h2hva - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > h2hva		Structured version Visualization version GIF version

Theorem h2hva 28745

Description: The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
h2h.1	⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
h2h.2	⊢ 𝑈 ∈ NrmCVec

**Assertion**
Ref	Expression
h2hva	⊢ +_ℎ = ( +_𝑣 ‘𝑈)

**Proof of Theorem h2hva**
Step	Hyp	Ref	Expression
1		eqid 2821	. . . 4 ⊢ ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
2	1	vafval 28374	. . 3 ⊢ ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (1^st ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
3		opex 5348	. . . . 5 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
4		h2h.1	. . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
5		h2h.2	. . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec
6	4, 5	eqeltrri 2910	. . . . . . 7 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
7		nvex 28382	. . . . . . 7 ⊢ (⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec → ( +_ℎ ∈ V ∧ ·_ℎ ∈ V ∧ norm_ℎ ∈ V))
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ( +_ℎ ∈ V ∧ ·_ℎ ∈ V ∧ norm_ℎ ∈ V)
9	8	simp3i 1137	. . . . 5 ⊢ norm_ℎ ∈ V
10	3, 9	op1st 7691	. . . 4 ⊢ (1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ⟨ +_ℎ , ·_ℎ ⟩
11	10	fveq2i 6667	. . 3 ⊢ (1^st ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)) = (1^st ‘⟨ +_ℎ , ·_ℎ ⟩)
12	8	simp1i 1135	. . . 4 ⊢ +_ℎ ∈ V
13	8	simp2i 1136	. . . 4 ⊢ ·_ℎ ∈ V
14	12, 13	op1st 7691	. . 3 ⊢ (1^st ‘⟨ +_ℎ , ·_ℎ ⟩) = +_ℎ
15	2, 11, 14	3eqtrri 2849	. 2 ⊢ +_ℎ = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
16	4	fveq2i 6667	. 2 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
17	15, 16	eqtr4i 2847	1 ⊢ +_ℎ = ( +_𝑣 ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⟨cop 4566 ‘cfv 6349 1^st c1st 7681 NrmCVeccnv 28355 +_𝑣 cpv 28356 +_ℎ cva 28691 ·_ℎ csm 28692 norm_ℎcno 28694

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-oprab 7154 df-1st 7683 df-vc 28330 df-nv 28363 df-va 28366

This theorem is referenced by: h2hvs 28748 axhfvadd-zf 28753 axhvcom-zf 28754 axhvass-zf 28755 axhvaddid-zf 28757 axhvdistr1-zf 28761 axhvdistr2-zf 28762 axhis2-zf 28766 hhva 28937

W3C validator