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 ) )