Theorem hosd2i

Description: Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hosd1.2	⊢ 𝑇 : ℋ ⟶ ℋ
	hosd1.3	⊢ 𝑈 : ℋ ⟶ ℋ
Assertion	hosd2i	⊢ ( 𝑇 +op 𝑈 ) = ( 𝑇 −op ( ( 𝑈 −op 𝑈 ) −op 𝑈 ) )

Proof

Step	Hyp	Ref	Expression
1		hosd1.2	⊢ 𝑇 : ℋ ⟶ ℋ
2		hosd1.3	⊢ 𝑈 : ℋ ⟶ ℋ
3	1 2	hosd1i	⊢ ( 𝑇 +op 𝑈 ) = ( 𝑇 −op ( 0hop −op 𝑈 ) )
4	2	hodidi	⊢ ( 𝑈 −op 𝑈 ) = 0hop
5	4	oveq1i	⊢ ( ( 𝑈 −op 𝑈 ) −op 𝑈 ) = ( 0hop −op 𝑈 )
6	5	oveq2i	⊢ ( 𝑇 −op ( ( 𝑈 −op 𝑈 ) −op 𝑈 ) ) = ( 𝑇 −op ( 0hop −op 𝑈 ) )
7	3 6	eqtr4i	⊢ ( 𝑇 +op 𝑈 ) = ( 𝑇 −op ( ( 𝑈 −op 𝑈 ) −op 𝑈 ) )