Theorem vsfval 8206

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 8166	. . . . . . . 8 |- +v = (1st o. 1st)
2	1	dmeqi 3307	. . . . . . 7 |- dom +v = dom (1st o. 1st)
3		fo1st 4081	. . . . . . . . . . 11 |- 1st:V-onto->V
4		fof 3663	. . . . . . . . . . 11 |- (1st:V-onto->V -> 1st:V-->V)
5	3, 4	ax-mp 7	. . . . . . . . . 10 |- 1st:V-->V
6		fdm 3623	. . . . . . . . . 10 |- (1st:V-->V -> dom 1st = V)
7	5, 6	ax-mp 7	. . . . . . . . 9 |- dom 1st = V
8		forn 3665	. . . . . . . . . 10 |- (1st:V-onto->V -> ran 1st = V)
9	3, 8	ax-mp 7	. . . . . . . . 9 |- ran 1st = V
10	7, 9	eqtr4 1495	. . . . . . . 8 |- dom 1st = ran 1st
11		dmcoeq 3358	. . . . . . . 8 |- (dom 1st = ran 1st -> dom (1st o. 1st) = dom 1st)
12	10, 11	ax-mp 7	. . . . . . 7 |- dom (1st o. 1st) = dom 1st
13	2, 12, 7	3eqtr 1496	. . . . . 6 |- dom +v = V
14	13	eleq2i 1535	. . . . 5 |- (U e. dom +v <-> U e. V)
15		visset 1809	. . . . . . . . . 10 |- g e. V
16		rnexg 3353	. . . . . . . . . 10 |- (g e. V -> ran g e. V)
17	15, 16	ax-mp 7	. . . . . . . . 9 |- ran g e. V
18		eqid 1473	. . . . . . . . 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)))}
19	17, 17, 18	oprabex2 4012	. . . . . . . 8 |- {<.<.x, y>., z>. | ((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y)))} e. V
20		df-gdiv 7990	. . . . . . . 8 |- /g = {<.g, f>. | (g e. Grp /\ f = {<.<.x, y>., z>. | ((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y)))})}
21	19, 20	fnopab2 3610	. . . . . . 7 |- /g Fn Grp
22		fnfun 3577	. . . . . . 7 |- ( /g Fn Grp -> Fun /g )
23	21, 22	ax-mp 7	. . . . . 6 |- Fun /g
24		fofun 3664	. . . . . . . . 9 |- (1st:V-onto->V -> Fun 1st)
25	3, 24	ax-mp 7	. . . . . . . 8 |- Fun 1st
26		funco 3542	. . . . . . . 8 |- ((Fun 1st /\ Fun 1st) -> Fun (1st o. 1st))
27	25, 25, 26	mp2an 696	. . . . . . 7 |- Fun (1st o. 1st)
28		funeq 3527	. . . . . . . 8 |- (+v = (1st o. 1st) -> (Fun +v <-> Fun (1st o. 1st)))
29	1, 28	ax-mp 7	. . . . . . 7 |- (Fun +v <-> Fun (1st o. 1st))
30	27, 29	mpbir 190	. . . . . 6 |- Fun +v
31		fvco 3765	. . . . . 6 |- ((Fun /g /\ Fun +v /\ U e. dom +v) -> (( /g o. +v)` U) = ( /g ` (+v` U)))
32	23, 30, 31	mp3an12 904	. . . . 5 |- (U e. dom +v -> (( /g o. +v)` U) = ( /g ` (+v` U)))
33	14, 32	sylbir 201	. . . 4 |- (U e. V -> (( /g o. +v)` U) = ( /g ` (+v` U)))
34		df-vs 8170	. . . . 5 |- -v = ( /g o. +v)
35	34	fveq1i 3716	. . . 4 |- (-v` U) = (( /g o. +v)` U)
36	33, 35	syl5eq 1516	. . 3 |- (U e. V -> (-v` U) = ( /g ` (+v` U)))
37		0ngrp 8005	. . . . . . 7 |- -. (/) e. Grp
38	19, 20	dmopab2 3611	. . . . . . . 8 |- dom /g = Grp
39	38	eleq2i 1535	. . . . . . 7 |- ((/) e. dom /g <-> (/) e. Grp)
40	37, 39	mtbir 192	. . . . . 6 |- -. (/) e. dom /g
41		ndmfv 3736	. . . . . 6 |- (-. (/) e. dom /g -> ( /g ` (/)) = (/))
42	40, 41	ax-mp 7	. . . . 5 |- ( /g ` (/)) = (/)
43	42	a1i 8	. . . 4 |- (-. U e. V -> ( /g ` (/)) = (/))
44		fvprc 3712	. . . . 5 |- (-. U e. V -> (+v` U) = (/))
45	44	fveq2d 3719	. . . 4 |- (-. U e. V -> ( /g ` (+v` U)) = ( /g ` (/)))
46		fvprc 3712	. . . 4 |- (-. U e. V -> (-v` U) = (/))
47	43, 45, 46	3eqtr4rd 1515	. . 3 |- (-. U e. V -> (-v` U) = ( /g ` (+v` U)))
48	36, 47	pm2.61i 126	. 2 |- (-v` U) = ( /g ` (+v` U))
49		vsfval.3	. 2 |- M = (-v` U)
50		vsfval.2	. . 3 |- G = (+v` U)
51	50	fveq2i 3718	. 2 |- ( /g ` G) = ( /g ` (+v` U))
52	48, 49, 51	3eqtr4 1502	1 |- M = ( /g ` G)

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  Vcvv 1807  (/)c0 2276  dom cdm 3165  ran crn 3166   o. ccom 3169  Fun wfun 3171   Fn wfn 3172  -->wf 3173  -onto->wfo 3175  ` cfv 3177  (class class class)co 3954  {copab2 3955  1stc1st 4067  Grpcgr 7983  invcgn 7985   /g cgs 7986  +vcpv 8156  -vcnsb 8160

This theorem is referenced by:  nvm 8214  nvmfval 8216  nvnnncan1 8220  nvnnncan2 8221  nvaddsubass 8230  nvaddsub 8231  nvmtri2 8252  va1cnlem 8292

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193  df-opr 3956  df-oprab 3957  df-1st 4069  df-grp 7987  df-gdiv 7990  df-va 8166  df-vs 8170

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