Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 136 | 137 | theorem mul_transvection_apply_of_ne (a b : n) (hb : b ≠ j) (c : R) (M : Matrix n n R) :
(M * transvection i j c) a b = M a b := by | simp [transvection, Matrix.mul_add, hb]
| 1,272 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 141 | 142 | theorem det_transvection_of_ne (h : i ≠ j) (c : R) : det (transvection i j c) = 1 := by |
rw [← updateRow_eq_transvection i j, det_updateRow_add_smul_self _ h, det_one]
| 1,272 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 184 | 188 | theorem det_toMatrix_prod [Fintype n] (L : List (TransvectionStruct n 𝕜)) :
det (L.map toMatrix).prod = 1 := by |
induction' L with t L IH
· simp
· simp [IH]
| 1,272 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 205 | 207 | theorem inv_mul (t : TransvectionStruct n R) : t.inv.toMatrix * t.toMatrix = 1 := by |
rcases t with ⟨_, _, t_hij⟩
simp [toMatrix, transvection_mul_transvection_same, t_hij]
| 1,272 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 210 | 212 | theorem mul_inv (t : TransvectionStruct n R) : t.toMatrix * t.inv.toMatrix = 1 := by |
rcases t with ⟨_, _, t_hij⟩
simp [toMatrix, transvection_mul_transvection_same, t_hij]
| 1,272 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 371 | 380 | theorem listTransvecCol_mul_last_row_drop (i : Sum (Fin r) Unit) {k : ℕ} (hk : k ≤ r) :
(((listTransvecCol M).drop k).prod * M) (inr unit) i = M (inr unit) i := by |
-- Porting note: `apply` didn't work anymore, because of the implicit arguments
refine Nat.decreasingInduction' ?_ hk ?_
· intro n hn _ IH
have hn' : n < (listTransvecCol M).length := by simpa [listTransvecCol] using hn
rw [List.drop_eq_get_cons hn']
simpa [listTransvecCol, Matrix.mul_assoc]
· simp... | 1,272 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_import group_theory.specific_groups.alternating from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
-- An example on how to de... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 77 | 80 | theorem prod_list_swap_mem_alternatingGroup_iff_even_length {l : List (Perm α)}
(hl : ∀ g ∈ l, IsSwap g) : l.prod ∈ alternatingGroup α ↔ Even l.length := by |
rw [mem_alternatingGroup, sign_prod_list_swap hl, neg_one_pow_eq_one_iff_even]
decide
| 1,273 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_import group_theory.specific_groups.alternating from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
-- An example on how to de... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 89 | 91 | theorem finRotate_bit1_mem_alternatingGroup {n : ℕ} :
finRotate (bit1 n) ∈ alternatingGroup (Fin (bit1 n)) := by |
rw [mem_alternatingGroup, bit1, sign_finRotate, pow_bit0', Int.units_mul_self, one_pow]
| 1,273 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_import group_theory.specific_groups.alternating from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
-- An example on how to de... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 96 | 101 | theorem two_mul_card_alternatingGroup [Nontrivial α] :
2 * card (alternatingGroup α) = card (Perm α) := by |
let this := (QuotientGroup.quotientKerEquivOfSurjective _ (sign_surjective α)).toEquiv
rw [← Fintype.card_units_int, ← Fintype.card_congr this]
simp only [← Nat.card_eq_fintype_card]
apply (Subgroup.card_eq_card_quotient_mul_card_subgroup _).symm
| 1,273 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_import group_theory.specific_groups.alternating from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
-- An example on how to de... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 219 | 224 | theorem nontrivial_of_three_le_card (h3 : 3 ≤ card α) : Nontrivial (alternatingGroup α) := by |
haveI := Fintype.one_lt_card_iff_nontrivial.1 (lt_trans (by decide) h3)
rw [← Fintype.one_lt_card_iff_nontrivial]
refine lt_of_mul_lt_mul_left ?_ (le_of_lt Nat.prime_two.pos)
rw [two_mul_card_alternatingGroup, card_perm, ← Nat.succ_le_iff]
exact le_trans h3 (card α).self_le_factorial
| 1,273 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace List
variable [D... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 58 | 70 | theorem formPerm_disjoint_iff (hl : Nodup l) (hl' : Nodup l') (hn : 2 ≤ l.length)
(hn' : 2 ≤ l'.length) : Perm.Disjoint (formPerm l) (formPerm l') ↔ l.Disjoint l' := by |
rw [disjoint_iff_eq_or_eq, List.Disjoint]
constructor
· rintro h x hx hx'
specialize h x
rw [formPerm_apply_mem_eq_self_iff _ hl _ hx, formPerm_apply_mem_eq_self_iff _ hl' _ hx'] at h
omega
· intro h x
by_cases hx : x ∈ l
on_goal 1 => by_cases hx' : x ∈ l'
· exact (h hx hx').elim
al... | 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace List
variable [D... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 73 | 86 | theorem isCycle_formPerm (hl : Nodup l) (hn : 2 ≤ l.length) : IsCycle (formPerm l) := by |
