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.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.LinearAlgebra.AffineSpace.Restrict
import Mathlib.Tactic.FailIfNoProgress
#align_import analysis.normed_space.affine_isomet... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 329 | 331 | theorem linear_eq_linear_isometry : e.linear = e.linearIsometryEquiv.toLinearEquiv := by |
ext
rfl
| 1,634 |
import Mathlib.Topology.Instances.RealVectorSpace
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import analysis.normed_space.mazur_ulam from "leanprover-community/mathlib"@"78261225eb5cedc61c5c74ecb44e5b385d13b733"
variable {E PE F PF : Type*} [NormedAddCommGroup E] [NormedSpace β E] [MetricSpace PE]
... | Mathlib/Analysis/NormedSpace/MazurUlam.lean | 45 | 83 | theorem midpoint_fixed {x y : PE} :
β e : PE βα΅’ PE, e x = x β e y = y β e (midpoint β x y) = midpoint β x y := by |
set z := midpoint β x y
-- Consider the set of `e : E βα΅’ E` such that `e x = x` and `e y = y`
set s := { e : PE βα΅’ PE | e x = x β§ e y = y }
haveI : Nonempty s := β¨β¨IsometryEquiv.refl PE, rfl, rflβ©β©
-- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
have h_bdd : BddAbove (range fun e :... | 1,635 |
import Mathlib.Topology.Instances.RealVectorSpace
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import analysis.normed_space.mazur_ulam from "leanprover-community/mathlib"@"78261225eb5cedc61c5c74ecb44e5b385d13b733"
variable {E PE F PF : Type*} [NormedAddCommGroup E] [NormedSpace β E] [MetricSpace PE]
... | Mathlib/Analysis/NormedSpace/MazurUlam.lean | 87 | 96 | theorem map_midpoint (f : PE βα΅’ PF) (x y : PE) : f (midpoint β x y) = midpoint β (f x) (f y) := by |
set e : PE βα΅’ PE :=
((f.trans <| (pointReflection β <| midpoint β (f x) (f y)).toIsometryEquiv).trans f.symm).trans
(pointReflection β <| midpoint β x y).toIsometryEquiv
have hx : e x = x := by simp [e]
have hy : e y = y := by simp [e]
have hm := e.midpoint_fixed hx hy
simp only [e, trans_apply] at... | 1,635 |
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
namespace Con... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 66 | 67 | theorem coe_contLinear_eq_linear (f : P βᴬ[R] Q) :
(f.contLinear : V ββ[R] W) = (f : P βα΅[R] Q).linear := by | ext; rfl
| 1,636 |
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
namespace Con... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 102 | 114 | theorem contLinear_eq_zero_iff_exists_const (f : P βᴬ[R] Q) :
f.contLinear = 0 β β q, f = const R P q := by |
have hβ : f.contLinear = 0 β (f : P βα΅[R] Q).linear = 0 := by
refine β¨fun h => ?_, fun h => ?_β© <;> ext
Β· rw [β coe_contLinear_eq_linear, h]; rfl
Β· rw [β coe_linear_eq_coe_contLinear, h]; rfl
have hβ : β q : Q, f = const R P q β (f : P βα΅[R] Q) = AffineMap.const R P q := by
intro q
refine β¨fun ... | 1,636 |
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
namespace Con... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 118 | 120 | theorem to_affine_map_contLinear (f : V βL[R] W) : f.toContinuousAffineMap.contLinear = f := by |
ext
rfl
| 1,636 |
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
namespace Con... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 148 | 151 | theorem decomp (f : V βᴬ[R] W) : (f : V β W) = f.contLinear + Function.const V (f 0) := by |
rcases f with β¨f, hβ©
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
β Function.const_def]
| 1,636 |
import Mathlib.Analysis.NormedSpace.ContinuousAffineMap
import Mathlib.Analysis.Calculus.ContDiff.Basic
#align_import analysis.calculus.affine_map from "leanprover-community/mathlib"@"839b92fedff9981cf3fe1c1f623e04b0d127f57c"
namespace ContinuousAffineMap
variable {π V W : Type*} [NontriviallyNormedField π]
va... | Mathlib/Analysis/Calculus/AffineMap.lean | 30 | 33 | theorem contDiff {n : ββ} (f : V βᴬ[π] W) : ContDiff π n f := by |
rw [f.decomp]
apply f.contLinear.contDiff.add
exact contDiff_const
| 1,637 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 118 | 120 | theorem norm_self_mul_star {x : E} : βx * xββ = βxβ * βxβ := by |
nth_rw 1 [β star_star x]
simp only [norm_star_mul_self, norm_star]
| 1,638 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 123 | 123 | theorem norm_star_mul_self' {x : E} : βxβ * xβ = βxββ * βxβ := by | rw [norm_star_mul_self, norm_star]
| 1,638 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 135 | 137 | theorem star_mul_self_eq_zero_iff (x : E) : xβ * x = 0 β x = 0 := by |
rw [β norm_eq_zero, norm_star_mul_self]
exact mul_self_eq_zero.trans norm_eq_zero
| 1,638 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 140 | 141 | theorem star_mul_self_ne_zero_iff (x : E) : xβ * x β 0 β x β 0 := by |
simp only [Ne, star_mul_self_eq_zero_iff]
| 1,638 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 145 | 146 | theorem mul_star_self_eq_zero_iff (x : E) : x * xβ = 0 β x = 0 := by |
simpa only [star_eq_zero, star_star] using @star_mul_self_eq_zero_iff _ _ _ _ (star x)
| 1,638 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 149 | 150 | theorem mul_star_self_ne_zero_iff (x : E) : x * xβ β 0 β x β 0 := by |
simp only [Ne, mul_star_self_eq_zero_iff]
| 1,638 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 203 | 205 | theorem norm_one [Nontrivial E] : β(1 : E)β = 1 := by |
have : 0 < β(1 : E)β := norm_pos_iff.mpr one_ne_zero
rw [β mul_left_inj' this.ne', β norm_star_mul_self, mul_one, star_one, one_mul]
| 1,638 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 212 | 214 | theorem norm_coe_unitary [Nontrivial E] (U : unitary E) : β(U : E)β = 1 := by |
