mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : Fintype ι
l : (i : ι) → G i
i : ι
v : G i
⊢ ⟪(V i) v, ∑ j, (V j) (l j)⟫ = ⟪v, l i⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
ι' : Type u_5
v : ι → E
hv : Orthonormal 𝕜 v
f : ι' → ι
hf : Function.Injective f
⊢ Orthonormal 𝕜 (v ∘ f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Function.Injective v
⊢ Orthonormal 𝕜 Subtype.val ↔ Orthonormal 𝕜 v

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
l₁ l₂ : (i : ι) → G i
s : Finset ι
⊢ ⟪∑ i ∈ s, (V i) (l₁ i), ∑ j ∈ s, (V j) (l₂ j)⟫ = ∑ j ∈ s, ∑ i ∈ s, ⟪(V i) (l₁ i), (V j) (l₂ j)⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
s : Set ι
i : ι
hi : i ∉ s
l : ι →₀ 𝕜
hl : l ∈ supported 𝕜 𝕜 s
⊢ ⟪(linearCombination 𝕜 v) l, v i⟫ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
l₁ l₂ : (i : ι) → G i
s : Finset ι
⊢ ∑ j ∈ s, ∑ i ∈ s, ⟪(V i) (l₁ i), (V j) (l₂ j)⟫ = ∑ j ∈ s, ∑ i ∈ s, if i = j then ⟪(V i) (l₁ i), (V j) (l₂ j)⟫ else 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
l₁ l₂ : (i : ι) → G i
s : Finset ι
⊢ (∑ j ∈ s, ∑ i ∈ s, if i = j then ⟪(V i) (l₁ i), (V j) (l₂ j)⟫ else 0) = ∑ i ∈ s, ⟪l₁ i, l₂ i⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: V : Type u_4
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
⊢ (∀ (x : V), ⟪T x, x⟫_ℂ = 0) ↔ T = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
l₁ l₂ : (i : ι) → G i
s : Finset ι
⊢ ⟪∑ i ∈ s, (V i) (l₁ i), ∑ j ∈ s, (V j) (l₂ j)⟫ = ∑ i ∈ s, ⟪l₁ i, l₂ i⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h✝ : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : ↑K
this : ⟪u - v, ↑w - v⟫_ℝ ≤ 0
⊢ ‖u - v‖ * ‖u - v‖ ≤ ‖u - v‖ * ‖u - v‖ - 2 * ⟪u - v, ↑w - v⟫_ℝ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
l : (i : ι) → G i
s : Finset ι
⊢ ↑‖∑ i ∈ s, (V i) (l i)‖ ^ 2 = ∑ i ∈ s, ↑‖l i‖ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: V : Type u_4
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
S T : V →ₗ[ℂ] V
⊢ (∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S = T

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h✝ : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : ↑K
this : ⟪u - v, ↑w - v⟫_ℝ ≤ 0
⊢ ‖u - v‖ * ‖u - v‖ - 2 * ⟪u - v, ↑w - v⟫_ℝ ≤ ‖u - v‖ ^ 2 - 2 * ⟪u - v, ↑w - v⟫_ℝ + ‖↑w - v‖ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h✝ : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : ↑K
this : ⟪u - v, ↑w - v⟫_ℝ ≤ 0
⊢ u - v - (↑w - v) = u - ↑w

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
E' : Type u_7
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
f : E →ₗᵢ[𝕜] E'
x y : E
⊢ ⟪f x, f y⟫ = ⟪x, y⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h✝ : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : ↑K
this : ⟪u - v, ↑w - v⟫_ℝ ≤ 0
⊢ ‖u - v - (↑w - v)‖ ^ 2 = ‖u - ↑w‖ * ‖u - ↑w‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
l : (i : ι) → G i
s : Finset ι
⊢ ‖∑ i ∈ s, (V i) (l i)‖ ^ 2 = ∑ i ∈ s, ‖l i‖ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v w : ι → E
hv : Orthonormal 𝕜 v
hw : ∀ (i : ι), w i = v i ∨ w i = -v i
⊢ Orthonormal 𝕜 w