cases' l with x l
· set_option tactic.skipAssignedInstances false in norm_num at hn
induction' l with y l generalizing x
· set_option tactic.skipAssignedInstances false in norm_num at hn
· use x
constructor
· rwa [formPerm_apply_mem_ne_self_iff _ hl _ (mem_cons_self _ _)]
· intro w hw
have ... | 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace List
variable [D... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 97 | 102 | theorem cycleOf_formPerm (hl : Nodup l) (hn : 2 ≤ l.length) (x) :
cycleOf l.attach.formPerm x = l.attach.formPerm :=
have hn : 2 ≤ l.attach.length := by | rwa [← length_attach] at hn
have hl : l.attach.Nodup := by rwa [← nodup_attach] at hl
(isCycle_formPerm hl hn).cycleOf_eq
((formPerm_apply_mem_ne_self_iff _ hl _ (mem_attach _ _)).mpr hn)
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace List
variable [D... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 105 | 117 | theorem cycleType_formPerm (hl : Nodup l) (hn : 2 ≤ l.length) :
cycleType l.attach.formPerm = {l.length} := by |
rw [← length_attach] at hn
rw [← nodup_attach] at hl
rw [cycleType_eq [l.attach.formPerm]]
· simp only [map, Function.comp_apply]
rw [support_formPerm_of_nodup _ hl, card_toFinset, dedup_eq_self.mpr hl]
· simp
· intro x h
simp [h, Nat.succ_le_succ_iff] at hn
· simp
· simpa using isCycle_f... | 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace List
variable [D... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 120 | 123 | theorem formPerm_apply_mem_eq_next (hl : Nodup l) (x : α) (hx : x ∈ l) :
formPerm l x = next l x hx := by |
obtain ⟨k, rfl⟩ := get_of_mem hx
rw [next_get _ hl, formPerm_apply_get _ hl]
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Cycle
variable [... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 149 | 156 | theorem formPerm_subsingleton (s : Cycle α) (h : Subsingleton s) : formPerm s h.nodup = 1 := by |
induction' s using Quot.inductionOn with s
simp only [formPerm_coe, mk_eq_coe]
simp only [length_subsingleton_iff, length_coe, mk_eq_coe] at h
cases' s with hd tl
· simp
· simp only [length_eq_zero, add_le_iff_nonpos_left, List.length, nonpos_iff_eq_zero] at h
simp [h]
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 221 | 221 | theorem toList_one : toList (1 : Perm α) x = [] := by | simp [toList, cycleOf_one]
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 225 | 225 | theorem toList_eq_nil_iff {p : Perm α} {x} : toList p x = [] ↔ x ∉ p.support := by | simp [toList]
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 229 | 229 | theorem length_toList : length (toList p x) = (cycleOf p x).support.card := by | simp [toList]
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 232 | 234 | theorem toList_ne_singleton (y : α) : toList p x ≠ [y] := by |
intro H
simpa [card_support_ne_one] using congr_arg length H
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 237 | 238 | theorem two_le_length_toList_iff_mem_support {p : Perm α} {x : α} :
2 ≤ length (toList p x) ↔ x ∈ p.support := by | simp
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 245 | 246 | theorem get_toList (n : ℕ) (hn : n < length (toList p x)) :
(toList p x).get ⟨n, hn⟩ = (p ^ n) x := by | simp [toList]
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 248 | 249 | theorem toList_get_zero (h : x ∈ p.support) :
(toList p x).get ⟨0, (length_toList_pos_of_mem_support _ _ h)⟩ = x := by | simp [toList]
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 253 | 254 | theorem nthLe_toList (n : ℕ) (hn : n < length (toList p x)) :
(toList p x).nthLe n hn = (p ^ n) x := by | simp [toList]
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 259 | 260 | theorem toList_nthLe_zero (h : x ∈ p.support) :
(toList p x).nthLe 0 (length_toList_pos_of_mem_support _ _ h) = x := by | simp [toList]
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 265 | 274 | theorem mem_toList_iff {y : α} : y ∈ toList p x ↔ SameCycle p x y ∧ x ∈ p.support := by |
simp only [toList, mem_range, mem_map]
constructor
· rintro ⟨n, hx, rfl⟩
refine ⟨⟨n, rfl⟩, ?_⟩
contrapose! hx
rw [← support_cycleOf_eq_nil_iff] at hx
simp [hx]
· rintro ⟨h, hx⟩
simpa using h.exists_pow_eq_of_mem_support hx
| 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 278 | 308 | theorem nodup_toList (p : Perm α) (x : α) : Nodup (toList p x) := by |
by_cases hx : p x = x
· rw [← not_mem_support, ← toList_eq_nil_iff] at hx
simp [hx]
have hc : IsCycle (cycleOf p x) := isCycle_cycleOf p hx
rw [nodup_iff_nthLe_inj]
rintro n m hn hm
rw [length_toList, ← hc.orderOf] at hm hn
rw [← cycleOf_apply_self, ← Ne, ← mem_support] at hx
rw [nthLe_toList, nthL... | 1,274 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 312 | 320 | theorem next_toList_eq_apply (p : Perm α) (x y : α) (hy : y ∈ toList p x) :
next (toList p x) y hy = p y := by |
rw [mem_toList_iff] at hy
