lnopaddi - Hilbert Space Explorer

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Theorem lnopaddi 28214

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 9873	. . 3 ⊢ 1 ∈ ℂ
2		lnopl.1	. . . 4 ⊢ 𝑇 ∈ LinOp
3	2	lnopli 28211	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)))
4	1, 3	mp3an1 1403	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)))
5		ax-hvmulid 27247	. . . . 5 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
6	5	oveq1d 6564	. . . 4 ⊢ (𝐴 ∈ ℋ → ((1 ·_ℎ 𝐴) +_ℎ 𝐵) = (𝐴 +_ℎ 𝐵))
7	6	fveq2d 6107	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
8	7	adantr 480	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
9	2	lnopfi 28212	. . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ
10	9	ffvelrni 6266	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
11		ax-hvmulid 27247	. . . . 5 ⊢ ((𝑇‘𝐴) ∈ ℋ → (1 ·_ℎ (𝑇‘𝐴)) = (𝑇‘𝐴))
12	10, 11	syl 17	. . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ (𝑇‘𝐴)) = (𝑇‘𝐴))
13	12	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 ·_ℎ (𝑇‘𝐴)) = (𝑇‘𝐴))
14	13	oveq1d 6564	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +_ℎ (𝑇‘𝐵)))
15	4, 8, 14	3eqtr3d 2652	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 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 1c1 9816 ℋchil 27160 +_ℎ cva 27161 ·_ℎ csm 27162 LinOpclo 27188

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-1cn 9873 ax-hilex 27240 ax-hvmulid 27247

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-lnop 28084

This theorem is referenced by: lnopaddmuli 28216 lnophsi 28244 lnopeq0lem1 28248 lnophmlem2 28260 imaelshi 28301 cnlnadjlem2 28311

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