Theorem 0vfval 8309

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 1860	. . . . . . . . . 10 |- x e. V
2	1	rnex 3418	. . . . . . . . 9 |- ran x e. V
3	2	rabex 2780	. . . . . . . 8 |- {z e. ran x | A.w e. ran x(zxw) = w} e. V
4	3	uniex 2926	. . . . . . 7 |- U.{z e. ran x | A.w e. ran x(zxw) = w} e. V
5		df-gid 8123	. . . . . . 7 |- Id = {<.x, y>. | (x e. Grp /\ y = U.{z e. ran x | A.w e. ran x(zxw) = w})}
6	4, 5	fnopab2 3675	. . . . . 6 |- Id Fn Grp
7		fnfun 3642	. . . . . 6 |- (Id Fn Grp -> Fun Id)
8	6, 7	ax-mp 7	. . . . 5 |- Fun Id
9		fo1st 4149	. . . . . . . . 9 |- 1st:V-onto->V
10		fof 3729	. . . . . . . . 9 |- (1st:V-onto->V -> 1st:V-->V)
11	9, 10	ax-mp 7	. . . . . . . 8 |- 1st:V-->V
12		ffn 3684	. . . . . . . 8 |- (1st:V-->V -> 1st Fn V)
13	11, 12	ax-mp 7	. . . . . . 7 |- 1st Fn V
14		ssv 2132	. . . . . . 7 |- ran 1st (_ V
15		fnco 3652	. . . . . . 7 |- ((1st Fn V /\ 1st Fn V /\ ran 1st (_ V) -> (1st o. 1st) Fn V)
16	13, 13, 14, 15	mp3an 928	. . . . . 6 |- (1st o. 1st) Fn V
17		df-va 8298	. . . . . . 7 |- +v = (1st o. 1st)
18		fneq1 3639	. . . . . . 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 197	. . . . 5 |- +v Fn V
21		fvco2 3832	. . . . 5 |- ((Fun Id /\ +v Fn V /\ U e. V) -> ((Id o. +v)` U) = (Id` (+v` U)))
22	8, 20, 21	mp3an12 918	. . . 4 |- (U e. V -> ((Id o. +v)` U) = (Id` (+v` U)))
23		df-0v 8301	. . . . 5 |- 0v = (Id o. +v)
24	23	fveq1i 3782	. . . 4 |- (0v` U) = ((Id o. +v)` U)
25	22, 24	syl5eq 1566	. . 3 |- (U e. V -> (0v` U) = (Id` (+v` U)))
26		fvprc 3778	. . . 4 |- (-. U e. V -> (0v` U) = (/))
27		fvprc 3778	. . . . . 6 |- (-. U e. V -> (+v` U) = (/))
28	27	fveq2d 3785	. . . . 5 |- (-. U e. V -> (Id` (+v` U)) = (Id` (/)))
29		0ngrp 8140	. . . . . . 7 |- -. (/) e. Grp
30	4, 5	dmopab2 3676	. . . . . . . 8 |- dom Id = Grp
31	30	eleq2i 1585	. . . . . . 7 |- ((/) e. dom Id <-> (/) e. Grp)
32	29, 31	mtbir 199	. . . . . 6 |- -. (/) e. dom Id
33		ndmfv 3802	. . . . . 6 |- (-. (/) e. dom Id -> (Id` (/)) = (/))
34	32, 33	ax-mp 7	. . . . 5 |- (Id` (/)) = (/)
35	28, 34	syl6req 1571	. . . 4 |- (-. U e. V -> (/) = (Id` (+v` U)))
36	26, 35	eqtrd 1554	. . 3 |- (-. U e. V -> (0v` U) = (Id` (+v` U)))
37	25, 36	pm2.61i 132	. 2 |- (0v` U) = (Id` (+v` U))
38		0vfval.5	. 2 |- Z = (0v` U)
39		0vfval.2	. . 3 |- G = (+v` U)
40	39	fveq2i 3784	. 2 |- (Id` G) = (Id` (+v` U))
41	37, 38, 40	3eqtr4i 1552	1 |- Z = (Id` G)

Syntax hints:  -. wn 2   <-> wb 153   = wceq 997   e. wcel 999  A.wral 1692  {crab 1695  Vcvv 1858   (_ wss 2098  (/)c0 2331  U.cuni 2557  dom cdm 3227  ran crn 3228   o. ccom 3231  Fun wfun 3233   Fn wfn 3234  -->wf 3235  -onto->wfo 3237  ` cfv 3239  (class class class)co 4021  1stc1st 4135  Grpcgr 8118  Idcgi 8119  +vcpv 8288  0vcn0v 8291

This theorem is referenced by:  nvi 8317  nvvc 8318  nvzcl 8339  nv0rid 8340  nv0lid 8341  nv0 8342  nvsz 8343  nvrinv 8357  nvlinv 8358  nvtri 8382  hh0v 9118  hhssabli 9215

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-rab 1699  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-id 2891  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-f 3251  df-fo 3253  df-fv 3255  df-opr 4023  df-1st 4137  df-grp 8122  df-gid 8123  df-va 8298  df-0v 8301

Copyright terms: Public domain