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Metamath Proof Explorer

Theorem hosubcl

Description: Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion hosubcl ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 −op 𝑇 ) : ℋ ⟶ ℋ )

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 1 feq1d ⊢ ( 𝑆 = if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) → ( ( 𝑆 −op 𝑇 ) : ℋ ⟶ ℋ ↔ ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) −op 𝑇 ) : ℋ ⟶ ℋ ) )
3 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 ) ) )
4 3 feq1d ⊢ ( 𝑇 = if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) → ( ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) −op 𝑇 ) : ℋ ⟶ ℋ ↔ ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) −op if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) ) : ℋ ⟶ ℋ ) )
5 ho0f ⊢ 0hop : ℋ ⟶ ℋ
6 5 elimf ⊢ if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) : ℋ ⟶ ℋ
7 5 elimf ⊢ if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) : ℋ ⟶ ℋ
8 6 7 hosubcli ⊢ ( if ( 𝑆 : ℋ ⟶ ℋ , 𝑆 , 0hop ) −op if ( 𝑇 : ℋ ⟶ ℋ , 𝑇 , 0hop ) ) : ℋ ⟶ ℋ
9 2 4 8 dedth2h ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 −op 𝑇 ) : ℋ ⟶ ℋ )