-- Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/PosPart.lean
/-- The positive and negative parts of a selfadjoint element `a` are unique. That is, if
`a = b - c` is the difference of nonnegative elements whose product is zero, then these are
precisely `a⁺` and `a⁻`. -/
lemma posPart_negPart_unique {a b c : A} (habc : a = b - c) (hbc : b * c = 0)
    (hb : 0 ≤ b

Proof state:
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : autoParam (0 ≤ b) _auto✝
hc : autoParam (0 ≤ c) _auto✝
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
⊢ a⁺ = b ∧ a⁻ = c

Tactic:
have ha : IsSelfAdjoint a := habc ▸ hb'.sub <| .of_nonneg hc

Result:
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : autoParam (0 ≤ b) _auto✝
hc : autoParam (0 ≤ c) _auto✝
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
⊢ a⁺ = b ∧ a⁻ = c

Proof state:
case intro.intro
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : autoParam (0 ≤ b) _auto✝
hc : autoParam (0 ≤ c) _auto✝
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
has : σₙ ℝ a ⊆ s
hbs : σₙ ℝ b ⊆ s
hcs : σₙ ℝ (-c) ⊆ s
zero : Zero ↑s := { zero := ⟨0, ⋯⟩ }
⊢ a⁺ = b

Tactic:
have s0 : (0 : s) = (0 : ℝ) := rfl

Result:
case intro.intro
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : autoParam (0 ≤ b) _auto✝
hc : autoParam (0 ≤ c) _auto✝
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
has : σₙ ℝ a ⊆ s
hbs : σₙ ℝ b ⊆ s
hcs : σₙ ℝ (-c) ⊆ s
zero : Zero ↑s := { zero := ⟨0, ⋯⟩ }
s0 : ↑0 = 0
⊢ a⁺ = b

Proof state:
case star_id
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : autoParam (0 ≤ b) _auto✝
hc : autoParam (0 ≤ c) _auto✝
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
has : σₙ ℝ a ⊆ s
hbs : σₙ ℝ b ⊆ s
hcs : σₙ ℝ (-c) ⊆ s
zero : Zero ↑s := { zero := ⟨0, ⋯⟩ }
s0 : ↑0 = 0
f g : C(↑s, ℝ)₀
⊢ (cfcₙHomSuperset hb' hbs) (ContinuousMapZero.id s0) * (cfcₙHomSuperset hc' hcs) g = 0

case id
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : autoParam (0 ≤ b) _auto✝
hc : autoParam (0 ≤ c) _auto✝
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
has : σₙ ℝ a ⊆ s
hbs : σₙ ℝ b ⊆ s
hcs : σₙ ℝ (-c) ⊆ s
zero : Zero ↑s := { zero := ⟨0, ⋯⟩ }
s0 : ↑0 = 0
f g : C(↑s, ℝ)₀
⊢ (cfcₙHomSuperset hb' hbs) (ContinuousMapZero.id s0) * (cfcₙHomSuperset hc' hcs) g = 0

Tactic:
all_goals
  refine g.mul_nonUnitalStarAlgHom_apply_eq_zero s0 _ _ ?_ ?_
    (cfcₙHomSuperset_continuous hc' hcs)
  all_goals simp only [zero, star_trivial, cfcₙHomSuperset_id' hb' hbs,
    cfcₙHomSuperset_id' hc' hcs, mul_neg, hbc, neg_zero]

Result:
no goals

Proof state:
case intro.intro
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : autoParam (0 ≤ b) _auto✝
hc : autoParam (0 ≤ c) _auto✝
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
has : σₙ ℝ a ⊆ s
hbs : σₙ ℝ b ⊆ s
hcs : σₙ ℝ (-c) ⊆ s
zero : Zero ↑s := { zero := ⟨0, ⋯⟩ }
s0 : ↑0 = 0
mul₁ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hb' hbs) f * (cfcₙHomSuperset hc' hcs) g = 0
mul₂ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hc' hcs) f * (cfcₙHomSuperset hb' hbs) g = 0
ψ : C(↑s, ℝ)₀ →⋆ₙₐ[ℝ] A :=
  let __src := ↑(cfcₙHomSuperset hb' hbs) + ↑(cfcₙHomSuperset hc' hcs);
  { toFun := ⇑(cfcₙHomSuperset hb' hbs) + ⇑(cfcₙHomSuperset hc' hcs), map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯,
    map_mul' := ⋯, map_star' := ⋯ }
key : cfcₙHomSuperset ha has = ψ
f : C(↑s, ℝ)₀ := { toFun := fun x => (↑x)⁺, continuous_toFun := ⋯, map_zero' := ⋯ }
⊢ a⁺ = b

Tactic:
replace key := congr($key f)

Result:
case intro.intro
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : autoParam (0 ≤ b) _auto✝
hc : autoParam (0 ≤ c) _auto✝
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
has : σₙ ℝ a ⊆ s
hbs : σₙ ℝ b ⊆ s
hcs : σₙ ℝ (-c) ⊆ s
zero : Zero ↑s := { zero := ⟨0, ⋯⟩ }
s0 : ↑0 = 0
mul₁ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hb' hbs) f * (cfcₙHomSuperset hc' hcs) g = 0
mul₂ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hc' hcs) f * (cfcₙHomSuperset hb' hbs) g = 0
ψ : C(↑s, ℝ)₀ →⋆ₙₐ[ℝ] A :=
  let __src := ↑(cfcₙHomSuperset hb' hbs) + ↑(cfcₙHomSuperset hc' hcs);
  { toFun := ⇑(cfcₙHomSuperset hb' hbs) + ⇑(cfcₙHomSuperset hc' hcs), map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯,
    map_mul' := ⋯, map_star' := ⋯ }
f : C(↑s, ℝ)₀ := { toFun := fun x => (↑x)⁺, continuous_toFun := ⋯, map_zero' := ⋯ }
key : (cfcₙHomSuperset ha has) f = ψ f
⊢ a⁺ = b