obtain ⟨k, hk, hk'⟩ := hy.left.exists_pow_eq_of_mem_support hy.right
rw [← nthLe_toList p x k (by simpa using hk)] at hk'
simp_rw [← hk']
rw [next_nthLe _ (nodup_toList _ _), nthLe_toList, nthLe_toList, ← mul_apply, ← pow_succ',
length_toList, ← pow_mod_orderOf_cycleOf_apply p (... | 1,274 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 33 | 40 | theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by |
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 42 | 43 | theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by |
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 50 | 54 | theorem Nat.Coprime.cast {R : Type*} [CommRing R] {a b : ℕ} (h : Nat.Coprime a b) :
IsCoprime (a : R) (b : R) := by |
rw [← isCoprime_iff_coprime] at h
rw [← Int.cast_natCast a, ← Int.cast_natCast b]
exact IsCoprime.intCast h
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 61 | 66 | theorem IsCoprime.prod_left : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x := by |
classical
refine Finset.induction_on t (fun _ ↦ isCoprime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 69 | 70 | theorem IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by |
simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 73 | 76 | theorem IsCoprime.prod_left_iff : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x := by |
classical
refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsCoprime.mul_left_iff, ih, Finset.forall_mem_insert]
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 79 | 80 | theorem IsCoprime.prod_right_iff : IsCoprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsCoprime x (s i) := by |
simpa only [isCoprime_comm] using IsCoprime.prod_left_iff (R := R)
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 94 | 108 | theorem Finset.prod_dvd_of_coprime :
(t : Set I).Pairwise (IsCoprime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by |
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsCoprime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir)... | 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 120 | 175 | theorem exists_sum_eq_one_iff_pairwise_coprime [DecidableEq I] (h : t.Nonempty) :
(∃ μ : I → R, (∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j) = 1) ↔
Pairwise (IsCoprime on fun i : t ↦ s i) := by |
induction h using Finset.Nonempty.cons_induction with
| singleton =>
simp [exists_apply_eq, Pairwise, Function.onFun]
| cons a t hat h ih =>
rw [pairwise_cons']
have mem : ∀ x ∈ t, a ∈ insert a t \ {x} := fun x hx ↦ by
rw [mem_sdiff, mem_singleton]
exact ⟨mem_insert_self _ _, fun ha ↦ hat... | 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 235 | 240 | theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by |
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 242 | 243 | theorem IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by |
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α)
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 245 | 248 | theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by |
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 250 | 251 | theorem IsRelPrime.prod_right_iff : IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i) := by |
simpa only [isRelPrime_comm] using IsRelPrime.prod_left_iff (α := α)
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 261 | 275 | theorem Finset.prod_dvd_of_isRelPrime :
(t : Set I).Pairwise (IsRelPrime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by |
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsRelPrime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir... | 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 281 | 289 | theorem pairwise_isRelPrime_iff_isRelPrime_prod [DecidableEq I] :
Pairwise (IsRelPrime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \ {i}, s j) := by |
refine ⟨fun hp i hi ↦ IsRelPrime.prod_right_iff.mpr fun j hj ↦ ?_, fun hp ↦ ?_⟩
· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain ⟨hj, ji⟩ := hj
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ fun h ↦ ji (congrArg Subtype.val h).symm
· rintro ⟨i, hi⟩ ⟨j, hj⟩ h
apply IsRelPrime.prod_right_iff.mp (hp i hi)
exac... | 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 295 | 297 | theorem pow_left (H : IsRelPrime x y) : IsRelPrime (x ^ m) y := by |
rw [← Finset.card_range m, ← Finset.prod_const]
exact IsRelPrime.prod_left fun _ _ ↦ H
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 299 | 301 | theorem pow_right (H : IsRelPrime x y) : IsRelPrime x (y ^ n) := by |
rw [← Finset.card_range n, ← Finset.prod_const]
exact IsRelPrime.prod_right fun _ _ ↦ H
| 1,275 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 306 | 309 | theorem pow_left_iff (hm : 0 < m) : IsRelPrime (x ^ m) y ↔ IsRelPrime x y := by |
refine ⟨fun h ↦ ?_, IsRelPrime.pow_left⟩
rw [← Finset.card_range m, ← Finset.prod_const] at h
