Theorem vafval 25923

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 = ( 1st ` ( 1st ` U ) )

Proof of Theorem vafval

Step	Hyp	Ref	Expression
1		vafval.2	. 2 |- G = ( +v ` U )
2		df-va 25915	. . . . 5 |- +v = ( 1st o. 1st )
3	2	fveq1i 5852	. . . 4 |- ( +v ` U ) = ( ( 1st o. 1st ) ` U )
4		fo1st 6806	. . . . . 6 |- 1st : _V -onto-> _V
5		fof 5780	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
6	4, 5	ax-mp 5	. . . . 5 |- 1st : _V --> _V
7		fvco3 5928	. . . . 5 |- ( ( 1st : _V --> _V /\ U e. _V ) -> ( ( 1st o. 1st ) ` U ) = ( 1st ` ( 1st ` U ) ) )
8	6, 7	mpan 670	. . . 4 |- ( U e. _V -> ( ( 1st o. 1st ) ` U ) = ( 1st ` ( 1st ` U ) ) )
9	3, 8	syl5eq 2457	. . 3 |- ( U e. _V -> ( +v ` U ) = ( 1st ` ( 1st ` U ) ) )
10		fvprc 5845	. . . 4 |- ( -. U e. _V -> ( +v ` U ) = (/) )
11		fvprc 5845	. . . . . 6 |- ( -. U e. _V -> ( 1st ` U ) = (/) )
12	11	fveq2d 5855	. . . . 5 |- ( -. U e. _V -> ( 1st ` ( 1st ` U ) ) = ( 1st ` (/) ) )
13		1st0 6792	. . . . 5 |- ( 1st ` (/) ) = (/)
14	12, 13	syl6req 2462	. . . 4 |- ( -. U e. _V -> (/) = ( 1st ` ( 1st ` U ) ) )
15	10, 14	eqtrd 2445	. . 3 |- ( -. U e. _V -> ( +v ` U ) = ( 1st ` ( 1st ` U ) ) )
16	9, 15	pm2.61i 166	. 2 |- ( +v ` U ) = ( 1st ` ( 1st ` U ) )
17	1, 16	eqtri 2433	1 |- G = ( 1st ` ( 1st ` U ) )

Syntax hints:   -. wn 3   = wceq 1407   e. wcel 1844   _Vcvv 3061   (/)c0 3740   o. ccom 4829   -->wf 5567   -onto->wfo 5569   ` cfv 5571   1stc1st 6784   +vcpv 25905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fo 5577  df-fv 5579  df-1st 6786  df-va 25915

This theorem is referenced by:  nvvop  25929  nvablo  25936  nvsf  25939  nvscl  25948  nvsid  25949  nvsass  25950  nvdi  25952  nvdir  25953  nv2  25954  nv0  25959  nvsz  25960  nvinv  25961  cnnvg  26010  phop  26160  phpar  26166  ip0i  26167  ipdirilem  26171  h2hva  26318  hhssva  26602  hhshsslem1  26610