mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
f_cont : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ f a = (cfcₙHom ha) (mkD ((σₙ R a).domRestrict f) 0)

Tactic:
·

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
f_cont : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ f a = (cfcₙHom ha) (mkD ((σₙ R a).domRestrict f) 0)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, 0 ≤ f x
hf : ContinuousOn f (spectrum R a)
⊢ 0 ≤ cfc f a

pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, 0 ≤ f x
hf : ¬ContinuousOn f (spectrum R a)
⊢ 0 ≤ cfc f a

Tactic:
·

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, 0 ≤ f x
hf : ContinuousOn f (spectrum R a)
⊢ 0 ≤ cfc f a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ContinuousOn f (σₙ R a) ∧ f 0 = 0 ∧ p a
⊢ P (cfcₙ f a)

pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬(ContinuousOn f (σₙ R a) ∧ f 0 = 0 ∧ p a)
⊢ P (cfcₙ f a)

Tactic:
·

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ContinuousOn f (σₙ R a) ∧ f 0 = 0 ∧ p a
⊢ P (cfcₙ f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, 0 ≤ f x
hf : ¬ContinuousOn f (spectrum R a)
⊢ 0 ≤ cfc f a

Tactic:
·

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, 0 ≤ f x
hf : ¬ContinuousOn f (spectrum R a)
⊢ 0 ≤ cfc f a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬(ContinuousOn f (σₙ R a) ∧ f 0 = 0 ∧ p a)
⊢ P (cfcₙ f a)

Tactic:
·

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬(ContinuousOn f (σₙ R a) ∧ f 0 = 0 ∧ p a)
⊢ P (cfcₙ f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 0
hf : ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0

pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 0
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0

Tactic:
·

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 0
hf : ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬ContinuousOn f (σₙ R a)
⊢ P (cfcₙ f a)

pp: case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬f 0 = 0
⊢ P (cfcₙ f a)

pp: case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬p a
⊢ P (cfcₙ f a)

Tactic:
·

Goal state after:
case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬ContinuousOn f (σₙ R a)
⊢ P (cfcₙ f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 0
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0

Tactic:
·

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 0
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0
Module: Mathlib.Topology.Algebra.StarSubalgebra

Goal state before:
pp: case mem.inl
R : Type u_1
A : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Semiring A
inst✝⁴ : StarRing A
inst✝³ : IsSemitopologicalSemiring A
inst✝² : ContinuousStar A
inst✝¹ : Algebra R A
inst✝ : StarModule R A
y✝ y u : A
hy✝ : y✝ ∈ elemental R u
P : (u_1 : A) → u_1 ∈ elemental R u → Prop
self : P u ⋯
star_self : P (star u) ⋯
algebraMap : ∀ (r : R), P ((Algebra.algebraMap R A) r) ⋯
add : ∀ (u_1 : A) (hu : u_1 ∈ elemental R u) (v : A) (hv : v ∈ elemental R u), P u_1 hu → P v hv → P (u_1 + v) ⋯
mul : ∀ (u_1 : A) (hu : u_1 ∈ elemental R u) (v : A) (hv : v ∈ elemental R u), P u_1 hu → P v hv → P (u_1 * v) ⋯
closure :
  ∀ (s : Set A) (hs : s ⊆ ↑(elemental R u)),
    (∀ (u : A) (hu : u ∈ s), P u ⋯) → ∀ (v : A) (hv : v ∈ closure[inst✝⁶] s), P v ⋯
hu : u ∈ {u} ∪ star {u}
hy : u ∈ ↑(adjoin R {u})
⊢ P u ⋯

pp: case mem.inr
R : Type u_1
A : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Semiring A
inst✝⁴ : StarRing A
inst✝³ : IsSemitopologicalSemiring A
inst✝² : ContinuousStar A
inst✝¹ : Algebra R A
inst✝ : StarModule R A
x y✝ : A
hy✝ : y✝ ∈ elemental R x
P : (u : A) → u ∈ elemental R x → Prop
self : P x ⋯
star_self : P (star x) ⋯
algebraMap : ∀ (r : R), P ((Algebra.algebraMap R A) r) ⋯
add : ∀ (u : A) (hu : u ∈ elemental R x) (v : A) (hv : v ∈ elemental R x), P u hu → P v hv → P (u + v) ⋯
mul : ∀ (u : A) (hu : u ∈ elemental R x) (v : A) (hv : v ∈ elemental R x), P u hu → P v hv → P (u * v) ⋯
closure :
  ∀ (s : Set A) (hs : s ⊆ ↑(elemental R x)),
    (∀ (u : A) (hu : u ∈ s), P u ⋯) → ∀ (v : A) (hv : v ∈ closure[inst✝⁶] s), P v ⋯
y u : A
hu✝ : u ∈ {x} ∪ star {x}
hy : u ∈ ↑(adjoin R {x})
hu : star u = x
⊢ P u ⋯