rw [β sq_eq_sq (norm_nonneg _) zero_le_one, one_pow 2, sq, β CstarRing.norm_star_mul_self,
unitary.coe_star_mul_self, CstarRing.norm_one]
| 1,638 |
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Unitization
#align_import analysis.normed_space.star.mul from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open ContinuousLinearMap
local postfix:max "β" => star
variable (π : Type*) {E : Type*}
varia... | Mathlib/Analysis/NormedSpace/Star/Unitization.lean | 87 | 124 | theorem Unitization.norm_splitMul_snd_sq (x : Unitization π E) :
β(Unitization.splitMul π E x).sndβ ^ 2 β€ β(Unitization.splitMul π E (star x * x)).sndβ := by |
/- The key idea is that we can use `sSup_closed_unit_ball_eq_norm` to make this about
applying this linear map to elements of norm at most one. There is a bit of `sqrt` and `sq`
shuffling that needs to occur, which is primarily just an annoyance. -/
refine (Real.le_sqrt (norm_nonneg _) (norm_nonneg _)).mp ?_
... | 1,639 |
import Mathlib.Algebra.Module.MinimalAxioms
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Topolo... | Mathlib/Topology/ContinuousFunction/Bounded.lean | 158 | 162 | theorem dist_set_exists : β C, 0 β€ C β§ β x : Ξ±, dist (f x) (g x) β€ C := by |
rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with β¨C, hCβ©
refine β¨max 0 C, le_max_left _ _, fun x => (hC ?_ ?_).trans (le_max_right _ _)β©
<;> [left; right]
<;> apply mem_range_self
| 1,640 |
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory.integral.riesz_markov_kakutani from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
noncomputable section
open BoundedContinuousFunction NNReal ENNReal
open Set Functio... | Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean | 51 | 56 | theorem rieszContentAux_image_nonempty (K : Compacts X) :
(Ξ '' { f : X βα΅ ββ₯0 | β x β K, (1 : ββ₯0) β€ f x }).Nonempty := by |
rw [image_nonempty]
use (1 : X βα΅ ββ₯0)
intro x _
simp only [BoundedContinuousFunction.coe_one, Pi.one_apply]; rfl
| 1,641 |
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
import Mathlib.Analysis.Normed.Group.InfiniteSum
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6d... | Mathlib/Topology/ContinuousFunction/Compact.lean | 132 | 133 | theorem dist_apply_le_dist (x : Ξ±) : dist (f x) (g x) β€ dist f g := by |
simp only [β dist_mkOfCompact, dist_coe_le_dist, β mkOfCompact_apply]
| 1,642 |
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
import Mathlib.Analysis.Normed.Group.InfiniteSum
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6d... | Mathlib/Topology/ContinuousFunction/Compact.lean | 137 | 138 | theorem dist_le (C0 : (0 : β) β€ C) : dist f g β€ C β β x : Ξ±, dist (f x) (g x) β€ C := by |
simp only [β dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
| 1,642 |
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
import Mathlib.Analysis.Normed.Group.InfiniteSum
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6d... | Mathlib/Topology/ContinuousFunction/Compact.lean | 141 | 143 | theorem dist_le_iff_of_nonempty [Nonempty Ξ±] : dist f g β€ C β β x, dist (f x) (g x) β€ C := by |
simp only [β dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
| 1,642 |
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
import Mathlib.Analysis.Normed.Group.InfiniteSum
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6d... | Mathlib/Topology/ContinuousFunction/Compact.lean | 146 | 147 | theorem dist_lt_iff_of_nonempty [Nonempty Ξ±] : dist f g < C β β x : Ξ±, dist (f x) (g x) < C := by |
simp only [β dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
| 1,642 |
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
import Mathlib.Analysis.Normed.Group.InfiniteSum
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6d... | Mathlib/Topology/ContinuousFunction/Compact.lean | 154 | 156 | theorem dist_lt_iff (C0 : (0 : β) < C) : dist f g < C β β x : Ξ±, dist (f x) (g x) < C := by |
rw [β dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
| 1,642 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 58 | 66 | theorem continuous_thickenedIndicatorAux {Ξ΄ : β} (Ξ΄_pos : 0 < Ξ΄) (E : Set Ξ±) :
Continuous (thickenedIndicatorAux Ξ΄ E) := by |
unfold thickenedIndicatorAux
let f := fun x : Ξ± => (β¨1, infEdist x E / ENNReal.ofReal Ξ΄β© : ββ₯0 Γ ββ₯0β)
let sub := fun p : ββ₯0 Γ ββ₯0β => (p.1 : ββ₯0β) - p.2
rw [show (fun x : Ξ± => (1 : ββ₯0β) - infEdist x E / ENNReal.ofReal Ξ΄) = sub β f by rfl]
apply (@ENNReal.continuous_nnreal_sub 1).comp
apply (ENNReal.cont... | 1,643 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 69 | 71 | theorem thickenedIndicatorAux_le_one (Ξ΄ : β) (E : Set Ξ±) (x : Ξ±) :
thickenedIndicatorAux Ξ΄ E x β€ 1 := by |
apply @tsub_le_self _ _ _ _ (1 : ββ₯0β)
| 1,643 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 79 | 81 | theorem thickenedIndicatorAux_closure_eq (Ξ΄ : β) (E : Set Ξ±) :
thickenedIndicatorAux Ξ΄ (closure E) = thickenedIndicatorAux Ξ΄ E := by |
simp (config := { unfoldPartialApp := true }) only [thickenedIndicatorAux, infEdist_closure]
| 1,643 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 84 | 86 | theorem thickenedIndicatorAux_one (Ξ΄ : β) (E : Set Ξ±) {x : Ξ±} (x_in_E : x β E) :
thickenedIndicatorAux Ξ΄ E x = 1 := by |
simp [thickenedIndicatorAux, infEdist_zero_of_mem x_in_E, tsub_zero]
| 1,643 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 89 | 91 | theorem thickenedIndicatorAux_one_of_mem_closure (Ξ΄ : β) (E : Set Ξ±) {x : Ξ±}
(x_mem : x β closure E) : thickenedIndicatorAux Ξ΄ E x = 1 := by |
rw [β thickenedIndicatorAux_closure_eq, thickenedIndicatorAux_one Ξ΄ (closure E) x_mem]
