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Theorem lnopaddi 29419

Description: Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
lnopl.1	⊢ 𝑇 ∈ LinOp

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

**Proof of Theorem lnopaddi**
Step	Hyp	Ref	Expression
1		ax-1cn 10332	. . 3 ⊢ 1 ∈ ℂ
2		lnopl.1	. . . 4 ⊢ 𝑇 ∈ LinOp
3	2	lnopli 29416	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)))
4	1, 3	mp3an1 1521	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)))
5		ax-hvmulid 28452	. . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
6	5	fvoveq1d 6946	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
7	6	adantr 474	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
8	2	lnopfi 29417	. . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ
9	8	ffvelrni 6624	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
10		ax-hvmulid 28452	. . . . 5 ⊢ ((𝑇‘𝐴) ∈ ℋ → (1 ·_ℎ (𝑇‘𝐴)) = (𝑇‘𝐴))
11	9, 10	syl 17	. . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ (𝑇‘𝐴)) = (𝑇‘𝐴))
12	11	adantr 474	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 ·_ℎ (𝑇‘𝐴)) = (𝑇‘𝐴))
13	12	oveq1d 6939	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +_ℎ (𝑇‘𝐵)))
14	4, 7, 13	3eqtr3d 2822	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ 𝐵)) = ((𝑇‘𝐴) +_ℎ (𝑇‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 1c1 10275 ℋchba 28365 +_ℎ cva 28366 ·_ℎ csm 28367 LinOpclo 28393

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-1cn 10332 ax-hilex 28445 ax-hvmulid 28452

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-map 8144 df-lnop 29289

This theorem is referenced by: lnopaddmuli 29421 lnophsi 29449 lnopeq0lem1 29453 lnophmlem2 29465 imaelshi 29506 cnlnadjlem2 29516

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