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Theorem h2hva 31049

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 30678	. . 3 ⊢ ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (1^st ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
3		opex 5412	. . . . 5 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
4		h2h.1	. . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
5		h2h.2	. . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec
6	4, 5	eqeltrri 2833	. . . . . . 7 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
7		nvex 30686	. . . . . . 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 7941	. . . 4 ⊢ (1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ⟨ +_ℎ , ·_ℎ ⟩
11	10	fveq2i 6837	. . 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 7941	. . 3 ⊢ (1^st ‘⟨ +_ℎ , ·_ℎ ⟩) = +_ℎ
15	2, 11, 14	3eqtrri 2764	. 2 ⊢ +_ℎ = ( +_𝑣 ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
16	4	fveq2i 6837	. 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 1541 ∈ wcel 2113 Vcvv 3440 ⟨cop 4586 ‘cfv 6492 1^st c1st 7931 NrmCVeccnv 30659 +_𝑣 cpv 30660 +_ℎ cva 30995 ·_ℎ csm 30996 norm_ℎcno 30998

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fo 6498 df-fv 6500 df-oprab 7362 df-1st 7933 df-vc 30634 df-nv 30667 df-va 30670

This theorem is referenced by: h2hvs 31052 axhfvadd-zf 31057 axhvcom-zf 31058 axhvass-zf 31059 axhvaddid-zf 31061 axhvdistr1-zf 31065 axhvdistr2-zf 31066 axhis2-zf 31070 hhva 31241

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