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Metamath Proof Explorer

Theorem hmopadj2

Description: An operator is Hermitian iff it is self-adjoint. Definition of Hermitian in Halmos p. 41. (Contributed by NM, 9-Apr-2006) (New usage is discouraged.)

Ref Expression
Assertion hmopadj2 ( 𝑇 ∈ dom adjℎ → ( 𝑇 ∈ HrmOp ↔ ( adjℎ ‘ 𝑇 ) = 𝑇 ) )

Proof

Step Hyp Ref Expression
1 hmopadj ⊢ ( 𝑇 ∈ HrmOp → ( adjℎ ‘ 𝑇 ) = 𝑇 )
2 dmadjop ⊢ ( 𝑇 ∈ dom adjℎ → 𝑇 : ℋ ⟶ ℋ )
3 2 adantr ⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( adjℎ ‘ 𝑇 ) = 𝑇 ) → 𝑇 : ℋ ⟶ ℋ )
4 adj1 ⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( ( adjℎ ‘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑦 ) )
5 4 3expb ⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( ( adjℎ ‘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑦 ) )
6 5 adantlr ⊢ ( ( ( 𝑇 ∈ dom adjℎ ∧ ( adjℎ ‘ 𝑇 ) = 𝑇 ) ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( ( adjℎ ‘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑦 ) )
7 fveq1 ⊢ ( ( adjℎ ‘ 𝑇 ) = 𝑇 → ( ( adjℎ ‘ 𝑇 ) ‘ 𝑥 ) = ( 𝑇 ‘ 𝑥 ) )
8 7 oveq1d ⊢ ( ( adjℎ ‘ 𝑇 ) = 𝑇 → ( ( ( adjℎ ‘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑦 ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) )
9 8 ad2antlr ⊢ ( ( ( 𝑇 ∈ dom adjℎ ∧ ( adjℎ ‘ 𝑇 ) = 𝑇 ) ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( ( ( adjℎ ‘ 𝑇 ) ‘ 𝑥 ) ·ih 𝑦 ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) )
10 6 9 eqtrd ⊢ ( ( ( 𝑇 ∈ dom adjℎ ∧ ( adjℎ ‘ 𝑇 ) = 𝑇 ) ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) )
11 10 ralrimivva ⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( adjℎ ‘ 𝑇 ) = 𝑇 ) → ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) )
12 elhmop ⊢ ( 𝑇 ∈ HrmOp ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) ) )
13 3 11 12 sylanbrc ⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( adjℎ ‘ 𝑇 ) = 𝑇 ) → 𝑇 ∈ HrmOp )
14 13 ex ⊢ ( 𝑇 ∈ dom adjℎ → ( ( adjℎ ‘ 𝑇 ) = 𝑇 → 𝑇 ∈ HrmOp ) )
15 1 14 impbid2 ⊢ ( 𝑇 ∈ dom adjℎ → ( 𝑇 ∈ HrmOp ↔ ( adjℎ ‘ 𝑇 ) = 𝑇 ) )