Theorem lnfnmuli

Description: Multiplicative 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	lnfnmuli	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lnfnl.1	⊢ 𝑇 ∈ LinFn
2		ax-hv0cl	⊢ 0ℎ ∈ ℋ
3	1	lnfnli	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 0ℎ ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 0ℎ ) ) )
4	2 3	mp3an3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 0ℎ ) ) )
5		hvmulcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ )
6		ax-hvaddid	⊢ ( ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ → ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) = ( 𝐴 ·ℎ 𝐵 ) )
7	5 6	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) = ( 𝐴 ·ℎ 𝐵 ) )
8	7	fveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) ) = ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) )
9	1	lnfn0i	⊢ ( 𝑇 ‘ 0ℎ ) = 0
10	9	oveq2i	⊢ ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 0ℎ ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + 0 )
11	1	lnfnfi	⊢ 𝑇 : ℋ ⟶ ℂ
12	11	ffvelrni	⊢ ( 𝐵 ∈ ℋ → ( 𝑇 ‘ 𝐵 ) ∈ ℂ )
13		mulcl	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝑇 ‘ 𝐵 ) ∈ ℂ ) → ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) ∈ ℂ )
14	12 13	sylan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) ∈ ℂ )
15	14	addid1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + 0 ) = ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) )
16	10 15	syl5eq	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 0ℎ ) ) = ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) )
17	4 8 16	3eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) )