Theorem 0vfval 8225

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

Hypotheses

Ref	Expression
0vfval.2	|- G = (+v` U)
0vfval.5	|- Z = (0v` U)

Assertion

Ref	Expression
0vfval	|- Z = (Id` G)

Proof of Theorem 0vfval

Step	Hyp	Ref	Expression
1		visset 1813	. . . . . . . . . 10 |- x e. V
2	1	rnex 3361	. . . . . . . . 9 |- ran x e. V
3	2	rabex 2725	. . . . . . . 8 |- {z e. ran x | A.w e. ran x(zxw) = w} e. V
4	3	uniex 2870	. . . . . . 7 |- U.{z e. ran x | A.w e. ran x(zxw) = w} e. V
5		df-gid 8038	. . . . . . 7 |- Id = {<.x, y>. | (x e. Grp /\ y = U.{z e. ran x | A.w e. ran x(zxw) = w})}
6	4, 5	fnopab2 3618	. . . . . 6 |- Id Fn Grp
7		fnfun 3585	. . . . . 6 |- (Id Fn Grp -> Fun Id)
8	6, 7	ax-mp 7	. . . . 5 |- Fun Id
9		fo1st 4091	. . . . . . . . 9 |- 1st:V-onto->V
10		fof 3672	. . . . . . . . 9 |- (1st:V-onto->V -> 1st:V-->V)
11	9, 10	ax-mp 7	. . . . . . . 8 |- 1st:V-->V
12		ffn 3627	. . . . . . . 8 |- (1st:V-->V -> 1st Fn V)
13	11, 12	ax-mp 7	. . . . . . 7 |- 1st Fn V
14		ssv 2081	. . . . . . 7 |- ran 1st (_ V
15		fnco 3595	. . . . . . 7 |- ((1st Fn V /\ 1st Fn V /\ ran 1st (_ V) -> (1st o. 1st) Fn V)
16	13, 13, 14, 15	mp3an 916	. . . . . 6 |- (1st o. 1st) Fn V
17		df-va 8214	. . . . . . 7 |- +v = (1st o. 1st)
18		fneq1 3582	. . . . . . 7 |- (+v = (1st o. 1st) -> (+v Fn V <-> (1st o. 1st) Fn V))
19	17, 18	ax-mp 7	. . . . . 6 |- (+v Fn V <-> (1st o. 1st) Fn V)
20	16, 19	mpbir 190	. . . . 5 |- +v Fn V
21		fvco2 3775	. . . . 5 |- ((Fun Id /\ +v Fn V /\ U e. V) -> ((Id o. +v)` U) = (Id` (+v` U)))
22	8, 20, 21	mp3an12 906	. . . 4 |- (U e. V -> ((Id o. +v)` U) = (Id` (+v` U)))
23		df-0v 8217	. . . . 5 |- 0v = (Id o. +v)
24	23	fveq1i 3725	. . . 4 |- (0v` U) = ((Id o. +v)` U)
25	22, 24	syl5eq 1519	. . 3 |- (U e. V -> (0v` U) = (Id` (+v` U)))
26		fvprc 3721	. . . 4 |- (-. U e. V -> (0v` U) = (/))
27		fvprc 3721	. . . . . 6 |- (-. U e. V -> (+v` U) = (/))
28	27	fveq2d 3728	. . . . 5 |- (-. U e. V -> (Id` (+v` U)) = (Id` (/)))
29		0ngrp 8055	. . . . . . 7 |- -. (/) e. Grp
30	4, 5	dmopab2 3619	. . . . . . . 8 |- dom Id = Grp
31	30	eleq2i 1538	. . . . . . 7 |- ((/) e. dom Id <-> (/) e. Grp)
32	29, 31	mtbir 192	. . . . . 6 |- -. (/) e. dom Id
33		ndmfv 3745	. . . . . 6 |- (-. (/) e. dom Id -> (Id` (/)) = (/))
34	32, 33	ax-mp 7	. . . . 5 |- (Id` (/)) = (/)
35	28, 34	syl6req 1524	. . . 4 |- (-. U e. V -> (/) = (Id` (+v` U)))
36	26, 35	eqtrd 1507	. . 3 |- (-. U e. V -> (0v` U) = (Id` (+v` U)))
37	25, 36	pm2.61i 126	. 2 |- (0v` U) = (Id` (+v` U))
38		0vfval.5	. 2 |- Z = (0v` U)
39		0vfval.2	. . 3 |- G = (+v` U)
40	39	fveq2i 3727	. 2 |- (Id` G) = (Id` (+v` U))
41	37, 38, 40	3eqtr4 1505	1 |- Z = (Id` G)

Syntax hints:  -. wn 2   <-> wb 146   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648  Vcvv 1811   (_ wss 2047  (/)c0 2280  U.cuni 2503  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  1stc1st 4077  Grpcgr 8033  Idcgi 8034  +vcpv 8204  0vcn0v 8207

This theorem is referenced by:  nvi 8233  nvvc 8234  nvzcl 8255  nv0rid 8256  nv0lid 8257  nv0 8258  nvsz 8259  nvrinv 8273  nvlinv 8274  nvtri 8298  hh0v 9035  hhssabl 9132

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-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-rab 1652  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-1st 4079  df-grp 8037  df-gid 8038  df-va 8214  df-0v 8217

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