exact h.of_prod_left 0 (Finset.mem_range.mpr hm)
| 1,275 |
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.NormNum.GCD
namespace Tactic
namespace NormNum
open Qq Lean Elab.Tactic Mathlib.Meta.NormNum
| Mathlib/Tactic/NormNum/IsCoprime.lean | 23 | 26 | theorem int_not_isCoprime_helper (x y : ℤ) (d : ℕ) (hd : Int.gcd x y = d)
(h : Nat.beq d 1 = false) : ¬ IsCoprime x y := by |
rw [Int.isCoprime_iff_gcd_eq_one, hd]
exact Nat.ne_of_beq_eq_false h
| 1,276 |
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Data.ZMod.Basic
import Mathlib.Data.Nat.PrimeFin
import Mathlib.RingTheory.Coprime.Lemmas
namespace ZMod
variable {n m : ℕ}
def unitsMap (hm : n ∣ m) : (ZMod m)ˣ →* (ZMod n)ˣ := Units.map (castHom hm (ZMod n))
lemma unitsMap_def (hm : n ∣ m) : unitsM... | Mathlib/Data/ZMod/Units.lean | 38 | 63 | theorem unitsMap_surjective [hm : NeZero m] (h : n ∣ m) :
Function.Surjective (unitsMap h) := by |
suffices ∀ x : ℕ, x.Coprime n → ∃ k : ℕ, (x + k * n).Coprime m by
intro x
have ⟨k, hk⟩ := this x.val.val (val_coe_unit_coprime x)
refine ⟨unitOfCoprime _ hk, Units.ext ?_⟩
have : NeZero n := ⟨fun hn ↦ hm.out (eq_zero_of_zero_dvd (hn ▸ h))⟩
simp [unitsMap_def]
intro x hx
let ps := m.primeFacto... | 1,277 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.RingTheory.Coprime.Lemmas
#align_import algebra.char_p.char_and_card from "leanprover-community/mathlib"@"2fae5fd7f90711febdadf19c44dc60fae8834d1b"
| Mathlib/Algebra/CharP/CharAndCard.lean | 24 | 47 | theorem isUnit_iff_not_dvd_char_of_ringChar_ne_zero (R : Type*) [CommRing R] (p : ℕ) [Fact p.Prime]
(hR : ringChar R ≠ 0) : IsUnit (p : R) ↔ ¬p ∣ ringChar R := by |
have hch := CharP.cast_eq_zero R (ringChar R)
have hp : p.Prime := Fact.out
constructor
· rintro h₁ ⟨q, hq⟩
rcases IsUnit.exists_left_inv h₁ with ⟨a, ha⟩
have h₃ : ¬ringChar R ∣ q := by
rintro ⟨r, hr⟩
rw [hr, ← mul_assoc, mul_comm p, mul_assoc] at hq
nth_rw 1 [← mul_one (ringChar R)] ... | 1,278 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.RingTheory.Coprime.Lemmas
#align_import algebra.char_p.char_and_card from "leanprover-community/mathlib"@"2fae5fd7f90711febdadf19c44dc60fae8834d1b"
theorem isUnit_iff_not_dvd_char_of_ringChar_ne_zero (R : Type*) [CommRin... | Mathlib/Algebra/CharP/CharAndCard.lean | 59 | 75 | theorem prime_dvd_char_iff_dvd_card {R : Type*} [CommRing R] [Fintype R] (p : ℕ) [Fact p.Prime] :
p ∣ ringChar R ↔ p ∣ Fintype.card R := by |
refine
⟨fun h =>
h.trans <|
Int.natCast_dvd_natCast.mp <|
(CharP.intCast_eq_zero_iff R (ringChar R) (Fintype.card R)).mp <|
mod_cast Nat.cast_card_eq_zero R,
fun h => ?_⟩
by_contra h₀
rcases exists_prime_addOrderOf_dvd_card p h with ⟨r, hr⟩
have hr₁ := addOrderOf_n... | 1,278 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import algebra.big_operators.finprod from "leanprover-community/mathlib"@"d6fad0e5bf... | Mathlib/Algebra/BigOperators/Finprod.lean | 171 | 176 | theorem finprod_eq_prod_plift_of_mulSupport_toFinset_subset {f : α → M}
(hf : (mulSupport (f ∘ PLift.down)).Finite) {s : Finset (PLift α)} (hs : hf.toFinset ⊆ s) :
∏ᶠ i, f i = ∏ i ∈ s, f i.down := by |
rw [finprod, dif_pos]
refine Finset.prod_subset hs fun x _ hxf => ?_
rwa [hf.mem_toFinset, nmem_mulSupport] at hxf
| 1,279 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Algebra.Group.ULift
#align_import topology.algebra.monoid from "leanprover-community/mathli... | Mathlib/Topology/Algebra/Monoid.lean | 150 | 152 | theorem le_nhds_mul (a b : M) : 𝓝 a * 𝓝 b ≤ 𝓝 (a * b) := by |
rw [← map₂_mul, ← map_uncurry_prod, ← nhds_prod_eq]
exact continuous_mul.tendsto _
| 1,280 |
import Mathlib.Topology.Constructions
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Order.Filter.ListTraverse
import Mathlib.Tactic.AdaptationNote
#align_import topology.list from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
open TopologicalSpace Set Filter
open Topology Filt... | Mathlib/Topology/List.lean | 28 | 66 | theorem nhds_list (as : List α) : 𝓝 as = traverse 𝓝 as := by |
refine nhds_mkOfNhds _ _ ?_ ?_
· intro l
induction l with
| nil => exact le_rfl
| cons a l ih =>
suffices List.cons <$> pure a <*> pure l ≤ List.cons <$> 𝓝 a <*> traverse 𝓝 l by
simpa only [functor_norm] using this
exact Filter.seq_mono (Filter.map_mono <| pure_le_nhds a) ih
· i... | 1,281 |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_zero from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862b9fb9a992fef2"
open Topology Filter Function
variable {α β G₀ : Type*}
section DivConst... | Mathlib/Topology/Algebra/GroupWithZero.lean | 52 | 54 | theorem Filter.Tendsto.div_const {x : G₀} (hf : Tendsto f l (𝓝 x)) (y : G₀) :
Tendsto (fun a => f a / y) l (𝓝 (x / y)) := by |
simpa only [div_eq_mul_inv] using hf.mul tendsto_const_nhds
| 1,282 |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_zero from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862b9fb9a992fef2"
open Topology Filter Function
variable {α β G₀ : Type*}
section DivConst... | Mathlib/Topology/Algebra/GroupWithZero.lean | 69 | 71 | theorem ContinuousOn.div_const (hf : ContinuousOn f s) (y : G₀) :
ContinuousOn (fun x => f x / y) s := by |
simpa only [div_eq_mul_inv] using hf.mul continuousOn_const
| 1,282 |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_zero from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862b9fb9a992fef2"
open Topology Filter Function
variable {α β G₀ : Type*}
section DivConst... | Mathlib/Topology/Algebra/GroupWithZero.lean | 75 | 76 | theorem Continuous.div_const (hf : Continuous f) (y : G₀) : Continuous fun x => f x / y := by |
simpa only [div_eq_mul_inv] using hf.mul continuous_const
| 1,282 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 47 | 49 | theorem nhds_eq_update : (𝓝 : Γ₀ → Filter Γ₀) = update pure 0 (⨅ γ ≠ 0, 𝓟 (Iio γ)) := by |
rw [nhds_nhdsAdjoint, sup_of_le_right]
exact le_iInf₂ fun γ hγ ↦ le_principal_iff.2 <| zero_lt_iff.2 hγ
| 1,283 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 56 | 57 | theorem nhds_zero : 𝓝 (0 : Γ₀) = ⨅ γ ≠ 0, 𝓟 (Iio γ) := by |
rw [nhds_eq_update, update_same]
| 1,283 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 62 | 65 | theorem hasBasis_nhds_zero : (𝓝 (0 : Γ₀)).HasBasis (fun γ : Γ₀ => γ ≠ 0) Iio := by |
rw [nhds_zero]
refine hasBasis_biInf_principal ?_ ⟨1, one_ne_zero⟩
exact directedOn_iff_directed.2 (Monotone.directed_ge fun a b hab => Iio_subset_Iio hab)
| 1,283 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 78 | 79 | theorem tendsto_zero : Tendsto f l (𝓝 (0 : Γ₀)) ↔ ∀ (γ₀) (_ : γ₀ ≠ 0), ∀ᶠ x in l, f x < γ₀ := by |
simp [nhds_zero]
| 1,283 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 101 | 101 | theorem singleton_mem_nhds_of_units (γ : Γ₀ˣ) : ({↑γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) := by | simp
| 1,283 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 106 | 106 | theorem singleton_mem_nhds_of_ne_zero (h : γ ≠ 0) : ({γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) := by | simp [h]
| 1,283 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 109 | 112 | theorem hasBasis_nhds_of_ne_zero {x : Γ₀} (h : x ≠ 0) :
HasBasis (𝓝 x) (fun _ : Unit => True) fun _ => {x} := by |
rw [nhds_of_ne_zero h]
exact hasBasis_pure _
| 1,283 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 120 | 121 | theorem tendsto_of_ne_zero {γ : Γ₀} (h : γ ≠ 0) : Tendsto f l (𝓝 γ) ↔ ∀ᶠ x in l, f x = γ := by |
rw [nhds_of_ne_zero h, tendsto_pure]
| 1,283 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 128 | 129 | theorem Iio_mem_nhds (h : γ₁ < γ₂) : Iio γ₂ ∈ 𝓝 γ₁ := by |
rcases eq_or_ne γ₁ 0 with (rfl | h₀) <;> simp [*, h.ne', Iio_mem_nhds_zero]
| 1,283 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 136 | 139 | theorem isOpen_iff {s : Set Γ₀} : IsOpen s ↔ (0 : Γ₀) ∉ s ∨ ∃ γ, γ ≠ 0 ∧ Iio γ ⊆ s := by |
rw [isOpen_iff_mem_nhds, ← and_forall_ne (0 : Γ₀)]
simp (config := { contextual := true }) [nhds_of_ne_zero, imp_iff_not_or,
hasBasis_nhds_zero.mem_iff]
| 1,283 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 142 | 144 | theorem isClosed_iff {s : Set Γ₀} : IsClosed s ↔ (0 : Γ₀) ∈ s ∨ ∃ γ, γ ≠ 0 ∧ s ⊆ Ici γ := by |
simp only [← isOpen_compl_iff, isOpen_iff, mem_compl_iff, not_not, ← compl_Ici,
compl_subset_compl]
| 1,283 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Set.Card
import Mathlib.GroupTheory.Subgroup.Center
open MulAction ConjClasses
variable (G : Type*) [Group G]
| Mathlib/GroupTheory/ClassEquation.lean | 31 | 35 | theorem sum_conjClasses_card_eq_card [Fintype <| ConjClasses G] [Fintype G]
[∀ x : ConjClasses G, Fintype x.carrier] :
∑ x : ConjClasses G, x.carrier.toFinset.card = Fintype.card G := by |
suffices (Σ x : ConjClasses G, x.carrier) ≃ G by simpa using (Fintype.card_congr this)
simpa [carrier_eq_preimage_mk] using Equiv.sigmaFiberEquiv ConjClasses.mk
| 1,284 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Set.Card
import Mathlib.GroupTheory.Subgroup.Center
open MulAction ConjClasses
variable (G : Type*) [Group G]
theorem sum_conjClasses_card_eq_card [Fintype <| Conj... | Mathlib/GroupTheory/ClassEquation.lean | 38 | 43 | theorem Group.sum_card_conj_classes_eq_card [Finite G] :
∑ᶠ x : ConjClasses G, x.carrier.ncard = Nat.card G := by |
classical
cases nonempty_fintype G
rw [Nat.card_eq_fintype_card, ← sum_conjClasses_card_eq_card, finsum_eq_sum_of_fintype]
simp [Set.ncard_eq_toFinset_card']
| 1,284 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Set.Card
import Mathlib.GroupTheory.Subgroup.Center
open MulAction ConjClasses
variable (G : Type*) [Group G]
theorem sum_conjClasses_card_eq_card [Fintype <| Conj... | Mathlib/GroupTheory/ClassEquation.lean | 47 | 70 | theorem Group.nat_card_center_add_sum_card_noncenter_eq_card [Finite G] :