| 1,643 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 94 | 102 | theorem thickenedIndicatorAux_zero {Ξ΄ : β} (Ξ΄_pos : 0 < Ξ΄) (E : Set Ξ±) {x : Ξ±}
(x_out : x β thickening Ξ΄ E) : thickenedIndicatorAux Ξ΄ E x = 0 := by |
rw [thickening, mem_setOf_eq, not_lt] at x_out
unfold thickenedIndicatorAux
apply le_antisymm _ bot_le
have key := tsub_le_tsub
(@rfl _ (1 : ββ₯0β)).le (ENNReal.div_le_div x_out (@rfl _ (ENNReal.ofReal Ξ΄ : ββ₯0β)).le)
rw [ENNReal.div_self (ne_of_gt (ENNReal.ofReal_pos.mpr Ξ΄_pos)) ofReal_ne_top] at key
si... | 1,643 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 110 | 115 | theorem indicator_le_thickenedIndicatorAux (Ξ΄ : β) (E : Set Ξ±) :
(E.indicator fun _ => (1 : ββ₯0β)) β€ thickenedIndicatorAux Ξ΄ E := by |
intro a
by_cases h : a β E
Β· simp only [h, indicator_of_mem, thickenedIndicatorAux_one Ξ΄ E h, le_refl]
Β· simp only [h, indicator_of_not_mem, not_false_iff, zero_le]
| 1,643 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 130 | 153 | theorem thickenedIndicatorAux_tendsto_indicator_closure {Ξ΄seq : β β β}
(Ξ΄seq_lim : Tendsto Ξ΄seq atTop (π 0)) (E : Set Ξ±) :
Tendsto (fun n => thickenedIndicatorAux (Ξ΄seq n) E) atTop
(π (indicator (closure E) fun _ => (1 : ββ₯0β))) := by |
rw [tendsto_pi_nhds]
intro x
by_cases x_mem_closure : x β closure E
Β· simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
rw [show (indicator (closure E) fun _ => (1 : ββ₯0β)) x = 1 by
simp only [x_mem_closure, indicator_of_mem]]
exact tendsto_const_nhds
Β· rw [show (closure E)... | 1,643 |
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "οΏ½... | Mathlib/Analysis/RCLike/Basic.lean | 105 | 106 | theorem real_smul_eq_coe_smul [AddCommGroup E] [Module K E] [Module β E] [IsScalarTower β K E]
(r : β) (x : E) : r β’ x = (r : K) β’ x := by | rw [RCLike.ofReal_alg, smul_one_smul]
| 1,644 |
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "οΏ½... | Mathlib/Analysis/RCLike/Basic.lean | 162 | 162 | theorem one_re : re (1 : K) = 1 := by | rw [β ofReal_one, ofReal_re]
| 1,644 |
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "οΏ½... | Mathlib/Analysis/RCLike/Basic.lean | 166 | 166 | theorem one_im : im (1 : K) = 0 := by | rw [β ofReal_one, ofReal_im]
| 1,644 |
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.normed_space.extend from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable {π : Ty... | Mathlib/Analysis/NormedSpace/Extend.lean | 88 | 90 | theorem extendToπ'_apply_re (fr : F ββ[β] β) (x : F) : re (fr.extendToπ' x : π) = fr x := by |
simp only [extendToπ'_apply, map_sub, zero_mul, mul_zero, sub_zero,
rclike_simps]
| 1,645 |
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.normed_space.extend from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable {π : Ty... | Mathlib/Analysis/NormedSpace/Extend.lean | 93 | 99 | theorem norm_extendToπ'_apply_sq (fr : F ββ[β] β) (x : F) :
β(fr.extendToπ' x : π)β ^ 2 = fr (conj (fr.extendToπ' x : π) β’ x) :=
calc
β(fr.extendToπ' x : π)β ^ 2 = re (conj (fr.extendToπ' x) * fr.extendToπ' x : π) := by |
rw [RCLike.conj_mul, β ofReal_pow, ofReal_re]
_ = fr (conj (fr.extendToπ' x : π) β’ x) := by
rw [β smul_eq_mul, β map_smul, extendToπ'_apply_re]
| 1,645 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.locally_convex.continuous_of_bounded from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open TopologicalSpace Bornology Filter Topology Pointwise
variable {π π' E F : Type*}
var... | Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean | 96 | 166 | theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E βββ[Ο] F)
(hf : β s, IsVonNBounded π s β IsVonNBounded π' (f '' s)) : ContinuousAt f 0 := by |
-- Assume that f is not continuous at 0
by_contra h
-- We use a decreasing balanced basis for 0 : E and a balanced basis for 0 : F
-- and reformulate non-continuity in terms of these bases
rcases (nhds_basis_balanced π E).exists_antitone_subbasis with β¨b, bE1, bEβ©
simp only [_root_.id] at bE
have bE' : ... | 1,646 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 66 | 68 | theorem gauge_def' : gauge s x = sInf {r β Set.Ioi (0 : β) | rβ»ΒΉ β’ x β s} := by |
congrm sInf {r | ?_}
exact and_congr_right fun hr => mem_smul_set_iff_inv_smul_memβ hr.ne' _ _
| 1,647 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 86 | 89 | theorem exists_lt_of_gauge_lt (absorbs : Absorbent β s) (h : gauge s x < a) :
β b, 0 < b β§ b < a β§ x β b β’ s := by |
obtain β¨b, β¨hb, hxβ©, hbaβ© := exists_lt_of_csInf_lt absorbs.gauge_set_nonempty h
exact β¨b, hb, hba, hxβ©
| 1,647 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 95 | 99 | theorem gauge_zero : gauge s 0 = 0 := by |
rw [gauge_def']
by_cases h : (0 : E) β s
Β· simp only [smul_zero, sep_true, h, csInf_Ioi]
Β· simp only [smul_zero, sep_false, h, Real.sInf_empty]
| 1,647 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 103 | 110 | theorem gauge_zero' : gauge (0 : Set E) = 0 := by |
ext x
rw [gauge_def']
obtain rfl | hx := eq_or_ne x 0
Β· simp only [csInf_Ioi, mem_zero, Pi.zero_apply, eq_self_iff_true, sep_true, smul_zero]
Β· simp only [mem_zero, Pi.zero_apply, inv_eq_zero, smul_eq_zero]
convert Real.sInf_empty
exact eq_empty_iff_forall_not_mem.2 fun r hr => hr.2.elim (ne_of_gt hr... | 1,647 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 114 | 116 | theorem gauge_empty : gauge (β
: Set E) = 0 := by |
ext
simp only [gauge_def', Real.sInf_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
| 1,647 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 119 | 121 | theorem gauge_of_subset_zero (h : s β 0) : gauge s = 0 := by |
