lnopaddi - Hilbert Space Explorer

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Theorem lnopaddi 28830

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 9994	. . 3 ⊢ 1 ∈ ℂ
2		lnopl.1	. . . 4 ⊢ 𝑇 ∈ LinOp
3	2	lnopli 28827	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)))
4	1, 3	mp3an1 1411	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)))
5		ax-hvmulid 27863	. . . . 5 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
6	5	oveq1d 6665	. . . 4 ⊢ (𝐴 ∈ ℋ → ((1 ·_ℎ 𝐴) +_ℎ 𝐵) = (𝐴 +_ℎ 𝐵))
7	6	fveq2d 6195	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
8	7	adantr 481	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
9	2	lnopfi 28828	. . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ
10	9	ffvelrni 6358	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
11		ax-hvmulid 27863	. . . . 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 481	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 ·_ℎ (𝑇‘𝐴)) = (𝑇‘𝐴))
14	13	oveq1d 6665	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +_ℎ (𝑇‘𝐵)))
15	4, 8, 14	3eqtr3d 2664	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 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 1c1 9937 ℋchil 27776 +_ℎ cva 27777 ·_ℎ csm 27778 LinOpclo 27804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-1cn 9994 ax-hilex 27856 ax-hvmulid 27863

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-lnop 28700

This theorem is referenced by: lnopaddmuli 28832 lnophsi 28860 lnopeq0lem1 28864 lnophmlem2 28876 imaelshi 28917 cnlnadjlem2 28927

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