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Theorem lnfnaddi 30405

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 10929	. . 3 ⊢ 1 ∈ ℂ
2		lnfnl.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfnli 30402	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)))
4	1, 3	mp3an1 1447	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)))
5		ax-hvmulid 29368	. . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
6	5	fvoveq1d 7297	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
7	6	adantr 481	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
8	2	lnfnfi 30403	. . . . . 6 ⊢ 𝑇: ℋ⟶ℂ
9	8	ffvelrni 6960	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
10	9	mulid2d 10993	. . . 4 ⊢ (𝐴 ∈ ℋ → (1 · (𝑇‘𝐴)) = (𝑇‘𝐴))
11	10	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 · (𝑇‘𝐴)) = (𝑇‘𝐴))
12	11	oveq1d 7290	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵)))
13	4, 7, 12	3eqtr3d 2786	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 1c1 10872 + caddc 10874 · cmul 10876 ℋchba 29281 +_ℎ cva 29282 ·_ℎ csm 29283 LinFnclf 29316

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-mulcl 10933 ax-mulcom 10935 ax-mulass 10937 ax-distr 10938 ax-1rid 10941 ax-cnre 10944 ax-hilex 29361 ax-hvmulid 29368

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-lnfn 30210

This theorem is referenced by: lnfnaddmuli 30407 nlelshi 30422

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