obtain rfl | rfl := subset_singleton_iff_eq.1 h
exacts [gauge_empty, gauge_zero']
| 1,647 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 129 | 131 | theorem gauge_neg (symmetric : β x β s, -x β s) (x : E) : gauge s (-x) = gauge s x := by |
have : β x, -x β s β x β s := fun x => β¨fun h => by simpa using symmetric _ h, symmetric xβ©
simp_rw [gauge_def', smul_neg, this]
| 1,647 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 134 | 135 | theorem gauge_neg_set_neg (x : E) : gauge (-s) (-x) = gauge s x := by |
simp_rw [gauge_def', smul_neg, neg_mem_neg]
| 1,647 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 138 | 139 | theorem gauge_neg_set_eq_gauge_neg (x : E) : gauge (-s) x = gauge s (-x) := by |
rw [β gauge_neg_set_neg, neg_neg]
| 1,647 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 142 | 145 | theorem gauge_le_of_mem (ha : 0 β€ a) (hx : x β a β’ s) : gauge s x β€ a := by |
obtain rfl | ha' := ha.eq_or_lt
Β· rw [mem_singleton_iff.1 (zero_smul_set_subset _ hx), gauge_zero]
Β· exact csInf_le gauge_set_bddBelow β¨ha', hxβ©
| 1,647 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 148 | 163 | theorem gauge_le_eq (hsβ : Convex β s) (hsβ : (0 : E) β s) (hsβ : Absorbent β s) (ha : 0 β€ a) :
{ x | gauge s x β€ a } = β (r : β) (_ : a < r), r β’ s := by |
ext x
simp_rw [Set.mem_iInter, Set.mem_setOf_eq]
refine β¨fun h r hr => ?_, fun h => le_of_forall_pos_lt_add fun Ξ΅ hΞ΅ => ?_β©
Β· have hr' := ha.trans_lt hr
rw [mem_smul_set_iff_inv_smul_memβ hr'.ne']
obtain β¨Ξ΄, Ξ΄_pos, hΞ΄r, hΞ΄β© := exists_lt_of_gauge_lt hsβ (h.trans_lt hr)
suffices (rβ»ΒΉ * Ξ΄) β’ Ξ΄β»ΒΉ β’ x β... | 1,647 |
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Convex.Gauge
#align_import analysis.locally_convex.abs_convex from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open NormedField Set
open NNReal Pointwis... | Mathlib/Analysis/LocallyConvex/AbsConvex.lean | 52 | 60 | theorem nhds_basis_abs_convex :
(π (0 : E)).HasBasis (fun s : Set E => s β π (0 : E) β§ Balanced π s β§ Convex β s) id := by |
refine
(LocallyConvexSpace.convex_basis_zero β E).to_hasBasis (fun s hs => ?_) fun s hs =>
β¨s, β¨hs.1, hs.2.2β©, rfl.subsetβ©
refine β¨convexHull β (balancedCore π s), ?_, convexHull_min (balancedCore_subset s) hs.2β©
refine β¨Filter.mem_of_superset (balancedCore_mem_nhds_zero hs.1) (subset_convexHull β _),... | 1,648 |
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Convex.Gauge
#align_import analysis.locally_convex.abs_convex from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open NormedField Set
open NNReal Pointwis... | Mathlib/Analysis/LocallyConvex/AbsConvex.lean | 65 | 74 | theorem nhds_basis_abs_convex_open :
(π (0 : E)).HasBasis (fun s => (0 : E) β s β§ IsOpen s β§ Balanced π s β§ Convex β s) id := by |
refine (nhds_basis_abs_convex π E).to_hasBasis ?_ ?_
Β· rintro s β¨hs_nhds, hs_balanced, hs_convexβ©
refine β¨interior s, ?_, interior_subsetβ©
exact
β¨mem_interior_iff_mem_nhds.mpr hs_nhds, isOpen_interior,
hs_balanced.interior (mem_interior_iff_mem_nhds.mpr hs_nhds), hs_convex.interiorβ©
rintro... | 1,648 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {π : Type*} [RCLike π] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 36 | 36 | theorem RCLike.norm_coe_norm {z : E} : β(βzβ : π)β = βzβ := by | simp
| 1,649 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {π : Type*} [RCLike π] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 43 | 45 | theorem norm_smul_inv_norm {x : E} (hx : x β 0) : β(βxββ»ΒΉ : π) β’ xβ = 1 := by |
have : βxβ β 0 := by simp [hx]
field_simp [norm_smul]
| 1,649 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {π : Type*} [RCLike π] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 49 | 52 | theorem norm_smul_inv_norm' {r : β} (r_nonneg : 0 β€ r) {x : E} (hx : x β 0) :
β((r : π) * (βxβ : π)β»ΒΉ) β’ xβ = r := by |
have : βxβ β 0 := by simp [hx]
field_simp [norm_smul, r_nonneg, rclike_simps]
| 1,649 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {π : Type*} [RCLike π] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 55 | 75 | theorem LinearMap.bound_of_sphere_bound {r : β} (r_pos : 0 < r) (c : β) (f : E ββ[π] π)
(h : β z β sphere (0 : E) r, βf zβ β€ c) (z : E) : βf zβ β€ c / r * βzβ := by |
by_cases z_zero : z = 0
Β· rw [z_zero]
simp only [LinearMap.map_zero, norm_zero, mul_zero]
exact le_rfl
set zβ := ((r : π) * (βzβ : π)β»ΒΉ) β’ z with hzβ
have norm_f_zβ : βf zββ β€ c := by
apply h
rw [mem_sphere_zero_iff_norm]
exact norm_smul_inv_norm' r_pos.le z_zero
have r_ne_zero : (r : οΏ½... | 1,649 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {π : Type*} [RCLike π] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 85 | 93 | theorem ContinuousLinearMap.opNorm_bound_of_ball_bound {r : β} (r_pos : 0 < r) (c : β)
(f : E βL[π] π) (h : β z β closedBall (0 : E) r, βf zβ β€ c) : βfβ β€ c / r := by |
apply ContinuousLinearMap.opNorm_le_bound
Β· apply div_nonneg _ r_pos.le
exact
(norm_nonneg _).trans
(h 0 (by simp only [norm_zero, mem_closedBall, dist_zero_left, r_pos.le]))
apply LinearMap.bound_of_ball_bound' r_pos
exact fun z hz => h z hz
| 1,649 |
import Mathlib.Algebra.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis
#align_import algebra.direct_sum.module from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v w uβ
namespace DirectSum
open DirectSum
section General
variable {... | Mathlib/Algebra/DirectSum/Module.lean | 164 | 168 | theorem linearEquivFunOnFintype_lof [Fintype ΞΉ] [DecidableEq ΞΉ] (i : ΞΉ) (m : M i) :