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
⊢ Orthonormal 𝕜 fun x ↦ ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
E' : Type u_7
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
f : E ≃ₗᵢ[𝕜] E'
x : E
y : E'
⊢ ⟪f x, y⟫ = ⟪x, f.symm y⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
h : IsComplete ↑K
⊢ ∀ (u : E), ∃ v ∈ K, ‖u - v‖ = ⨅ w, ‖u - ↑w‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : RCLike 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
ι : Type u_4
ι' : Type u_5
ι'' : Type u_6
E' : Type u_7
inst✝³ : SeminormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
E'' : Type u_8
inst✝¹ : SeminormedAddCommGroup E''
inst✝ : InnerProductSpace 𝕜 E''
f : E →ₗ[𝕜] E'
h : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : E
⊢ ‖f x‖ = ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : E
⊢ ‖((innerₛₗ 𝕜) x) y‖ ≤ 1 * ‖x‖ * ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
⊢ (innerSL ℝ).flip = innerSL ℝ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
⊢ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
E' : Type u_4
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
f : E →L[𝕜] E'
v : E'
⊢ 0 ≤ ‖f‖ * ‖v‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : F
hw : w ∈ K
w' : F := w + v
this : w' ∈ K
h₁ : ⟪u - v, w' - v⟫_ℝ ≤ 0
⊢ w' - v = w

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
E' : Type u_4
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
f : E →L[𝕜] E'
v : E'
x : E
h₁ : ‖f x‖ ≤ ‖f‖ * ‖x‖
h₂ : ‖⟪v, f x⟫‖ ≤ ‖v‖ * ‖f x‖
⊢ ‖v‖ * ‖f x‖ ≤ ‖v‖ * (‖f‖ * ‖x‖)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : F
hw : w ∈ K
⊢ ⟪u - v, w⟫_ℝ ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
η : Type u_5
s : η → Set E
hs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
h : ∀ (i : η), Orthonormal 𝕜 fun x ↦ ↑x
⊢ Orthonormal 𝕜 fun x ↦ ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
E' : Type u_4
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
f : E →L[𝕜] E'
v : E'
x : E
h₁ : ‖f x‖ ≤ ‖f‖ * ‖x‖
h₂ : ‖⟪v, f x⟫‖ ≤ ‖v‖ * ‖f x‖
⊢ ‖v‖ * (‖f‖ * ‖x‖) = ‖f‖ * ‖v‖ * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : F
hw : w ∈ K
le : ⟪u - v, w⟫_ℝ ≤ 0
w'' : F := -w + v
this : w'' ∈ K
h₁ : ⟪u - v, w'' - v⟫_ℝ ≤ 0
⊢ w'' - v = -w

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
E' : Type u_4
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
f : E →L[𝕜] E'
v : E'
⊢ ‖(toSesqForm f) v‖ ≤ ‖f‖ * ‖v‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s : Set (Set E)
hs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s
h : ∀ a ∈ s, Orthonormal 𝕜 fun x ↦ ↑x
⊢ ∀ (i : ↑s), Orthonormal 𝕜 fun x ↦ ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : F
hw : w ∈ K
le : ⟪u - v, w⟫_ℝ ≤ 0
⊢ ⟪u - v, w⟫_ℝ ≥ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
α : ι → Type u_6
v_family : (i : ι) → α i → G i
hv_family : ∀ (i : ι), Orthonormal 𝕜 (v_family i)
i : ι
v w : α i
hvw : ⟨i, v⟩ ≠ ⟨i, w⟩
h : v = w
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s : Set (Set E)
hs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s
h : ∀ a ∈ s, Orthonormal 𝕜 fun x ↦ ↑x
⊢ Orthonormal 𝕜 fun x ↦ ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ → ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
h : 0 < ‖x‖
⊢ ‖x‖ * ‖x‖ = ‖⟪x, x⟫‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ ‖(innerSL 𝕜) x‖ = ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h : ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0
⊢ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
⊢ ∀ (x : E), ‖(innerSL 𝕜) x‖ ≤ 1 * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: G : Type u_4
inst✝¹ : SeminormedAddCommGroup G
inst✝ : InnerProductSpace ℝ G
f : G ≃ₗᵢ[ℝ] ℂ
x y : G
⊢ ⟪x, y⟫_ℝ = (f y * (starRingEnd ℂ) (f x)).re

