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Theorem h2hva 30960

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 2736	. . . 4 ⊢ ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
2	1	vafval 30589	. . 3 ⊢ ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (1^st ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
3		opex 5444	. . . . 5 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
4		h2h.1	. . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
5		h2h.2	. . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec
6	4, 5	eqeltrri 2832	. . . . . . 7 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
7		nvex 30597	. . . . . . 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 1141	. . . . 5 ⊢ norm_ℎ ∈ V
10	3, 9	op1st 8001	. . . 4 ⊢ (1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ⟨ +_ℎ , ·_ℎ ⟩
11	10	fveq2i 6884	. . 3 ⊢ (1^st ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)) = (1^st ‘⟨ +_ℎ , ·_ℎ ⟩)
12	8	simp1i 1139	. . . 4 ⊢ +_ℎ ∈ V
13	8	simp2i 1140	. . . 4 ⊢ ·_ℎ ∈ V
14	12, 13	op1st 8001	. . 3 ⊢ (1^st ‘⟨ +_ℎ , ·_ℎ ⟩) = +_ℎ
15	2, 11, 14	3eqtrri 2764	. 2 ⊢ +_ℎ = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
16	4	fveq2i 6884	. 2 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
17	15, 16	eqtr4i 2762	1 ⊢ +_ℎ = ( +_𝑣 ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3464 ⟨cop 4612 ‘cfv 6536 1^st c1st 7991 NrmCVeccnv 30570 +_𝑣 cpv 30571 +_ℎ cva 30906 ·_ℎ csm 30907 norm_ℎcno 30909

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-oprab 7414 df-1st 7993 df-vc 30545 df-nv 30578 df-va 30581

This theorem is referenced by: h2hvs 30963 axhfvadd-zf 30968 axhvcom-zf 30969 axhvass-zf 30970 axhvaddid-zf 30972 axhvdistr1-zf 30976 axhvdistr2-zf 30977 axhis2-zf 30981 hhva 31152

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