(linearEquivFunOnFintype R ΞΉ M) (lof R ΞΉ M i m) = Pi.single i m := by |
ext a
change (DFinsupp.equivFunOnFintype (lof R ΞΉ M i m)) a = _
convert _root_.congr_fun (DFinsupp.equivFunOnFintype_single i m) a
| 1,650 |
import Mathlib.Data.Finset.Order
import Mathlib.Algebra.DirectSum.Module
import Mathlib.RingTheory.FreeCommRing
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.SuppressCompilation
#align_import algebra.direct_limit from "leanprover-community/mathlib"@"f0c8bf9245297a... | Mathlib/Algebra/DirectLimit.lean | 164 | 170 | theorem lift_unique [IsDirected ΞΉ (Β· β€ Β·)] (F : DirectLimit G f ββ[R] P) (x) :
F x =
lift R ΞΉ G f (fun i => F.comp <| of R ΞΉ G f i)
(fun i j hij x => by rw [LinearMap.comp_apply, of_f]; rfl) x := by |
cases isEmpty_or_nonempty ΞΉ
Β· simp_rw [Subsingleton.elim x 0, _root_.map_zero]
Β· exact DirectLimit.induction_on x fun i x => by rw [lift_of]; rfl
| 1,651 |
import Mathlib.Init.Align
import Mathlib.Data.Fintype.Order
import Mathlib.Algebra.DirectLimit
import Mathlib.ModelTheory.Quotients
import Mathlib.ModelTheory.FinitelyGenerated
#align_import model_theory.direct_limit from "leanprover-community/mathlib"@"f53b23994ac4c13afa38d31195c588a1121d1860"
universe v w w' uβ... | Mathlib/ModelTheory/DirectLimit.lean | 67 | 76 | theorem coe_natLERec (m n : β) (h : m β€ n) :
(natLERec f' m n h : G' m β G' n) = Nat.leRecOn h (@fun k => f' k) := by |
obtain β¨k, rflβ© := Nat.exists_eq_add_of_le h
ext x
induction' k with k ih
Β· -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [natLERec, Nat.leRecOn_self, Embedding.refl_apply, Nat.leRecOn_self]
Β· -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [Nat... | 1,652 |
import Mathlib.ModelTheory.FinitelyGenerated
import Mathlib.ModelTheory.DirectLimit
import Mathlib.ModelTheory.Bundled
#align_import model_theory.fraisse from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"
universe u v w w'
open scoped FirstOrder
open Set CategoryTheory
namespace Fir... | Mathlib/ModelTheory/Fraisse.lean | 169 | 182 | theorem age.countable_quotient [h : Countable M] : (Quotient.mk' '' L.age M).Countable := by |
classical
refine (congr_arg _ (Set.ext <| Quotient.forall.2 fun N => ?_)).mp
(countable_range fun s : Finset M => β¦β¨closure L (s : Set M), inferInstanceβ©β§)
constructor
Β· rintro β¨s, hsβ©
use Bundled.of (closure L (s : Set M))
exact β¨β¨(fg_iff_structure_fg _).1 (fg_closure s.finite_toSet), β¨Substructur... | 1,653 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ΞΉ R M Ο : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 127 | 128 | theorem decompose_coe {i : ΞΉ} (x : β³ i) : decompose β³ (x : M) = DirectSum.of _ i x := by |
rw [β decompose_symm_of _, Equiv.apply_symm_apply]
| 1,654 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ΞΉ R M Ο : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 136 | 137 | theorem decompose_of_mem_same {x : M} {i : ΞΉ} (hx : x β β³ i) : (decompose β³ x i : M) = x := by |
rw [decompose_of_mem _ hx, DirectSum.of_eq_same, Subtype.coe_mk]
| 1,654 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ΞΉ R M Ο : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 140 | 142 | theorem decompose_of_mem_ne {x : M} {i j : ΞΉ} (hx : x β β³ i) (hij : i β j) :
(decompose β³ x j : M) = 0 := by |
rw [decompose_of_mem _ hx, DirectSum.of_eq_of_ne _ _ _ _ hij, ZeroMemClass.coe_zero]
| 1,654 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ΞΉ R M Ο : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 145 | 147 | theorem degree_eq_of_mem_mem {x : M} {i j : ΞΉ} (hxi : x β β³ i) (hxj : x β β³ j) (hx : x β 0) :
i = j := by |
contrapose! hx; rw [β decompose_of_mem_same β³ hxj, decompose_of_mem_ne β³ hxi hx]
| 1,654 |
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.Algebra.GradedMulAction
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algebra.Module.BigOperators
#align_import algebra.module.graded_module from "leanprover-community/mathlib"@"59cdeb0da2480abbc235b7e611ccd9a7e5603d7c"
section
open Dir... | Mathlib/Algebra/Module/GradedModule.lean | 99 | 102 | theorem smulAddMonoidHom_apply_of_of [DecidableEq ΞΉA] [DecidableEq ΞΉB] [GMonoid A] [Gmodule A M]
{i j} (x : A i) (y : M j) :
smulAddMonoidHom A M (DirectSum.of A i x) (of M j y) = of M (i +α΅₯ j) (GSMul.smul x y) := by |
simp [smulAddMonoidHom]
| 1,655 |
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.Algebra.DirectSum.Module
#align_import linear_algebra.direct_sum.tensor_product from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
suppress_compilation
universe u vβ vβ wβ wβ' wβ wβ'
section Ring
namespace TensorProduct
... | Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean | 150 | 153 | theorem directSum_lof_tmul_lof (iβ : ΞΉβ) (mβ : Mβ iβ) (iβ : ΞΉβ) (mβ : Mβ iβ) :
TensorProduct.directSum R S Mβ Mβ (DirectSum.lof S ΞΉβ Mβ iβ mβ ββ DirectSum.lof R ΞΉβ Mβ iβ mβ) =
DirectSum.lof S (ΞΉβ Γ ΞΉβ) (fun i => Mβ i.1 β[R] Mβ i.2) (iβ, iβ) (mβ ββ mβ) := by |
simp [TensorProduct.directSum]
| 1,656 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 102 | 107 | theorem finsuppLeft_apply (t : (ΞΉ ββ M) β[R] N) (i : ΞΉ) :
finsuppLeft R M N ΞΉ t i = rTensor N (Finsupp.lapply i) t := by |
induction t using TensorProduct.induction_on with
| zero => simp
| tmul f n => simp only [finsuppLeft_apply_tmul_apply, rTensor_tmul, Finsupp.lapply_apply]
| add x y hx hy => simp [map_add, hx, hy]
| 1,657 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 137 | 142 | theorem finsuppRight_apply (t : M β[R] (ΞΉ ββ N)) (i : ΞΉ) :
finsuppRight R M N ΞΉ t i = lTensor M (Finsupp.lapply i) t := by |
