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Theorem h2hva 31061

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 2737	. . . 4 ⊢ ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
2	1	vafval 30690	. . 3 ⊢ ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (1^st ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
3		opex 5419	. . . . 5 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
4		h2h.1	. . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
5		h2h.2	. . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec
6	4, 5	eqeltrri 2834	. . . . . . 7 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
7		nvex 30698	. . . . . . 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 1142	. . . . 5 ⊢ norm_ℎ ∈ V
10	3, 9	op1st 7951	. . . 4 ⊢ (1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ⟨ +_ℎ , ·_ℎ ⟩
11	10	fveq2i 6845	. . 3 ⊢ (1^st ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)) = (1^st ‘⟨ +_ℎ , ·_ℎ ⟩)
12	8	simp1i 1140	. . . 4 ⊢ +_ℎ ∈ V
13	8	simp2i 1141	. . . 4 ⊢ ·_ℎ ∈ V
14	12, 13	op1st 7951	. . 3 ⊢ (1^st ‘⟨ +_ℎ , ·_ℎ ⟩) = +_ℎ
15	2, 11, 14	3eqtrri 2765	. 2 ⊢ +_ℎ = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
16	4	fveq2i 6845	. 2 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
17	15, 16	eqtr4i 2763	1 ⊢ +_ℎ = ( +_𝑣 ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ⟨cop 4588 ‘cfv 6500 1^st c1st 7941 NrmCVeccnv 30671 +_𝑣 cpv 30672 +_ℎ cva 31007 ·_ℎ csm 31008 norm_ℎcno 31010

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 5243 ax-nul 5253 ax-pr 5379 ax-un 7690

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 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-oprab 7372 df-1st 7943 df-vc 30646 df-nv 30679 df-va 30682

This theorem is referenced by: h2hvs 31064 axhfvadd-zf 31069 axhvcom-zf 31070 axhvass-zf 31071 axhvaddid-zf 31073 axhvdistr1-zf 31077 axhvdistr2-zf 31078 axhis2-zf 31082 hhva 31253

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