Theorem lnopsubi

Description: Subtraction property for a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005) (New usage is discouraged.)

	Ref	Expression
Hypothesis	lnopl.1	⊢ 𝑇 ∈ LinOp
Assertion	lnopsubi	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 −ℎ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) −ℎ ( 𝑇 ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lnopl.1	⊢ 𝑇 ∈ LinOp
2		neg1cn	⊢ - 1 ∈ ℂ
3	1	lnopaddmuli	⊢ ( ( - 1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) = ( ( 𝑇 ‘ 𝐴 ) +ℎ ( - 1 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ) )
4	2 3	mp3an1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) = ( ( 𝑇 ‘ 𝐴 ) +ℎ ( - 1 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ) )
5		hvsubval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 −ℎ 𝐵 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) )
6	5	fveq2d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 −ℎ 𝐵 ) ) = ( 𝑇 ‘ ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) )
7	1	lnopfi	⊢ 𝑇 : ℋ ⟶ ℋ
8	7	ffvelrni	⊢ ( 𝐴 ∈ ℋ → ( 𝑇 ‘ 𝐴 ) ∈ ℋ )
9	7	ffvelrni	⊢ ( 𝐵 ∈ ℋ → ( 𝑇 ‘ 𝐵 ) ∈ ℋ )
10		hvsubval	⊢ ( ( ( 𝑇 ‘ 𝐴 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝐵 ) ∈ ℋ ) → ( ( 𝑇 ‘ 𝐴 ) −ℎ ( 𝑇 ‘ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) +ℎ ( - 1 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ) )
11	8 9 10	syl2an	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝑇 ‘ 𝐴 ) −ℎ ( 𝑇 ‘ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) +ℎ ( - 1 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ) )
12	4 6 11	3eqtr4d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 −ℎ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) −ℎ ( 𝑇 ‘ 𝐵 ) ) )