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
⊢ (∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0) → ‖u - v‖ = ⨅ w, ‖u - ↑w‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
x : E
c : 𝕜
⊢ T.reApplyInnerSelf (c • x) = ‖c‖ ^ 2 * T.reApplyInnerSelf x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
F : Type u_6
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
⊢ ‖((rankOne 𝕜) x) y‖ = ‖x‖ * ‖y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
α : ι → Type u_6
v_family : (i : ι) → α i → G i
hv_family : ∀ (i : ι), Orthonormal 𝕜 (v_family i)
⊢ Orthonormal 𝕜 fun a ↦ (V a.fst) (v_family a.fst a.snd)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
F : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
G : Type u_8
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
x : E
y : F
f : E →L[𝕜] G
⊢ f ∘SL ((rankOne 𝕜) x) y = ((rankOne 𝕜) (f x)) y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
F : Type u_6
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : F
hx : ‖x‖ = 1
⊢ IsIdempotentElem (((rankOne 𝕜) x) x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
H : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
A : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0
w : E
hw : w ∈ K
⊢ re ⟪u - v, -I • w⟫_𝕜 = re (-I * ⟪u - v, w⟫_𝕜)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
F : Type u_6
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : F
⊢ ((rankOne 𝕜) x) 1 = toSpanSingleton 𝕜 x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
H : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
A : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0
w : E
hw : w ∈ K
⊢ re (-I * ⟪u - v, w⟫_𝕜) = im ⟪u - v, w⟫_𝕜

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
F : Type u_6
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : F
⊢ ((rankOne 𝕜) 1) x = (innerSL 𝕜) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x : E
y z : F
w : G
⊢ ((rankOne 𝕜) x) y ∘SL ((rankOne 𝕜) z) w = ⟪y, z⟫_𝕜 • ((rankOne 𝕜) x) w

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
F : Type u_6
G : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x : F
y z : G
w : F
⊢ ⟪(((rankOne 𝕜) x) y) z, w⟫_𝕜 = ⟪z, y⟫_𝕜 * ⟪x, w⟫_𝕜

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
F : Type u_6
G : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x y : F
z w : G
⊢ ⟪x, (((rankOne 𝕜) y) z) w⟫_𝕜 = ⟪x, y⟫_𝕜 * ⟪z, w⟫_𝕜

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
H : ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0
⊢ ∀ w ∈ K', ⟪u - v, w⟫_ℝ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
⊢ ((rankOne 𝕜) x) y = 0 ↔ x = 0 ∨ y = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s : Set E
hs : Orthonormal 𝕜 Subtype.val
⊢ ∃ w ⊇ s, Orthonormal 𝕜 Subtype.val ∧ ∀ u ⊇ w, Orthonormal 𝕜 Subtype.val → u = w

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ ((rankOne 𝕜) x) y ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
i : ι
⊢ v i ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
n : ℕ
⊢ 2 ^ n ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
E' : Type u_7
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
v : ι → E
f : E →ₗᵢ[𝕜] E'
⊢ Orthonormal 𝕜 (⇑f ∘ v) ↔ Orthonormal 𝕜 v

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
inst✝² : RCLike 𝕜
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : F
hx : x ≠ 0
⊢ IsIdempotentElem (((rankOne 𝕜) x) x) ↔ ‖x‖ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
E' : Type u_7
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
v : ι → E
hv : Orthonormal 𝕜 v
f : E →ₗᵢ[𝕜] E'
⊢ Orthonormal 𝕜 (⇑f ∘ v)

Tactic:
by

Goal state after: