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Theorem vafval 29254

Description: Value of the function for the vector addition (group) operation on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
vafval.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)

**Assertion**
Ref	Expression
vafval	⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))

**Proof of Theorem vafval**
Step	Hyp	Ref	Expression
1		vafval.2	. 2 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
2		df-va 29246	. . . . 5 ⊢ +_𝑣 = (1^st ∘ 1^st )
3	2	fveq1i 6827	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ((1^st ∘ 1^st )‘𝑈)
4		fo1st 7920	. . . . . 6 ⊢ 1^st :V–onto→V
5		fof 6740	. . . . . 6 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1^st :V⟶V
7		fvco3 6924	. . . . 5 ⊢ ((1^st :V⟶V ∧ 𝑈 ∈ V) → ((1^st ∘ 1^st )‘𝑈) = (1^st ‘(1^st ‘𝑈)))
8	6, 7	mpan 687	. . . 4 ⊢ (𝑈 ∈ V → ((1^st ∘ 1^st )‘𝑈) = (1^st ‘(1^st ‘𝑈)))
9	3, 8	eqtrid 2788	. . 3 ⊢ (𝑈 ∈ V → ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈)))
10		fvprc 6818	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( +_𝑣 ‘𝑈) = ∅)
11		fvprc 6818	. . . . . 6 ⊢ (¬ 𝑈 ∈ V → (1^st ‘𝑈) = ∅)
12	11	fveq2d 6830	. . . . 5 ⊢ (¬ 𝑈 ∈ V → (1^st ‘(1^st ‘𝑈)) = (1^st ‘∅))
13		1st0 7906	. . . . 5 ⊢ (1^st ‘∅) = ∅
14	12, 13	eqtr2di 2793	. . . 4 ⊢ (¬ 𝑈 ∈ V → ∅ = (1^st ‘(1^st ‘𝑈)))
15	10, 14	eqtrd 2776	. . 3 ⊢ (¬ 𝑈 ∈ V → ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈)))
16	9, 15	pm2.61i 182	. 2 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
17	1, 16	eqtri 2764	1 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ∅c0 4270 ∘ ccom 5625 ⟶wf 6476 –onto→wfo 6478 ‘cfv 6480 1^st c1st 7898 +_𝑣 cpv 29236

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5244 ax-nul 5251 ax-pr 5373 ax-un 7651

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4271 df-if 4475 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-br 5094 df-opab 5156 df-mpt 5177 df-id 5519 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-fo 6486 df-fv 6488 df-1st 7900 df-va 29246

This theorem is referenced by: nvvop 29260 nvablo 29267 nvsf 29270 nvscl 29277 nvsid 29278 nvsass 29279 nvdi 29281 nvdir 29282 nv2 29283 nv0 29288 nvsz 29289 nvinv 29290 cnnvg 29329 phop 29469 ip0i 29476 ipdirilem 29480 h2hva 29625 hhssva 29908 hhshsslem1 29918

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