Theorem vafval 25172

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 25164	. . . . 5 |- +v = ( 1st o. 1st )
3	2	fveq1i 5865	. . . 4 |- ( +v ` U ) = ( ( 1st o. 1st ) ` U )
4		fo1st 6801	. . . . . 6 |- 1st : _V -onto-> _V
5		fof 5793	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
6	4, 5	ax-mp 5	. . . . 5 |- 1st : _V --> _V
7		fvco3 5942	. . . . 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 2520	. . 3 |- ( U e. _V -> ( +v ` U ) = ( 1st ` ( 1st ` U ) ) )
10		fvprc 5858	. . . 4 |- ( -. U e. _V -> ( +v ` U ) = (/) )
11		fvprc 5858	. . . . . 6 |- ( -. U e. _V -> ( 1st ` U ) = (/) )
12	11	fveq2d 5868	. . . . 5 |- ( -. U e. _V -> ( 1st ` ( 1st ` U ) ) = ( 1st ` (/) ) )
13		1st0 6787	. . . . 5 |- ( 1st ` (/) ) = (/)
14	12, 13	syl6req 2525	. . . 4 |- ( -. U e. _V -> (/) = ( 1st ` ( 1st ` U ) ) )
15	10, 14	eqtrd 2508	. . 3 |- ( -. U e. _V -> ( +v ` U ) = ( 1st ` ( 1st ` U ) ) )
16	9, 15	pm2.61i 164	. 2 |- ( +v ` U ) = ( 1st ` ( 1st ` U ) )
17	1, 16	eqtri 2496	1 |- G = ( 1st ` ( 1st ` U ) )

Syntax hints:   -. wn 3   = wceq 1379   e. wcel 1767   _Vcvv 3113   (/)c0 3785   o. ccom 5003   -->wf 5582   -onto->wfo 5584   ` cfv 5586   1stc1st 6779   +vcpv 25154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-1st 6781  df-va 25164

This theorem is referenced by:  nvvop  25178  nvablo  25185  nvsf  25188  nvscl  25197  nvsid  25198  nvsass  25199  nvdi  25201  nvdir  25202  nv2  25203  nv0  25208  nvsz  25209  nvinv  25210  cnnvg  25259  phop  25409  phpar  25415  ip0i  25416  ipdirilem  25420  h2hva  25567  hhssva  25851  hhshsslem1  25859