lnopaddi - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > lnopaddi		Structured version Visualization version GIF version

Theorem lnopaddi 32061

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 11091	. . 3 ⊢ 1 ∈ ℂ
2		lnopl.1	. . . 4 ⊢ 𝑇 ∈ LinOp
3	2	lnopli 32058	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)))
4	1, 3	mp3an1 1451	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)))
5		ax-hvmulid 31096	. . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
6	5	fvoveq1d 7384	. . 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	lnopfi 32059	. . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ
9	8	ffvelcdmi 7031	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
10		ax-hvmulid 31096	. . . . 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 480	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 ·_ℎ (𝑇‘𝐴)) = (𝑇‘𝐴))
13	12	oveq1d 7377	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 ·_ℎ (𝑇‘𝐴)) +_ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +_ℎ (𝑇‘𝐵)))
14	4, 7, 13	3eqtr3d 2780	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 395 = wceq 1542 ∈ wcel 2114 ‘cfv 6494 (class class class)co 7362 ℂcc 11031 1c1 11034 ℋchba 31009 +_ℎ cva 31010 ·_ℎ csm 31011 LinOpclo 31037

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 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-1cn 11091 ax-hilex 31089 ax-hvmulid 31096

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 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-id 5521 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-fv 6502 df-ov 7365 df-oprab 7366 df-mpo 7367 df-map 8770 df-lnop 31931

This theorem is referenced by: lnopaddmuli 32063 lnophsi 32091 lnopeq0lem1 32095 lnophmlem2 32107 imaelshi 32148 cnlnadjlem2 32158

W3C validator