induction t using TensorProduct.induction_on with
| zero => simp
| tmul m f => simp [finsuppRight_apply_tmul_apply]
| add x y hx hy => simp [map_add, hx, hy]
| 1,657 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 256 | 259 | theorem finsuppTensorFinsupp_single (i : ΞΉ) (m : M) (k : ΞΊ) (n : N) :
finsuppTensorFinsupp R S M N ΞΉ ΞΊ (Finsupp.single i m ββ Finsupp.single k n) =
Finsupp.single (i, k) (m ββ n) := by |
simp [finsuppTensorFinsupp]
| 1,657 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 263 | 277 | theorem finsuppTensorFinsupp_apply (f : ΞΉ ββ M) (g : ΞΊ ββ N) (i : ΞΉ) (k : ΞΊ) :
finsuppTensorFinsupp R S M N ΞΉ ΞΊ (f ββ g) (i, k) = f i ββ g k := by |
apply Finsupp.induction_linear f
Β· simp
Β· intro fβ fβ hfβ hfβ
simp [add_tmul, hfβ, hfβ]
intro i' m
apply Finsupp.induction_linear g
Β· simp
Β· intro gβ gβ hgβ hgβ
simp [tmul_add, hgβ, hgβ]
intro k' n
classical
simp_rw [finsuppTensorFinsupp_single, Finsupp.single_apply, Prod.mk.inj_iff, ite_an... | 1,657 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 293 | 295 | theorem finsuppTensorFinsuppLid_apply_apply (f : ΞΉ ββ R) (g : ΞΊ ββ N) (a : ΞΉ) (b : ΞΊ) :
finsuppTensorFinsuppLid R N ΞΉ ΞΊ (f ββ[R] g) (a, b) = f a β’ g b := by |
simp [finsuppTensorFinsuppLid]
| 1,657 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 298 | 301 | theorem finsuppTensorFinsuppLid_single_tmul_single (a : ΞΉ) (b : ΞΊ) (r : R) (n : N) :
finsuppTensorFinsuppLid R N ΞΉ ΞΊ (Finsupp.single a r ββ[R] Finsupp.single b n) =
Finsupp.single (a, b) (r β’ n) := by |
simp [finsuppTensorFinsuppLid]
| 1,657 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 315 | 317 | theorem finsuppTensorFinsuppRid_apply_apply (f : ΞΉ ββ M) (g : ΞΊ ββ R) (a : ΞΉ) (b : ΞΊ) :
finsuppTensorFinsuppRid R M ΞΉ ΞΊ (f ββ[R] g) (a, b) = g b β’ f a := by |
simp [finsuppTensorFinsuppRid]
| 1,657 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 320 | 323 | theorem finsuppTensorFinsuppRid_single_tmul_single (a : ΞΉ) (b : ΞΊ) (m : M) (r : R) :
finsuppTensorFinsuppRid R M ΞΉ ΞΊ (Finsupp.single a m ββ[R] Finsupp.single b r) =
Finsupp.single (a, b) (r β’ m) := by |
simp [finsuppTensorFinsuppRid]
| 1,657 |
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Types.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.CategoryTheory.Linear.LinearFunctor
#align_import algebra.category.Module.adjunctions from "leanpr... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 89 | 109 | theorem ΞΌ_natural {X Y X' Y' : Type u} (f : X βΆ Y) (g : X' βΆ Y') :
((free R).map f β (free R).map g) β« (ΞΌ R Y Y').hom = (ΞΌ R X X').hom β« (free R).map (f β g) := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply Finsupp.lhom_ext'
intro x
apply LinearMap.ext_ring
apply Finsupp.lhom_ext'
intro x'
apply LinearMap.ext_ring
apply Finsupp.ext
intro β¨y, y'β©
-- Porting note (#10934): used to be dsimp [ΞΌ]
change (finsuppTensorFinsupp' R Y Y')
... | 1,658 |
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Types.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.CategoryTheory.Linear.LinearFunctor
#align_import algebra.category.Module.adjunctions from "leanpr... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 112 | 129 | theorem left_unitality (X : Type u) :
(Ξ»_ ((free R).obj X)).hom =
(Ξ΅ R β π ((free R).obj X)) β« (ΞΌ R (π_ (Type u)) X).hom β« map (free R).obj (Ξ»_ X).hom := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply LinearMap.ext_ring
apply Finsupp.lhom_ext'
intro x
apply LinearMap.ext_ring
apply Finsupp.ext
intro x'
-- Porting note (#10934): used to be dsimp [Ξ΅, ΞΌ]
let q : X ββ R := ((Ξ»_ (of R (X ββ R))).hom) (1 ββ[R] Finsupp.single x 1)
cha... | 1,658 |
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Types.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.CategoryTheory.Linear.LinearFunctor
#align_import algebra.category.Module.adjunctions from "leanpr... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 132 | 149 | theorem right_unitality (X : Type u) :
(Ο_ ((free R).obj X)).hom =
(π ((free R).obj X) β Ξ΅ R) β« (ΞΌ R X (π_ (Type u))).hom β« map (free R).obj (Ο_ X).hom := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply Finsupp.lhom_ext'
intro x
apply LinearMap.ext_ring
apply LinearMap.ext_ring
apply Finsupp.ext
intro x'
-- Porting note (#10934): used to be dsimp [Ξ΅, ΞΌ]
let q : X ββ R := ((Ο_ (of R (X ββ R))).hom) (Finsupp.single x 1 ββ[R] 1)
cha... | 1,658 |
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.BilinearForm.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Algebra.Bilinear
open LinearMap (BilinForm)
universe u v w
variable {R : Type*} {M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]
variable {Rβ : Type*} {Mβ :... | Mathlib/LinearAlgebra/BilinearForm/Hom.lean | 90 | 92 | theorem sum_apply {Ξ±} (t : Finset Ξ±) (B : Ξ± β BilinForm R M) (v w : M) :
(β i β t, B i) v w = β i β t, B i v w := by |
simp only [coeFn_sum, Finset.sum_apply]
| 1,659 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import linear_algebra.multilinear.basis from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
open MultilinearMap
variable {R : Type*} {ΞΉ : Type*} {n : β} {M : Fin n β Type*} {Mβ : Type*} {Mβ : Type*... | Mathlib/LinearAlgebra/Multilinear/Basis.lean | 32 | 49 | theorem Basis.ext_multilinear_fin {f g : MultilinearMap R M Mβ} {ΞΉβ : Fin n β Type*}
(e : β i, Basis (ΞΉβ i) R (M i))
(h : β v : β i, ΞΉβ i, (f fun i => e i (v i)) = g fun i => e i (v i)) : f = g := by |
induction' n with m hm
Β· ext x
convert h finZeroElim
Β· apply Function.LeftInverse.injective uncurry_curryLeft
