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Theorem lnfnaddi 32123

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

**Hypothesis**
Ref	Expression
lnfnl.1	⊢ 𝑇 ∈ LinFn

**Assertion**
Ref	Expression
lnfnaddi	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵)))

**Proof of Theorem lnfnaddi**
Step	Hyp	Ref	Expression
1		ax-1cn 11089	. . 3 ⊢ 1 ∈ ℂ
2		lnfnl.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfnli 32120	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)))
4	1, 3	mp3an1 1451	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)))
5		ax-hvmulid 31086	. . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
6	5	fvoveq1d 7383	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
7	6	adantr 480	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
8	2	lnfnfi 32121	. . . . . 6 ⊢ 𝑇: ℋ⟶ℂ
9	8	ffvelcdmi 7030	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
10	9	mullidd 11155	. . . 4 ⊢ (𝐴 ∈ ℋ → (1 · (𝑇‘𝐴)) = (𝑇‘𝐴))
11	10	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 · (𝑇‘𝐴)) = (𝑇‘𝐴))
12	11	oveq1d 7376	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵)))
13	4, 7, 12	3eqtr3d 2780	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ‘cfv 6493 (class class class)co 7361 ℂcc 11029 1c1 11032 + caddc 11034 · cmul 11036 ℋchba 30999 +_ℎ cva 31000 ·_ℎ csm 31001 LinFnclf 31034

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-cnex 11087 ax-resscn 11088 ax-1cn 11089 ax-icn 11090 ax-addcl 11091 ax-mulcl 11093 ax-mulcom 11095 ax-mulass 11097 ax-distr 11098 ax-1rid 11101 ax-cnre 11104 ax-hilex 31079 ax-hvmulid 31086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3401 df-v 3443 df-sbc 3742 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8770 df-lnfn 31928

This theorem is referenced by: lnfnaddmuli 32125 nlelshi 32140

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