Theorem vsfval 27488

Description: Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
vsfval.2	|- G = ( +v ` U )
vsfval.3	|- M = ( -v ` U )

Assertion

Ref	Expression
vsfval	|- M = ( /g ` G )

Proof of Theorem vsfval

Dummy variables x g y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-vs 27454	. . . . 5 |- -v = ( /g o. +v )
2	1	fveq1i 6192	. . . 4 |- ( -v ` U ) = ( ( /g o. +v ) ` U )
3		fo1st 7188	. . . . . . . 8 |- 1st : _V -onto-> _V
4		fof 6115	. . . . . . . 8 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
5	3, 4	ax-mp 5	. . . . . . 7 |- 1st : _V --> _V
6		fco 6058	. . . . . . 7 |- ( ( 1st : _V --> _V /\ 1st : _V --> _V ) -> ( 1st o. 1st ) : _V --> _V )
7	5, 5, 6	mp2an 708	. . . . . 6 |- ( 1st o. 1st ) : _V --> _V
8		df-va 27450	. . . . . . 7 |- +v = ( 1st o. 1st )
9	8	feq1i 6036	. . . . . 6 |- ( +v : _V --> _V <-> ( 1st o. 1st ) : _V --> _V )
10	7, 9	mpbir 221	. . . . 5 |- +v : _V --> _V
11		fvco3 6275	. . . . 5 |- ( ( +v : _V --> _V /\ U e. _V ) -> ( ( /g o. +v ) ` U ) = ( /g ` ( +v ` U ) ) )
12	10, 11	mpan 706	. . . 4 |- ( U e. _V -> ( ( /g o. +v ) ` U ) = ( /g ` ( +v ` U ) ) )
13	2, 12	syl5eq 2668	. . 3 |- ( U e. _V -> ( -v ` U ) = ( /g ` ( +v ` U ) ) )
14		0ngrp 27365	. . . . . 6 |- -. (/) e. GrpOp
15		vex 3203	. . . . . . . . . 10 |- g e. _V
16	15	rnex 7100	. . . . . . . . 9 |- ran g e. _V
17	16, 16	mpt2ex 7247	. . . . . . . 8 |- ( x e. ran g , y e. ran g |-> ( x g ( ( inv ` g ) ` y ) ) ) e. _V
18		df-gdiv 27350	. . . . . . . 8 |- /g = ( g e. GrpOp |-> ( x e. ran g , y e. ran g |-> ( x g ( ( inv ` g ) ` y ) ) ) )
19	17, 18	dmmpti 6023	. . . . . . 7 |- dom /g = GrpOp
20	19	eleq2i 2693	. . . . . 6 |- ( (/) e. dom /g <-> (/) e. GrpOp )
21	14, 20	mtbir 313	. . . . 5 |- -. (/) e. dom /g
22		ndmfv 6218	. . . . 5 |- ( -. (/) e. dom /g -> ( /g ` (/) ) = (/) )
23	21, 22	mp1i 13	. . . 4 |- ( -. U e. _V -> ( /g ` (/) ) = (/) )
24		fvprc 6185	. . . . 5 |- ( -. U e. _V -> ( +v ` U ) = (/) )
25	24	fveq2d 6195	. . . 4 |- ( -. U e. _V -> ( /g ` ( +v ` U ) ) = ( /g ` (/) ) )
26		fvprc 6185	. . . 4 |- ( -. U e. _V -> ( -v ` U ) = (/) )
27	23, 25, 26	3eqtr4rd 2667	. . 3 |- ( -. U e. _V -> ( -v ` U ) = ( /g ` ( +v ` U ) ) )
28	13, 27	pm2.61i 176	. 2 |- ( -v ` U ) = ( /g ` ( +v ` U ) )
29		vsfval.3	. 2 |- M = ( -v ` U )
30		vsfval.2	. . 3 |- G = ( +v ` U )
31	30	fveq2i 6194	. 2 |- ( /g ` G ) = ( /g ` ( +v ` U ) )
32	28, 29, 31	3eqtr4i 2654	1 |- M = ( /g ` G )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   dom cdm 5114   ran crn 5115   o. ccom 5118   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   GrpOpcgr 27343   invcgn 27345   /g cgs 27346   +vcpv 27440   -vcnsb 27444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gdiv 27350  df-va 27450  df-vs 27454

This theorem is referenced by:  nvm  27496  nvmfval  27499  nvnnncan1  27502  nvaddsub  27510