Nat.card (Subgroup.center G) + ∑ᶠ x ∈ noncenter G, Nat.card x.carrier = Nat.card G := by |
classical
cases nonempty_fintype G
rw [@Nat.card_eq_fintype_card G, ← sum_conjClasses_card_eq_card, ←
Finset.sum_sdiff (ConjClasses.noncenter G).toFinset.subset_univ]
simp only [Nat.card_eq_fintype_card, Set.toFinset_card]
congr 1
swap
· convert finsum_cond_eq_sum_of_cond_iff _ _
simp [Set.mem_to... | 1,284 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Set.Card
import Mathlib.GroupTheory.Subgroup.Center
open MulAction ConjClasses
variable (G : Type*) [Group G]
theorem sum_conjClasses_card_eq_card [Fintype <| Conj... | Mathlib/GroupTheory/ClassEquation.lean | 72 | 81 | theorem Group.card_center_add_sum_card_noncenter_eq_card (G) [Group G]
[∀ x : ConjClasses G, Fintype x.carrier] [Fintype G] [Fintype <| Subgroup.center G]
[Fintype <| noncenter G] : Fintype.card (Subgroup.center G) +
∑ x ∈ (noncenter G).toFinset, x.carrier.toFinset.card = Fintype.card G := by |
convert Group.nat_card_center_add_sum_card_noncenter_eq_card G using 2
· simp
· rw [← finsum_set_coe_eq_finsum_mem (noncenter G), finsum_eq_sum_of_fintype,
← Finset.sum_set_coe]
simp
· simp
| 1,284 |
import Mathlib.Topology.Separation
import Mathlib.Algebra.BigOperators.Finprod
#align_import topology.algebra.infinite_sum.basic from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
noncomputable section
open Filter Function
open scoped Topology
variable {α β γ : Type*}
section HasP... | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 124 | 125 | theorem tprod_eq_one_of_not_multipliable (h : ¬Multipliable f) : ∏' b, f b = 1 := by |
simp [tprod_def, h]
| 1,285 |
import Mathlib.Topology.Separation
import Mathlib.Algebra.BigOperators.Finprod
#align_import topology.algebra.infinite_sum.basic from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
noncomputable section
open Filter Function
open scoped Topology
variable {α β γ : Type*}
section HasP... | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 129 | 131 | theorem Function.Injective.hasProd_iff {g : γ → β} (hg : Injective g)
(hf : ∀ x, x ∉ Set.range g → f x = 1) : HasProd (f ∘ g) a ↔ HasProd f a := by |
simp only [HasProd, Tendsto, comp_apply, hg.map_atTop_finset_prod_eq hf]
| 1,285 |
import Mathlib.Topology.Separation
import Mathlib.Algebra.BigOperators.Finprod
#align_import topology.algebra.infinite_sum.basic from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
noncomputable section
open Filter Function
open scoped Topology
variable {α β γ : Type*}
section HasP... | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 166 | 170 | theorem Multipliable.hasProd (ha : Multipliable f) : HasProd f (∏' b, f b) := by |
simp only [tprod_def, ha, dite_true]
by_cases H : (mulSupport f).Finite
· simp [H, hasProd_prod_of_ne_finset_one, finprod_eq_prod]
· simpa [H] using ha.choose_spec
| 1,285 |
import Mathlib.Topology.Separation
import Mathlib.Algebra.BigOperators.Finprod
#align_import topology.algebra.infinite_sum.basic from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
noncomputable section
open Filter Function
open scoped Topology
variable {α β γ : Type*}
section HasP... | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 174 | 175 | theorem HasProd.unique {a₁ a₂ : α} [T2Space α] : HasProd f a₁ → HasProd f a₂ → a₁ = a₂ := by |
classical exact tendsto_nhds_unique
| 1,285 |
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section HasProd
variable [CommMonoid α] [TopologicalSpace α]
variable {f g : β → α} ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 35 | 35 | theorem hasProd_one : HasProd (fun _ ↦ 1 : β → α) 1 := by | simp [HasProd, tendsto_const_nhds]
| 1,286 |
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section HasProd
variable [CommMonoid α] [TopologicalSpace α]
variable {f g : β → α} ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 39 | 40 | theorem hasProd_empty [IsEmpty β] : HasProd f 1 := by |
convert @hasProd_one α β _ _
| 1,286 |
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section HasProd
variable [CommMonoid α] [TopologicalSpace α]
variable {f g : β → α} ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 101 | 104 | theorem hasProd_subtype_iff_mulIndicator {s : Set β} :
HasProd (f ∘ (↑) : s → α) a ↔ HasProd (s.mulIndicator f) a := by |
rw [← Set.mulIndicator_range_comp, Subtype.range_coe,
hasProd_subtype_iff_of_mulSupport_subset Set.mulSupport_mulIndicator_subset]
| 1,286 |
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section HasProd
variable [CommMonoid α] [TopologicalSpace α]
variable {f g : β → α} ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 132 | 133 | theorem multipliable_of_finite_mulSupport (h : (mulSupport f).Finite) : Multipliable f := by |
apply multipliable_of_ne_finset_one (s := h.toFinset); simp
| 1,286 |
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section tprod
