Theorem grpidval 17260

Description: The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
grpidval.b	|- B = ( Base ` G )
grpidval.p	|- .+ = ( +g ` G )
grpidval.o	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
grpidval	|- .0. = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )

Distinct variable groups:   x, e, B   e, G, x

Allowed substitution hints:   .+ ( x, e)   .0. ( x, e)

Proof of Theorem grpidval

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpidval.o	. 2 |- .0. = ( 0g ` G )
2		fveq2 6191	. . . . . . . 8 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
3		grpidval.b	. . . . . . . 8 |- B = ( Base ` G )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( g = G -> ( Base ` g ) = B )
5	4	eleq2d 2687	. . . . . 6 |- ( g = G -> ( e e. ( Base ` g ) <-> e e. B ) )
6		fveq2 6191	. . . . . . . . . . 11 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		grpidval.p	. . . . . . . . . . 11 |- .+ = ( +g ` G )
8	6, 7	syl6eqr 2674	. . . . . . . . . 10 |- ( g = G -> ( +g ` g ) = .+ )
9	8	oveqd 6667	. . . . . . . . 9 |- ( g = G -> ( e ( +g ` g ) x ) = ( e .+ x ) )
10	9	eqeq1d 2624	. . . . . . . 8 |- ( g = G -> ( ( e ( +g ` g ) x ) = x <-> ( e .+ x ) = x ) )
11	8	oveqd 6667	. . . . . . . . 9 |- ( g = G -> ( x ( +g ` g ) e ) = ( x .+ e ) )
12	11	eqeq1d 2624	. . . . . . . 8 |- ( g = G -> ( ( x ( +g ` g ) e ) = x <-> ( x .+ e ) = x ) )
13	10, 12	anbi12d 747	. . . . . . 7 |- ( g = G -> ( ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) <-> ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
14	4, 13	raleqbidv 3152	. . . . . 6 |- ( g = G -> ( A. x e. ( Base ` g ) ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) <-> A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
15	5, 14	anbi12d 747	. . . . 5 |- ( g = G -> ( ( e e. ( Base ` g ) /\ A. x e. ( Base ` g ) ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) ) <-> ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) )
16	15	iotabidv 5872	. . . 4 |- ( g = G -> ( iota e ( e e. ( Base ` g ) /\ A. x e. ( Base ` g ) ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) ) ) = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) )
17		df-0g 16102	. . . 4 |- 0g = ( g e. _V |-> ( iota e ( e e. ( Base ` g ) /\ A. x e. ( Base ` g ) ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) ) ) )
18		iotaex 5868	. . . 4 |- ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) e. _V
19	16, 17, 18	fvmpt 6282	. . 3 |- ( G e. _V -> ( 0g ` G ) = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) )
20		fvprc 6185	. . . 4 |- ( -. G e. _V -> ( 0g ` G ) = (/) )
21		euex 2494	. . . . . . 7 |- ( E! e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) -> E. e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
22		n0i 3920	. . . . . . . . . 10 |- ( e e. B -> -. B = (/) )
23		fvprc 6185	. . . . . . . . . . 11 |- ( -. G e. _V -> ( Base ` G ) = (/) )
24	3, 23	syl5eq 2668	. . . . . . . . . 10 |- ( -. G e. _V -> B = (/) )
25	22, 24	nsyl2 142	. . . . . . . . 9 |- ( e e. B -> G e. _V )
26	25	adantr 481	. . . . . . . 8 |- ( ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) -> G e. _V )
27	26	exlimiv 1858	. . . . . . 7 |- ( E. e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) -> G e. _V )
28	21, 27	syl 17	. . . . . 6 |- ( E! e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) -> G e. _V )
29	28	con3i 150	. . . . 5 |- ( -. G e. _V -> -. E! e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
30		iotanul 5866	. . . . 5 |- ( -. E! e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) -> ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) = (/) )
31	29, 30	syl 17	. . . 4 |- ( -. G e. _V -> ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) = (/) )
32	20, 31	eqtr4d 2659	. . 3 |- ( -. G e. _V -> ( 0g ` G ) = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) )
33	19, 32	pm2.61i 176	. 2 |- ( 0g ` G ) = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
34	1, 33	eqtri 2644	1 |- .0. = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   A.wral 2912   _Vcvv 3200   (/)c0 3915   iotacio 5849   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100

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

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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-0g 16102

This theorem is referenced by:  grpidpropd  17261  0g0  17263  ismgmid  17264  oppgid  17786  dfur2  18504  oppr0  18633  oppr1  18634