Theorem lnopaddi

Description: Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		lnopl.1	⊢ 𝑇 ∈ LinOp
2		ax-1cn	⊢ 1 ∈ ℂ
3	1	lnopli	⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ( 1 ·ℎ 𝐴 ) +ℎ 𝐵 ) ) = ( ( 1 ·ℎ ( 𝑇 ‘ 𝐴 ) ) +ℎ ( 𝑇 ‘ 𝐵 ) ) )
4	2 3	mp3an1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ( 1 ·ℎ 𝐴 ) +ℎ 𝐵 ) ) = ( ( 1 ·ℎ ( 𝑇 ‘ 𝐴 ) ) +ℎ ( 𝑇 ‘ 𝐵 ) ) )
5		ax-hvmulid	⊢ ( 𝐴 ∈ ℋ → ( 1 ·ℎ 𝐴 ) = 𝐴 )
6	5	fvoveq1d	⊢ ( 𝐴 ∈ ℋ → ( 𝑇 ‘ ( ( 1 ·ℎ 𝐴 ) +ℎ 𝐵 ) ) = ( 𝑇 ‘ ( 𝐴 +ℎ 𝐵 ) ) )
7	6	adantr	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ( 1 ·ℎ 𝐴 ) +ℎ 𝐵 ) ) = ( 𝑇 ‘ ( 𝐴 +ℎ 𝐵 ) ) )
8	1	lnopfi	⊢ 𝑇 : ℋ ⟶ ℋ
9	8	ffvelcdmi	⊢ ( 𝐴 ∈ ℋ → ( 𝑇 ‘ 𝐴 ) ∈ ℋ )
10		ax-hvmulid	⊢ ( ( 𝑇 ‘ 𝐴 ) ∈ ℋ → ( 1 ·ℎ ( 𝑇 ‘ 𝐴 ) ) = ( 𝑇 ‘ 𝐴 ) )
11	9 10	syl	⊢ ( 𝐴 ∈ ℋ → ( 1 ·ℎ ( 𝑇 ‘ 𝐴 ) ) = ( 𝑇 ‘ 𝐴 ) )
12	11	adantr	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 1 ·ℎ ( 𝑇 ‘ 𝐴 ) ) = ( 𝑇 ‘ 𝐴 ) )
13	12	oveq1d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 1 ·ℎ ( 𝑇 ‘ 𝐴 ) ) +ℎ ( 𝑇 ‘ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐵 ) ) )
14	4 7 13	3eqtr3d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 +ℎ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐵 ) ) )