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Theorem h2hva 31003

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 2735	. . . 4 ⊢ ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
2	1	vafval 30632	. . 3 ⊢ ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (1^st ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
3		opex 5475	. . . . 5 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
4		h2h.1	. . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
5		h2h.2	. . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec
6	4, 5	eqeltrri 2836	. . . . . . 7 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
7		nvex 30640	. . . . . . 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 1140	. . . . 5 ⊢ norm_ℎ ∈ V
10	3, 9	op1st 8021	. . . 4 ⊢ (1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ⟨ +_ℎ , ·_ℎ ⟩
11	10	fveq2i 6910	. . 3 ⊢ (1^st ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)) = (1^st ‘⟨ +_ℎ , ·_ℎ ⟩)
12	8	simp1i 1138	. . . 4 ⊢ +_ℎ ∈ V
13	8	simp2i 1139	. . . 4 ⊢ ·_ℎ ∈ V
14	12, 13	op1st 8021	. . 3 ⊢ (1^st ‘⟨ +_ℎ , ·_ℎ ⟩) = +_ℎ
15	2, 11, 14	3eqtrri 2768	. 2 ⊢ +_ℎ = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
16	4	fveq2i 6910	. 2 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
17	15, 16	eqtr4i 2766	1 ⊢ +_ℎ = ( +_𝑣 ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⟨cop 4637 ‘cfv 6563 1^st c1st 8011 NrmCVeccnv 30613 +_𝑣 cpv 30614 +_ℎ cva 30949 ·_ℎ csm 30950 norm_ℎcno 30952

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-oprab 7435 df-1st 8013 df-vc 30588 df-nv 30621 df-va 30624

This theorem is referenced by: h2hvs 31006 axhfvadd-zf 31011 axhvcom-zf 31012 axhvass-zf 31013 axhvaddid-zf 31015 axhvdistr1-zf 31019 axhvdistr2-zf 31020 axhis2-zf 31024 hhva 31195

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