refine Basis.ext (e 0) ?_
intro i
apply hm (Fin.tail e)
intro j
convert h (Fin.cons i j)
iterate 2
rw [curryLeft_apply]
congr 1 with x
refine Fin.cases rfl (... | 1,660 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import linear_algebra.multilinear.basis from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
open MultilinearMap
variable {R : Type*} {ΞΉ : Type*} {n : β} {M : Fin n β Type*} {Mβ : Type*} {Mβ : Type*... | Mathlib/LinearAlgebra/Multilinear/Basis.lean | 56 | 61 | theorem Basis.ext_multilinear [Finite ΞΉ] {f g : MultilinearMap R (fun _ : ΞΉ => Mβ) Mβ} {ΞΉβ : Type*}
(e : Basis ΞΉβ R Mβ) (h : β v : ΞΉ β ΞΉβ, (f fun i => e (v i)) = g fun i => e (v i)) : f = g := by |
cases nonempty_fintype ΞΉ
exact
(domDomCongr_eq_iff (Fintype.equivFin ΞΉ) f g).mp
(Basis.ext_multilinear_fin (fun _ => e) fun i => h (i β _))
| 1,660 |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ΞΉ : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 55 | 57 | theorem stdBasis_apply' (i i' : ΞΉ) : (stdBasis R (fun _x : ΞΉ => R) i) 1 i' = ite (i = i') 1 0 := by |
rw [LinearMap.stdBasis_apply, Function.update_apply, Pi.zero_apply]
congr 1; rw [eq_iff_iff, eq_comm]
| 1,661 |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ΞΉ : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 73 | 77 | theorem stdBasis_eq_pi_diag (i : ΞΉ) : stdBasis R Ο i = pi (diag i) := by |
ext x j
-- Porting note: made types explicit
convert (update_apply (R := R) (Ο := Ο) (ΞΉ := ΞΉ) 0 x i j _).symm
rfl
| 1,661 |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ΞΉ : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 84 | 85 | theorem proj_comp_stdBasis (i j : ΞΉ) : (proj i).comp (stdBasis R Ο j) = diag j i := by |
rw [stdBasis_eq_pi_diag, proj_pi]
| 1,661 |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ΞΉ : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 96 | 103 | theorem iSup_range_stdBasis_le_iInf_ker_proj (I J : Set ΞΉ) (h : Disjoint I J) :
β¨ i β I, range (stdBasis R Ο i) β€ β¨
i β J, ker (proj i : (β i, Ο i) ββ[R] Ο i) := by |
refine iSup_le fun i => iSup_le fun hi => range_le_iff_comap.2 ?_
simp only [β ker_comp, eq_top_iff, SetLike.le_def, mem_ker, comap_iInf, mem_iInf]
rintro b - j hj
rw [proj_stdBasis_ne R Ο j i, zero_apply]
rintro rfl
exact h.le_bot β¨hi, hjβ©
| 1,661 |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ΞΉ : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 123 | 129 | theorem iSup_range_stdBasis_eq_iInf_ker_proj {I J : Set ΞΉ} (hd : Disjoint I J)
(hu : Set.univ β I βͺ J) (hI : Set.Finite I) :
β¨ i β I, range (stdBasis R Ο i) = β¨
i β J, ker (proj i : (β i, Ο i) ββ[R] Ο i) := by |
refine le_antisymm (iSup_range_stdBasis_le_iInf_ker_proj _ _ _ _ hd) ?_
have : Set.univ β βhI.toFinset βͺ J := by rwa [hI.coe_toFinset]
refine le_trans (iInf_ker_proj_le_iSup_range_stdBasis R Ο this) (iSup_mono fun i => ?_)
rw [Set.Finite.mem_toFinset]
| 1,661 |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ΞΉ : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 132 | 137 | theorem iSup_range_stdBasis [Finite ΞΉ] : β¨ i, range (stdBasis R Ο i) = β€ := by |
cases nonempty_fintype ΞΉ
convert top_unique (iInf_emptyset.ge.trans <| iInf_ker_proj_le_iSup_range_stdBasis R Ο _)
Β· rename_i i
exact ((@iSup_pos _ _ _ fun _ => range <| stdBasis R Ο i) <| Finset.mem_univ i).symm
Β· rw [Finset.coe_univ, Set.union_empty]
| 1,661 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 127 | 131 | theorem monomial_def [DecidableEq Ο] (n : Ο ββ β) :
(monomial R n) = LinearMap.stdBasis R (fun _ β¦ R) n := by |
rw [monomial]
-- unify the `Decidable` arguments
convert rfl
| 1,662 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 134 | 140 | theorem coeff_monomial [DecidableEq Ο] (m n : Ο ββ β) (a : R) :
coeff R m (monomial R n a) = if m = n then a else 0 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [coeff, monomial_def, LinearMap.proj_apply (i := m)]
dsimp only
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [LinearMap.stdBasis_apply, Function.update_apply, Pi.zero_apply]
| 1,662 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 144 | 147 | theorem coeff_monomial_same (n : Ο ββ β) (a : R) : coeff R n (monomial R n a) = a := by |
classical
rw [monomial_def]
exact LinearMap.stdBasis_same R (fun _ β¦ R) n a
| 1,662 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 150 | 153 | theorem coeff_monomial_ne {m n : Ο ββ β} (h : m β n) (a : R) : coeff R m (monomial R n a) = 0 := by |
classical
rw [monomial_def]
exact LinearMap.stdBasis_ne R (fun _ β¦ R) _ _ h a
| 1,662 |
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.Data.Finsupp.Interval
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
namespace MvPowerSeries
open Finsupp
variable {Ο R : Type*}
section Trunc
variable [CommSemiring R] (n : Ο ββ β)
def truncFun (Ο : MvPowerSeries Ο R) : MvPol... | Mathlib/RingTheory/MvPowerSeries/Trunc.lean | 43 | 46 | theorem coeff_truncFun (m : Ο ββ β) (Ο : MvPowerSeries Ο R) :
(truncFun n Ο).coeff m = if m < n then coeff R m Ο else 0 := by |
classical
simp [truncFun, MvPolynomial.coeff_sum]
| 1,663 |
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.Data.Finsupp.Interval
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
namespace MvPowerSeries
open Finsupp
variable {Ο R : Type*}
section Trunc
variable [CommSemiring R] (n : Ο ββ β)
def truncFun (Ο : MvPowerSeries Ο R) : MvPol... | Mathlib/RingTheory/MvPowerSeries/Trunc.lean | 71 | 73 | theorem coeff_trunc (m : Ο ββ β) (Ο : MvPowerSeries Ο R) :
(trunc R n Ο).coeff m = if m < n then coeff R m Ο else 0 := by |
