Theorem vsfval 9586

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 9546	. . . . . . . 8 |- +v = (1st o. 1st)
2	1	dmeqi 4158	. . . . . . 7 |- dom +v = dom (1st o. 1st)
3		fo1st 5032	. . . . . . . . . . 11 |- 1st:_V-onto->_V
4		fof 4617	. . . . . . . . . . 11 |- (1st:_V-onto->_V -> 1st:_V-->_V)
5	3, 4	ax-mp 7	. . . . . . . . . 10 |- 1st:_V-->_V
6	5	fdmi 4568	. . . . . . . . 9 |- dom 1st = _V
7		forn 4620	. . . . . . . . . 10 |- (1st:_V-onto->_V -> ran 1st = _V)
8	3, 7	ax-mp 7	. . . . . . . . 9 |- ran 1st = _V
9	6, 8	eqtr4i 1911	. . . . . . . 8 |- dom 1st = ran 1st
10		dmcoeq 4216	. . . . . . . 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	3eqtri 1912	. . . . . 6 |- dom +v = _V
13	12	eleq2i 1961	. . . . 5 |- (U e. dom +v <-> U e. _V)
14		visset 2295	. . . . . . . . . 10 |- g e. _V
15	14	rnex 4209	. . . . . . . . 9 |- ran g e. _V
16		eqid 1884	. . . . . . . . 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 4950	. . . . . . . 8 |- {<.<.x, y>., z>. | ((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y)))} e. _V
18		df-gdiv 9319	. . . . . . . 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 4549	. . . . . . 7 |- /g Fn Grp
20		fnfun 4510	. . . . . . 7 |- ( /g Fn Grp -> Fun /g )
21	19, 20	ax-mp 7	. . . . . 6 |- Fun /g
22		fofun 4618	. . . . . . . . 9 |- (1st:_V-onto->_V -> Fun 1st)
23	3, 22	ax-mp 7	. . . . . . . 8 |- Fun 1st
24		funco 4457	. . . . . . . 8 |- ((Fun 1st /\ Fun 1st) -> Fun (1st o. 1st))
25	23, 23, 24	mp2an 761	. . . . . . 7 |- Fun (1st o. 1st)
26		funeq 4441	. . . . . . . 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 207	. . . . . 6 |- Fun +v
29		fvco 4736	. . . . . 6 |- ((Fun /g /\ Fun +v /\ U e. dom +v) -> (( /g o. +v)` U) = ( /g ` (+v` U)))
30	21, 28, 29	mp3an12 1181	. . . . 5 |- (U e. dom +v -> (( /g o. +v)` U) = ( /g ` (+v` U)))
31	13, 30	sylbir 218	. . . 4 |- (U e. _V -> (( /g o. +v)` U) = ( /g ` (+v` U)))
32		df-vs 9550	. . . . 5 |- -v = ( /g o. +v)
33	32	fveq1i 4682	. . . 4 |- (-v` U) = (( /g o. +v)` U)
34	31, 33	syl5eq 1940	. . 3 |- (U e. _V -> (-v` U) = ( /g ` (+v` U)))
35		0ngrp 9335	. . . . . . 7 |- -. (/) e. Grp
36	17, 18	dmopab2 4550	. . . . . . . 8 |- dom /g = Grp
37	36	eleq2i 1961	. . . . . . 7 |- ((/) e. dom /g <-> (/) e. Grp)
38	35, 37	mtbir 209	. . . . . 6 |- -. (/) e. dom /g
39		ndmfv 4702	. . . . . 6 |- (-. (/) e. dom /g -> ( /g ` (/)) = (/))
40	38, 39	ax-mp 7	. . . . 5 |- ( /g ` (/)) = (/)
41	40	a1i 8	. . . 4 |- (-. U e. _V -> ( /g ` (/)) = (/))
42		fvprc 4678	. . . . 5 |- (-. U e. _V -> (+v` U) = (/))
43	42	fveq2d 4685	. . . 4 |- (-. U e. _V -> ( /g ` (+v` U)) = ( /g ` (/)))
44		fvprc 4678	. . . 4 |- (-. U e. _V -> (-v` U) = (/))
45	41, 43, 44	3eqtr4rd 1939	. . 3 |- (-. U e. _V -> (-v` U) = ( /g ` (+v` U)))
46	34, 45	pm2.61i 140	. 2 |- (-v` U) = ( /g ` (+v` U))
47		vsfval.3	. 2 |- M = (-v` U)
48		vsfval.2	. . 3 |- G = (+v` U)
49	48	fveq2i 4684	. 2 |- ( /g ` G) = ( /g ` (+v` U))
50	46, 47, 49	3eqtr4i 1921	1 |- M = ( /g ` G)

Syntax hints:  -. wn 2   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  dom cdm 3986  ran crn 3987   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  Grpcgr 9311  invcgn 9313   /g cgs 9314  +vcpv 9536  -vcnsb 9540

This theorem is referenced by:  nvm 9594  nvmfval 9596  nvnnncan1 9600  nvnnncan2 9601  nvaddsubass 9610  nvaddsub 9611  nvmtri2 9632  va1cnlem 9684

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-grp 9316  df-gdiv 9319  df-va 9546  df-vs 9550

Copyright terms: Public domain