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Metamath Proof Explorer

Theorem hoaddcom

Description: Commutativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion hoaddcom ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 +op 𝑇 ) = ( 𝑇 +op 𝑆 ) )

Proof

Step Hyp Ref Expression
1 oveq1 ⊢ ( 𝑆 = if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) → ( 𝑆 +op 𝑇 ) = ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) +op 𝑇 ) )
2 oveq2 ⊢ ( 𝑆 = if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) → ( 𝑇 +op 𝑆 ) = ( 𝑇 +op if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) ) )
3 1 2 eqeq12d ⊢ ( 𝑆 = if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) → ( ( 𝑆 +op 𝑇 ) = ( 𝑇 +op 𝑆 ) ↔ ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) +op 𝑇 ) = ( 𝑇 +op if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) ) ) )
4 oveq2 ⊢ ( 𝑇 = if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) → ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) +op 𝑇 ) = ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) +op if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) ) )
5 oveq1 ⊢ ( 𝑇 = if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) → ( 𝑇 +op if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) ) = ( if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) +op if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) ) )
6 4 5 eqeq12d ⊢ ( 𝑇 = if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) → ( ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) +op 𝑇 ) = ( 𝑇 +op if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) ) ↔ ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) +op if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) ) = ( if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) +op if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) ) ) )
7 ho0f ⊢ 0hop : ℋ ⟶ ℋ
8 7 elimf ⊢ if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) : ℋ ⟶ ℋ
9 7 elimf ⊢ if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) : ℋ ⟶ ℋ
10 8 9 hoaddcomi ⊢ ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) +op if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) ) = ( if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) +op if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) )
11 3 6 10 dedth2h ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 +op 𝑇 ) = ( 𝑇 +op 𝑆 ) )