classical simp [trunc, coeff_truncFun]
| 1,663 |
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.StdBasis
#align_import linear_algebra.finsupp_vector_space from "leanprover-community/mathlib"@"59628387770d82eb6f6dd7b7107308aa2509ec95"
noncomputable section
open Set LinearMap Submodule
open scoped Cardinal
universe u v w
namespace Finsupp
... | Mathlib/LinearAlgebra/FinsuppVectorSpace.lean | 34 | 51 | theorem linearIndependent_single {Ο : ΞΉ β Type*} {f : β ΞΉ, Ο ΞΉ β M}
(hf : β i, LinearIndependent R (f i)) :
LinearIndependent R fun ix : Ξ£i, Ο i => single ix.1 (f ix.1 ix.2) := by |
apply @linearIndependent_iUnion_finite R _ _ _ _ ΞΉ Ο fun i x => single i (f i x)
Β· intro i
have h_disjoint : Disjoint (span R (range (f i))) (ker (lsingle i)) := by
rw [ker_lsingle]
exact disjoint_bot_right
apply (hf i).map h_disjoint
Β· intro i t _ hit
refine (disjoint_lsingle_lsingle {i}... | 1,664 |
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.StdBasis
#align_import linear_algebra.finsupp_vector_space from "leanprover-community/mathlib"@"59628387770d82eb6f6dd7b7107308aa2509ec95"
noncomputable section
open Set LinearMap Submodule
open scoped Cardinal
universe u v w
namespace Finsupp
... | Mathlib/LinearAlgebra/FinsuppVectorSpace.lean | 161 | 164 | theorem _root_.Finset.sum_single_ite [Fintype n] (a : R) (i : n) :
(β x : n, Finsupp.single x (if i = x then a else 0)) = Finsupp.single i a := by |
simp only [apply_ite (Finsupp.single _), Finsupp.single_zero, Finset.sum_ite_eq,
if_pos (Finset.mem_univ _)]
| 1,664 |
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.StdBasis
#align_import linear_algebra.finsupp_vector_space from "leanprover-community/mathlib"@"59628387770d82eb6f6dd7b7107308aa2509ec95"
noncomputable section
open Set LinearMap Submodule
open scoped Cardinal
universe u v w
namespace Finsupp
... | Mathlib/LinearAlgebra/FinsuppVectorSpace.lean | 167 | 170 | theorem equivFun_symm_stdBasis [Finite n] (b : Basis n R M) (i : n) :
b.equivFun.symm (LinearMap.stdBasis R (fun _ => R) i 1) = b i := by |
cases nonempty_fintype n
simp
| 1,664 |
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Module.Projective
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.Data.Finsupp.Basic
#align_import algebra.category.Module.projective from "leanprover-community/mathlib"@"201a3f... | Mathlib/Algebra/Category/ModuleCat/Projective.lean | 31 | 41 | theorem IsProjective.iff_projective {R : Type u} [Ring R] {P : Type max u v} [AddCommGroup P]
[Module R P] : Module.Projective R P β Projective (ModuleCat.of R P) := by |
refine β¨fun h => ?_, fun h => ?_β©
Β· letI : Module.Projective R (ModuleCat.of R P) := h
exact β¨fun E X epi => Module.projective_lifting_property _ _
((ModuleCat.epi_iff_surjective _).mp epi)β©
Β· refine Module.Projective.of_lifting_property.{u,v} ?_
intro E X mE mX sE sX f g s
haveI : Epi (βf) := ... | 1,665 |
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.RepresentationTheory.Rep
#align_import representation_theory.group_cohomology.resolution from "leanprover-community/mathlib"@"cec81510e48e579bde6acd8568c06a87a... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 108 | 124 | theorem actionDiagonalSucc_hom_apply {G : Type u} [Group G] {n : β} (f : Fin (n + 1) β G) :
(actionDiagonalSucc G n).hom.hom f = (f 0, fun i => (f (Fin.castSucc i))β»ΒΉ * f i.succ) := by |
induction' n with n hn
Β· exact Prod.ext rfl (funext fun x => Fin.elim0 x)
Β· refine Prod.ext rfl (funext fun x => ?_)
/- Porting note (#11039): broken proof was
Β· dsimp only [actionDiagonalSucc]
simp only [Iso.trans_hom, comp_hom, types_comp_apply, diagonalSucc_hom_hom,
leftRegularTensorIso_hom_... | 1,666 |
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.RepresentationTheory.Rep
#align_import representation_theory.group_cohomology.resolution from "leanprover-community/mathlib"@"cec81510e48e579bde6acd8568c06a87a... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 128 | 153 | theorem actionDiagonalSucc_inv_apply {G : Type u} [Group G] {n : β} (g : G) (f : Fin n β G) :
(actionDiagonalSucc G n).inv.hom (g, f) = (g β’ Fin.partialProd f : Fin (n + 1) β G) := by |
revert g
induction' n with n hn
Β· intro g
funext (x : Fin 1)
simp only [Subsingleton.elim x 0, Pi.smul_apply, Fin.partialProd_zero, smul_eq_mul, mul_one]
rfl
Β· intro g
/- Porting note (#11039): broken proof was
ext
dsimp only [actionDiagonalSucc]
simp only [Iso.trans_inv, comp_hom, hn, ... | 1,666 |
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.FinsuppVectorSpace
#align_import linear_algebra.tensor_product_basis from "leanprover-community/mathlib"@"f784cc6142443d9ee623a20788c282112c322081"
noncomputable section
open Set LinearMap Submodule
section CommSemiring
variable {R : T... | Mathlib/LinearAlgebra/TensorProduct/Basis.lean | 39 | 41 | theorem Basis.tensorProduct_apply (b : Basis ΞΉ R M) (c : Basis ΞΊ R N) (i : ΞΉ) (j : ΞΊ) :
Basis.tensorProduct b c (i, j) = b i ββ c j := by |
simp [Basis.tensorProduct]
| 1,667 |
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.FinsuppVectorSpace
#align_import linear_algebra.tensor_product_basis from "leanprover-community/mathlib"@"f784cc6142443d9ee623a20788c282112c322081"
noncomputable section
open Set LinearMap Submodule
section CommSemiring
variable {R : T... | Mathlib/LinearAlgebra/TensorProduct/Basis.lean | 44 | 46 | theorem Basis.tensorProduct_apply' (b : Basis ΞΉ R M) (c : Basis ΞΊ R N) (i : ΞΉ Γ ΞΊ) :
Basis.tensorProduct b c i = b i.1 ββ c i.2 := by |
simp [Basis.tensorProduct]
| 1,667 |
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