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Metamath Proof Explorer

Theorem hoaddsubass

Description: Associative-type law for addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion hoaddsubass ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝑆 +op 𝑇 ) −op 𝑈 ) = ( 𝑆 +op ( 𝑇 −op 𝑈 ) ) )

Proof

Step Hyp Ref Expression
1 ho0f ⊢ 0hop : ℋ ⟶ ℋ
2 hosubcl ⊢ ( ( 0hop : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 0hop −op 𝑈 ) : ℋ ⟶ ℋ )
3 1 2 mpan ⊢ ( 𝑈 : ℋ ⟶ ℋ → ( 0hop −op 𝑈 ) : ℋ ⟶ ℋ )
4 hoaddass ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ ( 0hop −op 𝑈 ) : ℋ ⟶ ℋ ) → ( ( 𝑆 +op 𝑇 ) +op ( 0hop −op 𝑈 ) ) = ( 𝑆 +op ( 𝑇 +op ( 0hop −op 𝑈 ) ) ) )
5 3 4 syl3an3 ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝑆 +op 𝑇 ) +op ( 0hop −op 𝑈 ) ) = ( 𝑆 +op ( 𝑇 +op ( 0hop −op 𝑈 ) ) ) )
6 hoaddcl ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 +op 𝑇 ) : ℋ ⟶ ℋ )
7 ho0sub ⊢ ( ( ( 𝑆 +op 𝑇 ) : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝑆 +op 𝑇 ) −op 𝑈 ) = ( ( 𝑆 +op 𝑇 ) +op ( 0hop −op 𝑈 ) ) )
8 6 7 stoic3 ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝑆 +op 𝑇 ) −op 𝑈 ) = ( ( 𝑆 +op 𝑇 ) +op ( 0hop −op 𝑈 ) ) )
9 ho0sub ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑇 −op 𝑈 ) = ( 𝑇 +op ( 0hop −op 𝑈 ) ) )
10 9 3adant1 ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑇 −op 𝑈 ) = ( 𝑇 +op ( 0hop −op 𝑈 ) ) )
11 10 oveq2d ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( 𝑆 +op ( 𝑇 −op 𝑈 ) ) = ( 𝑆 +op ( 𝑇 +op ( 0hop −op 𝑈 ) ) ) )
12 5 8 11 3eqtr4d ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑈 : ℋ ⟶ ℋ ) → ( ( 𝑆 +op 𝑇 ) −op 𝑈 ) = ( 𝑆 +op ( 𝑇 −op 𝑈 ) ) )