variable [CommMonoid α] [TopologicalSpace α] {f g : β → α} {a a₁ a₂ : ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 387 | 388 | theorem tprod_congr_set_coe (f : β → α) {s t : Set β} (h : s = t) :
∏' x : s, f x = ∏' x : t, f x := by | rw [h]
| 1,286 |
import Mathlib.Algebra.TrivSqZeroExt
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.instances.triv_sq_zero_ext from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984e4"
open scoped Topology
variable {α S R M : Type*}
local... | Mathlib/Topology/Instances/TrivSqZeroExt.lean | 46 | 48 | theorem nhds_def (x : tsze R M) : 𝓝 x = (𝓝 x.fst).prod (𝓝 x.snd) := by |
cases x using Prod.rec
exact nhds_prod_eq
| 1,287 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Order.Group.WithTop
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.Valuation.Basic
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter... | Mathlib/RingTheory/HahnSeries/Summable.lean | 89 | 92 | theorem addVal_le_of_coeff_ne_zero {x : HahnSeries Γ R} {g : Γ} (h : x.coeff g ≠ 0) :
addVal Γ R x ≤ g := by |
rw [addVal_apply_of_ne (ne_zero_of_coeff_ne_zero h), WithTop.coe_le_coe]
exact order_le_of_coeff_ne_zero h
| 1,288 |
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsingleton
#align_import order.order_iso_nat from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
variable {α : Type*}
namespa... | Mathlib/Order/OrderIsoNat.lean | 58 | 62 | theorem exists_not_acc_lt_of_not_acc {a : α} {r} (h : ¬Acc r a) : ∃ b, ¬Acc r b ∧ r b a := by |
contrapose! h
refine ⟨_, fun b hr => ?_⟩
by_contra hb
exact h b hb hr
| 1,289 |
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsingleton
#align_import order.order_iso_nat from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
variable {α : Type*}
namespa... | Mathlib/Order/OrderIsoNat.lean | 66 | 81 | theorem acc_iff_no_decreasing_seq {x} :
Acc r x ↔ IsEmpty { f : ((· > ·) : ℕ → ℕ → Prop) ↪r r // x ∈ Set.range f } := by |
constructor
· refine fun h => h.recOn fun x _ IH => ?_
constructor
rintro ⟨f, k, hf⟩
exact IsEmpty.elim' (IH (f (k + 1)) (hf ▸ f.map_rel_iff.2 (lt_add_one k))) ⟨f, _, rfl⟩
· have : ∀ x : { a // ¬Acc r a }, ∃ y : { a // ¬Acc r a }, r y.1 x.1 := by
rintro ⟨x, hx⟩
cases exists_not_acc_lt_of_... | 1,289 |
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsingleton
#align_import order.order_iso_nat from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
variable {α : Type*}
namespa... | Mathlib/Order/OrderIsoNat.lean | 84 | 86 | theorem not_acc_of_decreasing_seq (f : ((· > ·) : ℕ → ℕ → Prop) ↪r r) (k : ℕ) : ¬Acc r (f k) := by |
rw [acc_iff_no_decreasing_seq, not_isEmpty_iff]
exact ⟨⟨f, k, rfl⟩⟩
| 1,289 |
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsingleton
#align_import order.order_iso_nat from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
variable {α : Type*}
namespa... | Mathlib/Order/OrderIsoNat.lean | 90 | 96 | theorem wellFounded_iff_no_descending_seq :
WellFounded r ↔ IsEmpty (((· > ·) : ℕ → ℕ → Prop) ↪r r) := by |
constructor
· rintro ⟨h⟩
exact ⟨fun f => not_acc_of_decreasing_seq f 0 (h _)⟩
· intro h
exact ⟨fun x => acc_iff_no_decreasing_seq.2 inferInstance⟩
| 1,289 |
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsingleton
#align_import order.order_iso_nat from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
variable {α : Type*}
namespa... | Mathlib/Order/OrderIsoNat.lean | 99 | 101 | theorem not_wellFounded_of_decreasing_seq (f : ((· > ·) : ℕ → ℕ → Prop) ↪r r) : ¬WellFounded r := by |
rw [wellFounded_iff_no_descending_seq, not_isEmpty_iff]
exact ⟨f⟩
| 1,289 |
import Mathlib.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Order.Zorn
import Mathlib.Data.Finset.Order
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.Finite.Set
import Mathlib.Tactic.TFAE
#alig... | Mathlib/Order/CompactlyGenerated/Basic.lean | 83 | 105 | theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
CompleteLattice.IsCompactElement k ↔
∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s := by |
classical
constructor
· intro H ι s hs
obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs
have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop
choose f hf using this
refine ⟨Finset.univ.image f, ht'.trans ?_⟩
rw [Finset.sup_le_iff]
intro b hb
rw [← show s (f ⟨b, hb⟩) = id b fro... | 1,290 |
import Mathlib.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Order.Zorn
import Mathlib.Data.Finset.Order
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.Finite.Set
import Mathlib.Tactic.TFAE
#alig... | Mathlib/Order/CompactlyGenerated/Basic.lean | 110 | 149 | theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
IsCompactElement k ↔
∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x := by |
classical
constructor
· intro hk s hne hdir hsup
obtain ⟨t, ht⟩ := hk s hsup
-- certainly every element of t is below something in s, since ↑t ⊆ s.
have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩
obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directe... | 1,290 |
import Mathlib.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Order.Zorn
import Mathlib.Data.Finset.Order
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.Finite.Set
import Mathlib.Tactic.TFAE
#alig... | Mathlib/Order/CompactlyGenerated/Basic.lean | 152 | 169 | theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type*}
(f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by |
classical
let g : Finset ι → α := fun s => ⨆ i ∈ s, f i
have h1 : DirectedOn (· ≤ ·) (Set.range g) := by
rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩
exact
⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset Finset.subset_union_left,
iSup_le_iSup_of_subset Finset.subset_union_right⟩
have h2 : ... | 1,290 |
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Order.CompactlyGenerated.Basic
variable {ι α : Type*} [CompleteLattice α]
namespace Set.Iic
| Mathlib/Order/CompactlyGenerated/Intervals.lean | 18 | 24 | theorem isCompactElement {a : α} {b : Iic a} (h : CompleteLattice.IsCompactElement (b : α)) :
CompleteLattice.IsCompactElement b := by |
simp only [CompleteLattice.isCompactElement_iff, Finset.sup_eq_iSup] at h ⊢
intro ι s hb
replace hb : (b : α) ≤ iSup ((↑) ∘ s) := le_trans hb <| (coe_iSup s) ▸ le_refl _
obtain ⟨t, ht⟩ := h ι ((↑) ∘ s) hb
exact ⟨t, (by simpa using ht : (b : α) ≤ _)⟩
| 1,291 |
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Submodule.RestrictScalars
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.LinearAlgebra.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_... | Mathlib/LinearAlgebra/Span.lean | 147 | 150 | theorem span_insert_zero : span R (insert (0 : M) s) = span R s := by |
refine le_antisymm ?_ (Submodule.span_mono (Set.subset_insert 0 s))
rw [span_le, Set.insert_subset_iff]
exact ⟨by simp only [SetLike.mem_coe, Submodule.zero_mem], Submodule.subset_span⟩
| 1,292 |
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Submodule.RestrictScalars
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.LinearAlgebra.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_... | Mathlib/LinearAlgebra/Span.lean | 154 | 157 | theorem span_preimage_le (f : F) (s : Set M₂) :
span R (f ⁻¹' s) ≤ (span R₂ s).comap f := by |
rw [span_le, comap_coe]
exact preimage_mono subset_span
| 1,292 |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w u₁ v₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 88 | 90 | theorem algebraMap_smul [SMul R M] [IsScalarTower R A M] (r : R) (x : M) :
algebraMap R A r • x = r • x := by |
rw [Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]
| 1,293 |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w u₁ v₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 94 | 96 | theorem of_algebraMap_smul [SMul R M] (h : ∀ (r : R) (x : M), algebraMap R A r • x = r • x) :
IsScalarTower R A M where
smul_assoc r a x := by | rw [Algebra.smul_def, mul_smul, h]
| 1,293 |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w u₁ v₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 130 | 131 | theorem algebraMap_apply (x : R) : algebraMap R A x = algebraMap S A (algebraMap R S x) := by |
rw [algebraMap_eq R S A, RingHom.comp_apply]
| 1,293 |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w u₁ v₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 162 | 164 | theorem _root_.AlgHom.map_algebraMap (f : A →ₐ[S] B) (r : R) :
f (algebraMap R A r) = algebraMap R B r := by |
rw [algebraMap_apply R S A r, f.commutes, ← algebraMap_apply R S B]
| 1,293 |
import Mathlib.Algebra.Algebra.Tower
#align_import algebra.algebra.restrict_scalars from "leanprover-community/mathlib"@"c310cfdc40da4d99a10a58c33a95360ef9e6e0bf"
variable (R S M A : Type*)
@[nolint unusedArguments]
def RestrictScalars (_R _S M : Type*) : Type _ := M
#align restrict_scalars RestrictScalars
ins... | Mathlib/Algebra/Algebra/RestrictScalars.lean | 175 | 179 | theorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :
(RestrictScalars.addEquiv R S M).symm ((r • s) • x) =
r • (RestrictScalars.addEquiv R S M).symm (s • x) := by |
rw [Algebra.smul_def, mul_smul]
rfl
| 1,294 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
variable (R A B : Type*) {σ : Type*}
namespace MvPolynomial
section Semiring
variable [CommSemiring R] [Com... | Mathlib/RingTheory/MvPolynomial/Tower.lean | 35 | 37 | theorem aeval_map_algebraMap (x : σ → B) (p : MvPolynomial σ R) :
aeval x (map (algebraMap R A) p) = aeval x p := by |
rw [aeval_def, aeval_def, eval₂_map, IsScalarTower.algebraMap_eq R A B]
| 1,295 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
variable (R A B : Type*) {σ : Type*}
namespace MvPolynomial
section CommSemiring
variable [CommSemiring R] ... | Mathlib/RingTheory/MvPolynomial/Tower.lean | 48 | 53 | theorem aeval_algebraMap_apply (x : σ → A) (p : MvPolynomial σ R) :
aeval (algebraMap A B ∘ x) p = algebraMap A B (MvPolynomial.aeval x p) := by |
rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ←
IsScalarTower.algebraMap_eq]
-- Porting note: added
simp only [Function.comp]
| 1,295 |
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