Theorem lnfn0i

Description: The value of a linear Hilbert space functional at zero is zero. Remark in Beran p. 99. (Contributed by NM, 11-Feb-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	lnfnl.1	⊢ 𝑇 ∈ LinFn
Assertion	lnfn0i	⊢ ( 𝑇 ‘ 0ℎ ) = 0

Proof

Step	Hyp	Ref	Expression
1		lnfnl.1	⊢ 𝑇 ∈ LinFn
2		ax-hv0cl	⊢ 0ℎ ∈ ℋ
3	1	lnfnfi	⊢ 𝑇 : ℋ ⟶ ℂ
4	3	ffvelcdmi	⊢ ( 0ℎ ∈ ℋ → ( 𝑇 ‘ 0ℎ ) ∈ ℂ )
5	2 4	ax-mp	⊢ ( 𝑇 ‘ 0ℎ ) ∈ ℂ
6	5 5	pncan3oi	⊢ ( ( ( 𝑇 ‘ 0ℎ ) + ( 𝑇 ‘ 0ℎ ) ) − ( 𝑇 ‘ 0ℎ ) ) = ( 𝑇 ‘ 0ℎ )
7		ax-1cn	⊢ 1 ∈ ℂ
8	1	lnfnli	⊢ ( ( 1 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 0ℎ ∈ ℋ ) → ( 𝑇 ‘ ( ( 1 ·ℎ 0ℎ ) +ℎ 0ℎ ) ) = ( ( 1 · ( 𝑇 ‘ 0ℎ ) ) + ( 𝑇 ‘ 0ℎ ) ) )
9	7 2 2 8	mp3an	⊢ ( 𝑇 ‘ ( ( 1 ·ℎ 0ℎ ) +ℎ 0ℎ ) ) = ( ( 1 · ( 𝑇 ‘ 0ℎ ) ) + ( 𝑇 ‘ 0ℎ ) )
10	7 2	hvmulcli	⊢ ( 1 ·ℎ 0ℎ ) ∈ ℋ
11		ax-hvaddid	⊢ ( ( 1 ·ℎ 0ℎ ) ∈ ℋ → ( ( 1 ·ℎ 0ℎ ) +ℎ 0ℎ ) = ( 1 ·ℎ 0ℎ ) )
12	10 11	ax-mp	⊢ ( ( 1 ·ℎ 0ℎ ) +ℎ 0ℎ ) = ( 1 ·ℎ 0ℎ )
13		ax-hvmulid	⊢ ( 0ℎ ∈ ℋ → ( 1 ·ℎ 0ℎ ) = 0ℎ )
14	2 13	ax-mp	⊢ ( 1 ·ℎ 0ℎ ) = 0ℎ
15	12 14	eqtri	⊢ ( ( 1 ·ℎ 0ℎ ) +ℎ 0ℎ ) = 0ℎ
16	15	fveq2i	⊢ ( 𝑇 ‘ ( ( 1 ·ℎ 0ℎ ) +ℎ 0ℎ ) ) = ( 𝑇 ‘ 0ℎ )
17	9 16	eqtr3i	⊢ ( ( 1 · ( 𝑇 ‘ 0ℎ ) ) + ( 𝑇 ‘ 0ℎ ) ) = ( 𝑇 ‘ 0ℎ )
18	5	mullidi	⊢ ( 1 · ( 𝑇 ‘ 0ℎ ) ) = ( 𝑇 ‘ 0ℎ )
19	18	oveq1i	⊢ ( ( 1 · ( 𝑇 ‘ 0ℎ ) ) + ( 𝑇 ‘ 0ℎ ) ) = ( ( 𝑇 ‘ 0ℎ ) + ( 𝑇 ‘ 0ℎ ) )
20	17 19	eqtr3i	⊢ ( 𝑇 ‘ 0ℎ ) = ( ( 𝑇 ‘ 0ℎ ) + ( 𝑇 ‘ 0ℎ ) )
21	20	oveq1i	⊢ ( ( 𝑇 ‘ 0ℎ ) − ( 𝑇 ‘ 0ℎ ) ) = ( ( ( 𝑇 ‘ 0ℎ ) + ( 𝑇 ‘ 0ℎ ) ) − ( 𝑇 ‘ 0ℎ ) )
22	5	subidi	⊢ ( ( 𝑇 ‘ 0ℎ ) − ( 𝑇 ‘ 0ℎ ) ) = 0
23	21 22	eqtr3i	⊢ ( ( ( 𝑇 ‘ 0ℎ ) + ( 𝑇 ‘ 0ℎ ) ) − ( 𝑇 ‘ 0ℎ ) ) = 0
24	6 23	eqtr3i	⊢ ( 𝑇 ‘ 0ℎ ) = 0