Theorem 0vfval 27461

Description: Value of the function for the zero vector on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
0vfval	|- ( U e. V -> Z = (GId ` G ) )

Proof of Theorem 0vfval

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( U e. V -> U e. _V )
2		fo1st 7188	. . . . . . 7 |- 1st : _V -onto-> _V
3		fofn 6117	. . . . . . 7 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
4	2, 3	ax-mp 5	. . . . . 6 |- 1st Fn _V
5		ssv 3625	. . . . . 6 |- ran 1st C_ _V
6		fnco 5999	. . . . . 6 |- ( ( 1st Fn _V /\ 1st Fn _V /\ ran 1st C_ _V ) -> ( 1st o. 1st ) Fn _V )
7	4, 4, 5, 6	mp3an 1424	. . . . 5 |- ( 1st o. 1st ) Fn _V
8		df-va 27450	. . . . . 6 |- +v = ( 1st o. 1st )
9	8	fneq1i 5985	. . . . 5 |- ( +v Fn _V <-> ( 1st o. 1st ) Fn _V )
10	7, 9	mpbir 221	. . . 4 |- +v Fn _V
11		fvco2 6273	. . . 4 |- ( ( +v Fn _V /\ U e. _V ) -> ( (GId o. +v ) ` U ) = (GId ` ( +v ` U ) ) )
12	10, 11	mpan 706	. . 3 |- ( U e. _V -> ( (GId o. +v ) ` U ) = (GId ` ( +v ` U ) ) )
13		0vfval.5	. . . 4 |- Z = ( 0vec ` U )
14		df-0v 27453	. . . . 5 |- 0vec = (GId o. +v )
15	14	fveq1i 6192	. . . 4 |- ( 0vec ` U ) = ( (GId o. +v ) ` U )
16	13, 15	eqtri 2644	. . 3 |- Z = ( (GId o. +v ) ` U )
17		0vfval.2	. . . 4 |- G = ( +v ` U )
18	17	fveq2i 6194	. . 3 |- (GId ` G ) = (GId ` ( +v ` U ) )
19	12, 16, 18	3eqtr4g 2681	. 2 |- ( U e. _V -> Z = (GId ` G ) )
20	1, 19	syl 17	1 |- ( U e. V -> Z = (GId ` G ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ran crn 5115   o. ccom 5118   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888   1stc1st 7166  GIdcgi 27344   +vcpv 27440   0veccn0v 27443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-va 27450  df-0v 27453

This theorem is referenced by:  nvi  27469  nvzcl  27489  nv0rid  27490  nv0lid  27491  nv0  27492  nvsz  27493  nvrinv  27506  nvlinv  27507  hh0v  28025