Theorem lnfnaddi

Description: Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	lnfnl.1	⊢ 𝑇 ∈ LinFn
Assertion	lnfnaddi	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 +ℎ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) + ( 𝑇 ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lnfnl.1	⊢ 𝑇 ∈ LinFn
2		ax-1cn	⊢ 1 ∈ ℂ
3	1	lnfnli	⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ( 1 ·ℎ 𝐴 ) +ℎ 𝐵 ) ) = ( ( 1 · ( 𝑇 ‘ 𝐴 ) ) + ( 𝑇 ‘ 𝐵 ) ) )
4	2 3	mp3an1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ( 1 ·ℎ 𝐴 ) +ℎ 𝐵 ) ) = ( ( 1 · ( 𝑇 ‘ 𝐴 ) ) + ( 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	lnfnfi	⊢ 𝑇 : ℋ ⟶ ℂ
9	8	ffvelrni	⊢ ( 𝐴 ∈ ℋ → ( 𝑇 ‘ 𝐴 ) ∈ ℂ )
10	9	mulid2d	⊢ ( 𝐴 ∈ ℋ → ( 1 · ( 𝑇 ‘ 𝐴 ) ) = ( 𝑇 ‘ 𝐴 ) )
11	10	adantr	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 1 · ( 𝑇 ‘ 𝐴 ) ) = ( 𝑇 ‘ 𝐴 ) )
12	11	oveq1d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 1 · ( 𝑇 ‘ 𝐴 ) ) + ( 𝑇 ‘ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) + ( 𝑇 ‘ 𝐵 ) ) )
13	4 7 12	3eqtr3d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 +ℎ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) + ( 𝑇 ‘ 𝐵 ) ) )