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Metamath Proof Explorer

Theorem hoaddsubassi

Description: Associativity of sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004) (New usage is discouraged.)

Ref Expression
Hypotheses hoaddsubass.1 ⊢ 𝑅 : ℋ ⟶ ℋ
hoaddsubass.2 ⊢ 𝑆 : ℋ ⟶ ℋ
hoaddsubass.3 ⊢ 𝑇 : ℋ ⟶ ℋ
Assertion hoaddsubassi ( ( 𝑅 +op 𝑆 ) −op 𝑇 ) = ( 𝑅 +op ( 𝑆 −op 𝑇 ) )

Proof

Step Hyp Ref Expression
1 hoaddsubass.1 ⊢ 𝑅 : ℋ ⟶ ℋ
2 hoaddsubass.2 ⊢ 𝑆 : ℋ ⟶ ℋ
3 hoaddsubass.3 ⊢ 𝑇 : ℋ ⟶ ℋ
4 hoaddsubass ⊢ ( ( 𝑅 : ℋ ⟶ ℋ ∧ 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ( 𝑅 +op 𝑆 ) −op 𝑇 ) = ( 𝑅 +op ( 𝑆 −op 𝑇 ) ) )
5 1 2 3 4 mp3an ⊢ ( ( 𝑅 +op 𝑆 ) −op 𝑇 ) = ( 𝑅 +op ( 𝑆 −op 𝑇 ) )