Theorem vafval 24102

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 24094	. . . . 5 |- +v = ( 1st o. 1st )
3	2	fveq1i 5776	. . . 4 |- ( +v ` U ) = ( ( 1st o. 1st ) ` U )
4		fo1st 6682	. . . . . 6 |- 1st : _V -onto-> _V
5		fof 5704	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
6	4, 5	ax-mp 5	. . . . 5 |- 1st : _V --> _V
7		fvco3 5853	. . . . 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 2502	. . 3 |- ( U e. _V -> ( +v ` U ) = ( 1st ` ( 1st ` U ) ) )
10		fvprc 5769	. . . 4 |- ( -. U e. _V -> ( +v ` U ) = (/) )
11		fvprc 5769	. . . . . 6 |- ( -. U e. _V -> ( 1st ` U ) = (/) )
12	11	fveq2d 5779	. . . . 5 |- ( -. U e. _V -> ( 1st ` ( 1st ` U ) ) = ( 1st ` (/) ) )
13		1st0 6669	. . . . 5 |- ( 1st ` (/) ) = (/)
14	12, 13	syl6req 2507	. . . 4 |- ( -. U e. _V -> (/) = ( 1st ` ( 1st ` U ) ) )
15	10, 14	eqtrd 2490	. . 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 2478	1 |- G = ( 1st ` ( 1st ` U ) )

Syntax hints:   -. wn 3   = wceq 1370   e. wcel 1757   _Vcvv 3054   (/)c0 3721   o. ccom 4928   -->wf 5498   -onto->wfo 5500   ` cfv 5502   1stc1st 6661   +vcpv 24084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-fo 5508  df-fv 5510  df-1st 6663  df-va 24094

This theorem is referenced by:  nvvop  24108  nvablo  24115  nvsf  24118  nvscl  24127  nvsid  24128  nvsass  24129  nvdi  24131  nvdir  24132  nv2  24133  nv0  24138  nvsz  24139  nvinv  24140  cnnvg  24189  phop  24339  phpar  24345  ip0i  24346  ipdirilem  24350  h2hva  24497  hhssva  24781  hhshsslem1  24789