Theorem vsfval 8254

Description: Value of the function for the vector subtraction operation on a normed complex vector space.

Hypotheses

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

Assertion

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

Proof of Theorem vsfval

Step	Hyp	Ref	Expression
1		df-va 8214	. . . . . . . 8 |- +v = (1st o. 1st)
2	1	dmeqi 3312	. . . . . . 7 |- dom +v = dom (1st o. 1st)
3		fo1st 4091	. . . . . . . . . . 11 |- 1st:V-onto->V
4		fof 3672	. . . . . . . . . . 11 |- (1st:V-onto->V -> 1st:V-->V)
5	3, 4	ax-mp 7	. . . . . . . . . 10 |- 1st:V-->V
6	5	fdmi 3632	. . . . . . . . 9 |- dom 1st = V
7		forn 3674	. . . . . . . . . 10 |- (1st:V-onto->V -> ran 1st = V)
8	3, 7	ax-mp 7	. . . . . . . . 9 |- ran 1st = V
9	6, 8	eqtr4 1498	. . . . . . . 8 |- dom 1st = ran 1st
10		dmcoeq 3366	. . . . . . . 8 |- (dom 1st = ran 1st -> dom (1st o. 1st) = dom 1st)
11	9, 10	ax-mp 7	. . . . . . 7 |- dom (1st o. 1st) = dom 1st
12	2, 11, 6	3eqtr 1499	. . . . . 6 |- dom +v = V
13	12	eleq2i 1538	. . . . 5 |- (U e. dom +v <-> U e. V)
14		visset 1813	. . . . . . . . . 10 |- g e. V
15	14	rnex 3361	. . . . . . . . 9 |- ran g e. V
16		eqid 1475	. . . . . . . . 9 |- {<.<.x, y>., z>. | ((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y)))} = {<.<.x, y>., z>. | ((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y)))}
17	15, 15, 16	oprabex2 4021	. . . . . . . 8 |- {<.<.x, y>., z>. | ((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y)))} e. V
18		df-gdiv 8040	. . . . . . . 8 |- /g = {<.g, f>. | (g e. Grp /\ f = {<.<.x, y>., z>. | ((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y)))})}
19	17, 18	fnopab2 3618	. . . . . . 7 |- /g Fn Grp
20		fnfun 3585	. . . . . . 7 |- ( /g Fn Grp -> Fun /g )
21	19, 20	ax-mp 7	. . . . . 6 |- Fun /g
22		fofun 3673	. . . . . . . . 9 |- (1st:V-onto->V -> Fun 1st)
23	3, 22	ax-mp 7	. . . . . . . 8 |- Fun 1st
24		funco 3550	. . . . . . . 8 |- ((Fun 1st /\ Fun 1st) -> Fun (1st o. 1st))
25	23, 23, 24	mp2an 697	. . . . . . 7 |- Fun (1st o. 1st)
26		funeq 3535	. . . . . . . 8 |- (+v = (1st o. 1st) -> (Fun +v <-> Fun (1st o. 1st)))
27	1, 26	ax-mp 7	. . . . . . 7 |- (Fun +v <-> Fun (1st o. 1st))
28	25, 27	mpbir 190	. . . . . 6 |- Fun +v
29		fvco 3774	. . . . . 6 |- ((Fun /g /\ Fun +v /\ U e. dom +v) -> (( /g o. +v)` U) = ( /g ` (+v` U)))
30	21, 28, 29	mp3an12 906	. . . . 5 |- (U e. dom +v -> (( /g o. +v)` U) = ( /g ` (+v` U)))
31	13, 30	sylbir 201	. . . 4 |- (U e. V -> (( /g o. +v)` U) = ( /g ` (+v` U)))
32		df-vs 8218	. . . . 5 |- -v = ( /g o. +v)
33	32	fveq1i 3725	. . . 4 |- (-v` U) = (( /g o. +v)` U)
34	31, 33	syl5eq 1519	. . 3 |- (U e. V -> (-v` U) = ( /g ` (+v` U)))
35		0ngrp 8055	. . . . . . 7 |- -. (/) e. Grp
36	17, 18	dmopab2 3619	. . . . . . . 8 |- dom /g = Grp
37	36	eleq2i 1538	. . . . . . 7 |- ((/) e. dom /g <-> (/) e. Grp)
38	35, 37	mtbir 192	. . . . . 6 |- -. (/) e. dom /g
39		ndmfv 3745	. . . . . 6 |- (-. (/) e. dom /g -> ( /g ` (/)) = (/))
40	38, 39	ax-mp 7	. . . . 5 |- ( /g ` (/)) = (/)
41	40	a1i 8	. . . 4 |- (-. U e. V -> ( /g ` (/)) = (/))
42		fvprc 3721	. . . . 5 |- (-. U e. V -> (+v` U) = (/))
43	42	fveq2d 3728	. . . 4 |- (-. U e. V -> ( /g ` (+v` U)) = ( /g ` (/)))
44		fvprc 3721	. . . 4 |- (-. U e. V -> (-v` U) = (/))
45	41, 43, 44	3eqtr4rd 1518	. . 3 |- (-. U e. V -> (-v` U) = ( /g ` (+v` U)))
46	34, 45	pm2.61i 126	. 2 |- (-v` U) = ( /g ` (+v` U))
47		vsfval.3	. 2 |- M = (-v` U)
48		vsfval.2	. . 3 |- G = (+v` U)
49	48	fveq2i 3727	. 2 |- ( /g ` G) = ( /g ` (+v` U))
50	46, 47, 49	3eqtr4 1505	1 |- M = ( /g ` G)

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  (/)c0 2280  dom cdm 3170  ran crn 3171   o. ccom 3174  Fun wfun 3176   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  ` cfv 3182  (class class class)co 3963  {copab2 3964  1stc1st 4077  Grpcgr 8033  invcgn 8035   /g cgs 8036  +vcpv 8204  -vcnsb 8208

This theorem is referenced by:  nvm 8262  nvmfval 8264  nvnnncan1 8268  nvnnncan2 8269  nvaddsubass 8278  nvaddsub 8279  nvmtri2 8300  va1cnlem 8345

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-oprab 3966  df-1st 4079  df-grp 8037  df-gdiv 8040  df-va 8214  df-vs 8218

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