Tactic:
·

Goal state after:
case mem.inl
R : Type u_1
A : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Semiring A
inst✝⁴ : StarRing A
inst✝³ : IsSemitopologicalSemiring A
inst✝² : ContinuousStar A
inst✝¹ : Algebra R A
inst✝ : StarModule R A
y✝ y u : A
hy✝ : y✝ ∈ elemental R u
P : (u_1 : A) → u_1 ∈ elemental R u → Prop
self : P u ⋯
star_self : P (star u) ⋯
algebraMap : ∀ (r : R), P ((Algebra.algebraMap R A) r) ⋯
add : ∀ (u_1 : A) (hu : u_1 ∈ elemental R u) (v : A) (hv : v ∈ elemental R u), P u_1 hu → P v hv → P (u_1 + v) ⋯
mul : ∀ (u_1 : A) (hu : u_1 ∈ elemental R u) (v : A) (hv : v ∈ elemental R u), P u_1 hu → P v hv → P (u_1 * v) ⋯
closure :
  ∀ (s : Set A) (hs : s ⊆ ↑(elemental R u)),
    (∀ (u : A) (hu : u ∈ s), P u ⋯) → ∀ (v : A) (hv : v ∈ closure[inst✝⁶] s), P v ⋯
hu : u ∈ {u} ∪ star {u}
hy : u ∈ ↑(adjoin R {u})
⊢ P u ⋯
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬f 0 = 0
⊢ P (cfcₙ f a)

pp: case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬p a
⊢ P (cfcₙ f a)

Tactic:
·

Goal state after:
case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬f 0 = 0
⊢ P (cfcₙ f a)
Module: Mathlib.Topology.Algebra.StarSubalgebra

Goal state before:
pp: case mem.inr
R : Type u_1
A : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Semiring A
inst✝⁴ : StarRing A
inst✝³ : IsSemitopologicalSemiring A
inst✝² : ContinuousStar A
inst✝¹ : Algebra R A
inst✝ : StarModule R A
x y✝ : A
hy✝ : y✝ ∈ elemental R x
P : (u : A) → u ∈ elemental R x → Prop
self : P x ⋯
star_self : P (star x) ⋯
algebraMap : ∀ (r : R), P ((Algebra.algebraMap R A) r) ⋯
add : ∀ (u : A) (hu : u ∈ elemental R x) (v : A) (hv : v ∈ elemental R x), P u hu → P v hv → P (u + v) ⋯
mul : ∀ (u : A) (hu : u ∈ elemental R x) (v : A) (hv : v ∈ elemental R x), P u hu → P v hv → P (u * v) ⋯
closure :
  ∀ (s : Set A) (hs : s ⊆ ↑(elemental R x)),
    (∀ (u : A) (hu : u ∈ s), P u ⋯) → ∀ (v : A) (hv : v ∈ closure[inst✝⁶] s), P v ⋯
y u : A
hu✝ : u ∈ {x} ∪ star {x}
hy : u ∈ ↑(adjoin R {x})
hu : star u = x
⊢ P u ⋯

Tactic:
·

Goal state after:
case mem.inr
R : Type u_1
A : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Semiring A
inst✝⁴ : StarRing A
inst✝³ : IsSemitopologicalSemiring A
inst✝² : ContinuousStar A
inst✝¹ : Algebra R A
inst✝ : StarModule R A
x y✝ : A
hy✝ : y✝ ∈ elemental R x
P : (u : A) → u ∈ elemental R x → Prop
self : P x ⋯
star_self : P (star x) ⋯
algebraMap : ∀ (r : R), P ((Algebra.algebraMap R A) r) ⋯
add : ∀ (u : A) (hu : u ∈ elemental R x) (v : A) (hv : v ∈ elemental R x), P u hu → P v hv → P (u + v) ⋯
mul : ∀ (u : A) (hu : u ∈ elemental R x) (v : A) (hv : v ∈ elemental R x), P u hu → P v hv → P (u * v) ⋯
closure :
  ∀ (s : Set A) (hs : s ⊆ ↑(elemental R x)),
    (∀ (u : A) (hu : u ∈ s), P u ⋯) → ∀ (v : A) (hv : v ∈ closure[inst✝⁶] s), P v ⋯
y u : A
hu✝ : u ∈ {x} ∪ star {x}
hy : u ∈ ↑(adjoin R {x})
hu : star u = x
⊢ P u ⋯
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬p a
⊢ P (cfcₙ f a)

Tactic:
·

Goal state after:
case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬p a
⊢ P (cfcₙ f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0
⊢ cfcₙ f a = cfcₙ g a

pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : ¬(p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0)
⊢ cfcₙ f a = cfcₙ g a

Tactic:
·

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0
⊢ cfcₙ f a = cfcₙ g a