e98916 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Algebra.Module.LocalizedModule.Submodule
public import Mathlib.RingTheory.Ideal.Colon
public import Mathlib.RingTheory.Localization.Finiteness... |
12,148 | lean_file_chunk | e813b90168357a2c93c8310c839da43220d7b1eaeef907ae40aad48230d7d7b0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
module
public import Mathlib.Algebra.Polynomial.Taylor
public import Mathlib.RingTheory.LocalRing.ResidueField.Basic
public import Mathlib.RingTheory.AdicCompletion.... |
14,885 | lean_file_chunk | d4a9111b30792ffe87ad8dce7d9c993c348d1fd4eb7e52a9ca056fd210678c1c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sihan Su. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Sihan Su, Wan Lin, Xiaoyang Su
-/
module
public import Mathlib.Algebra.MvPolynomial.Monad
public import Mathlib.Data.List.Indexes
public import Mathlib.RingTheory.IntegralC... |
8,816 | lean_file_chunk | 21da9664e1a1f86195c54265283b6a3c8dd44dd9abb83d2be39cae0fb34b8896 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes
-/
module
public import Mathlib.Algebra.Polynomial.Roots
public import Mathlib.Algebra.Ring.GeomSum
public import Mathlib.Data.Fintype.Inv
public import M... |
23,992 | lean_file_chunk | 6a9e2852445b99ccc70ed2dd7c9d255ad233fe864d571761b4dd0a4ac6b932ea | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Algebraic.StronglyTranscendental
public import Mathlib.RingTheory.Conductor
public import Mathlib.RingTheory.Ideal.Quotient.Nilpote... |
17,238 | lean_file_chunk | bbd80305884e20d28694b71613bf65301684e1b036be97ec145bba8221dc0f61 | lean_source_file | pkg_mathlib_source_file_chunks | variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
-- Subsumed by `ZariskisMainProperty.of_finiteType`.
private lemma ZariskisMainProperty.of_adjoin_eq_top
(p : Ideal S) [p.IsPrime] [Algebra.WeaklyQuasiFiniteAt R p]
(x : S) (hx : Algebra.adjoin R {x} = ⊤) : ZariskisMainProperty R p := by
wlog H ... |
6,837 | lean_file_chunk | 5880c837236f70a11b9c2ca646b277598ef91d7b0569ea7fa55577ddb1781b69 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Data.Matrix.Basis
public import Mathlib.Data.Matrix.Composition
public import Mathlib.RingTheory.MatrixAlgebra
public import Mathlib.RingThe... |
23,981 | lean_file_chunk | 4fbd3086b1399c0970d5594581ed16a2dc8abc8a5b94e0d93a379418273d9267 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric
public import Mathlib.CategoryTheory.Monoidal.Skeleton
public import Mathlib.LinearAlgebra.Contracti... |
17,654 | lean_file_chunk | baed7fab6efd442ce3cf2d71a6b2f7d9f20d250fd9148885fa07235a6163a9e1 | lean_source_file | pkg_mathlib_source_file_chunks | simp_rw [mk_tensor, mk_eq_self]
variable {R A B} [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem mapAlgebra_mapAlgebra {M : Pic R} : mapAlgebra A B (mapAlgebra R A M) = mapAlgebra R B M :=
mk_eq_mk_iff.mpr ⟨AlgebraTensorModule.congr (.refl ..) (mk.linearEquiv ..) ≪≫ₗ
AlgebraTensorModule.cancelBaseChan... |
10,360 | lean_file_chunk | 1481d0cf1d4a9d56bf11eb253102234c3fb7e8e109b9e3b6f7e63e0e007a600f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.RingTheory.Finiteness.Basic
public import Mathlib.RingTheory.Finiteness.Nakayama
public import Mathlib.RingTheory.Jacobson.Ideal
/-!
# Naka... |
23,978 | lean_file_chunk | 01a17aae13df3fc54283e5bbc35d7053bf5deaf229eec72e03549646a666f5b9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes, Daniel Weber
-/
module
public import Mathlib.Algebra.GroupWithZero.Associated
public import Mathlib.Algebra.Ring.Divisibility.Basic
public import Mathl... |
7,756 | lean_file_chunk | e4e9085268a5e2dad908d37010105b13eebb6176346e2c119d4205e0b46bc723 | lean_source_file | pkg_mathlib_source_file_chunks | multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | _ | hab)
· rw [add_comm, ha.multiplicity_add_of_gt hab, min_eq_left]
exact le_of_lt hab
· contradiction
· rw [hb.multiplicity_add_of_gt hab, min_eq_right]
exact... |
7,530 | lean_file_chunk | b508a882a8abc5a937664c5732c79a7954b1cc2a111b2fba958bfe8650aa0d0f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.Algebra.Algebra.Tower
public import Mathlib.LinearAlgebra.Basis.Basic
/-!
# Towers of algebras
We set up the basic theory of algebra towers.
An ... |
23,950 | lean_file_chunk | 1b0ec587d77bfe495860dc6839184cedc255343a96d256803b837eb326fd8b8a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.Polynomial.FieldDivision
public import Mathlib.FieldTheory.Minpoly.Basi... |
20,412 | lean_file_chunk | b180fd7a6e62b1420076886572d7fffa55704c5fc4183ef362ed37a79e24c7f0 | lean_source_file | pkg_mathlib_source_file_chunks | (powerBasisAux' hg).repr f i = (modByMonicHom hg f).coeff ↑i :=
rfl
/-- The power basis `1, root g, ..., root g ^ (d - 1)` for `AdjoinRoot g`,
where `g` is a monic polynomial of degree `d`. -/
@[simps]
def powerBasis' (hg : g.Monic) : PowerBasis R (AdjoinRoot g) where
gen := root g
dim := g.natDegree
basis := ... |
7,927 | lean_file_chunk | 26a824dc96bf7fe46b62c5b289de41d475bdec02d6009ce8b847eb0cd38561b4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Rodriguez
-/
module
public import Mathlib.Algebra.GroupWithZero.Action.Center
public import Mathlib.GroupTheory.ClassEquation
public import Mathlib.RingTheory.Pol... |
4,947 | lean_file_chunk | dc8a2c6d22e81fbe6454ab4a8b6125281dcf26f302ac17f31c66ba2e0d37ea44 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
module
public import Mathlib.LinearAlgebra.Quotient.Basic
public import Mathlib.RingTheory.Ideal.Colon
public import Mathlib.RingTheory.Ideal.Operations
/-!
# Pri... |
17,498 | lean_file_chunk | ba975d66246b9a00131d982e483c15d09312636f6e542f367c55acfa91a9b9ef | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Algebra.Polynomial.Module.Basic
public import Mathlib.RingTheory.Finiteness.Nakayama
public import Mathlib.RingTheory.LocalRing.MaximalIdeal.B... |
10,139 | lean_file_chunk | af819f9e0baf6d3b925b62209f5581483d7292fc874e23a344de5e9326c78a1e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Xavier Roblot, Andrew Yang
-/
module
public import Mathlib.RingTheory.Localization.AtPrime.Basic
public import Mathlib.RingTheory.Localization.Submodule
public import Mathli... |
14,145 | lean_file_chunk | 40ea3fe2d29753e2a37a228bd81f215f28ff28af2ab3627350781412ef1bfd5f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module
public import Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite
public import Mathlib.RingTheory.Norm.Transitivity
public import Mathlib.RingTheory.Tra... |
19,375 | lean_file_chunk | 247846bf2f35f31273533db4ecd046e1985ec075425301d18f4ba5e1f64f77eb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.RingTheory.DedekindDomain.Ideal.Basic
/-!
# The ideal class group
This file defines the ideal class group `ClassGroup R` of fractional ideal... |
3,596 | lean_file_chunk | 4a7dec54b843beb73dad38b37321ce58ea47530ea3a9b4deb738b21a0173a7a6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
module
public import Mathlib.Algebra.GCDMonoid.Basic
public import Mathlib.Algebra.EuclideanDomain.Basic
public import Mathlib.RingTheory.Ideal.Basic
p... |
21,310 | lean_file_chunk | e675abb2b96bef7078db655769c41485df7a36a6366f0fc810f209b301c80550 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
module
public import Mathlib.Algebra.EuclideanDomain.Basic
public import Mathlib.Algebra.EuclideanDomain.Field
public import Mathlib.Algebra.GCDMonoid.B... |
11,957 | lean_file_chunk | 8f4085d0156c3e51ecff8ffaf148455a3e8783b40ce03e20af6fc2b0cd46f77e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.FieldTheory.Finite.Basic
public import Mathlib.RingTheory.Invariant.Basic
public import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
public ... |
2,388 | lean_file_chunk | b975c6c9e3da2aac9666c73ae0e0c2dc8db806aa30a2369e11d66b6953ae5a40 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.PrincipalIdealDomain
/-!
# Bézout rings
A Bézout ring (Bezout ring) is a ring whose finitely generated ideals are principal.
Not... |
865 | lean_file_chunk | 31c4c96781a12f3bc9870b3424e2528ae442f2be2d90b1c5b7cecb67120c1f1d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.RingTheory.Norm.Defs
public import Mathlib.RingTheory.Trace.Defs
/-!
# Relation between norms and traces
-/
public section
open Module
lem... |
8,581 | lean_file_chunk | 725c64d06675df67549d02fb5401be14b7cc7752045ba4aabe13c498e544ee57 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.RingTheory.IsTensorProduct
/-!
# Base change of polynomial algebras
Given `[CommSemiri... |
16,302 | lean_file_chunk | be9a90ab4d86df0003c25d24badd02824f323a3cb9f187216fd20b823a870a8a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin
-/
module
public import Mathlib.Algebra.MvPolynomial.Equiv
public import Mathlib.Algebra.MvPolynomial.CommRing
public import Mathlib.Logic.Equiv.Functor
public im... |
20,575 | lean_file_chunk | 095a85b9c60e2991ee0a897a32f92f64451a9a851ba74736ad91b90c0b543f65 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.FieldTheory.Minpoly.Field
public import Mathlib.LinearAlgebra.SModEq.Basic
public import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power ba... |
23,943 | lean_file_chunk | 5e0b0773cc656e17dad32361cc021430dbe1986693b5b850bad6d17d138d2616 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
module
public import Mathlib.Algebra.FreeAlgebra
public import Mathlib.RingTheory.Adjoin.Polynomial.Basic
public import Mathlib.RingTheory.Adjoin.Tower
public import... |
1,802 | lean_file_chunk | d70785561b5ea86a9d802940c985ca4e3f7f4c07cb544f79709687cd2c9edf48 | lean_source_file | pkg_mathlib_source_file_chunks | fun ⟨x, hx⟩ ↦ ?_) ⟨n, (subset_span (by simp))⟩ (Subtype.val_injective hn)).1)
· induction hx using span_induction with
| mem x hx =>
change i x ∈ M'
simp only [Set.singleton_union, Set.mem_insert_iff, Set.mem_range] at hx
rcases hx with hx | ⟨j, rfl⟩
· rw [hx, ← hb, piEquiv_apply_apply]
... |
1,596 | lean_file_chunk | 8f915e4aba9c474330db74faa2fe5f80741898bebd1f19e2f0e3adcaba56ab89 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
module
public import Mathlib.LinearAlgebra.Complex.Module
public import Mathlib.RingTheory.Norm.Defs
public import Mathlib.RingTheory.Trace.Defs
/-! # Lemmas about `Algeb... |
2,254 | lean_file_chunk | 58d2d7b45b44645be028982e84814fd571cb883eec565fa5686e2c42ba167afb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Algebra.MvPolynomial.CommRing
public import Mathlib.LinearAlgebra.Dimension.Finite
public import Mathlib.LinearAlgebra.Dimension.StrongR... |
3,540 | lean_file_chunk | e56efc6d7054ecf2829e368e6108d986070d6c7fd027cb28997a049329863ff0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.RingTheory.Spectrum.Prime.FreeLocus
/-!
# Grassmannians
## Main definitions
- `Module.Grassmannian`: `G(k, M; R)` is the `k`ᵗʰ Grassmannian of ... |
11,536 | lean_file_chunk | a314d7958f8889f27f534b07ff2762ac076813912c6ee08f60c41109f5fc5f98 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Algebra.Category.Ring.Constructions
public import Mathlib.Algebra.Category.Ring.Colimits
public import Mathlib.CategoryTheory.Iso
public impor... |
5,049 | lean_file_chunk | 95b1f0ae70623819517dc6c03cd2a4aa57b9dd8872396b28b0bae7f758948438 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.RingTheory.DedekindDomain.IntegralClosure
/-!
# Normal closure of an extension of domains
We define the normal closure of an extension o... |
23,987 | lean_file_chunk | c2934935c59d59be7d6d24e368674342b853e62645a429c92acecbcdeb961090 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.MulOpposite
public import Mathlib.Algebra.Algebra.Subalgebra.Rank
public import Mathlib.Algebra.Polynomial.Basis
public impor... |
16,966 | lean_file_chunk | ca9062021e37f9625efda77a2ed92bc0c6ab864d5ec10805d9d33b137e6e1bba | lean_source_file | pkg_mathlib_source_file_chunks | end LinearDisjoint
end Ring
section CommRing
namespace LinearDisjoint
variable [CommRing R] [CommRing S] [Algebra R S]
variable {A B : Subalgebra R S}
/--
If `A` and `B` are subalgebras in a commutative algebra `S` over `R`, and if they are
linearly disjoint and such that `A ⊔ B = S`, then `trace` and `algebraMap... |
23,928 | lean_file_chunk | c6bab9bf4b35f2cd140803188ade8797b4690074cb8dc597c774af97bdc7f972 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
public import Mathlib.RingTheory.PowerBasis
public ... |
4,392 | lean_file_chunk | 513df73d262c96e13b7eab46608154d7e25273a5bcf65dceae1dd1932de6dbe6 | lean_source_file | pkg_mathlib_source_file_chunks | h'.lift i x hx (h.algEquiv h' z) = h.lift i x hx z := by rw [← h.map_repr z]; simp [-map_repr]
@[simp]
theorem liftHom_algEquiv [Algebra R U] (x : U) (hx z) :
h'.liftHom x hx (h.algEquiv h' z) = h.liftHom x hx z := h.lift_algEquiv h' _ _ hx _
end Equiv
end IsAdjoinRoot
namespace IsAdjoinRootMonic
variable (h :... |
6,175 | lean_file_chunk | 4a45c3e8631ca2ba8af05a4d5f3212a33ff8965a5bed4f36694f449e8d114b42 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Brendan Murphy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brendan Murphy
-/
module
public import Mathlib.LinearAlgebra.DFinsupp
public import Mathlib.LinearAlgebra.TensorProduct.Quotient
public import Mathlib.LinearAlgebra.TensorProduct.Righ... |
21,355 | lean_file_chunk | 3d3a91cd47c41894300df372c156b862814aafb49cf48eb3bbdce057b2f8fdb5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
module
public import Mathlib.Algebra.GCDMonoid.Basic
public import Mathlib.Algebra.IsPrimePow
public import Mathlib.RingTheory.UniqueFactorizationDomain.Multi... |
3,143 | lean_file_chunk | a3e18b1f4b35c699f7ada8a59a3b4f79190b3b795b5c30dbd1e2adad2aadc785 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.Algebra.Ring.Divisibility.Basic
public import Mathlib.Algebra.Order.Group.Unbundled.Abs
public import Mathlib.Algebra.Prime.Defs
public impo... |
2,600 | lean_file_chunk | 6f4ec57f1b8eaa2690b1f8598908d9416ce81abb1c03bc9ae43dcbf69a9cdc98 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.RingTheory.Etale.Pi
public import Mathlib.RingTheory.Smooth.StandardSmoothCotangent
public import Mathlib.RingTheory.TensorProduct.P... |
23,092 | lean_file_chunk | cb2e93b93c62b9ce0db96d8095b3c4513f7abaf2dd45c3b3b3354aa3c79ef9c9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
module
public import Mathlib.Data.Finite.Sum
public import Mathlib.RingTheory.FiniteType
public import Mathlib.RingTheory.Finiteness.Ideal
public import Mathlib.Ring... |
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