Theorem grpidvalg 13075

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
grpidvalg	|- ( G e. V -> .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)   V( x, e)   .0. ( x, e)

Proof of Theorem grpidvalg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpidval.o	. 2 |- .0. = ( 0g ` G )
2		df-0g 12960	. . 3 |- 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 ) ) ) )
3		fveq2 5561	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpidval.b	. . . . . . 7 |- B = ( Base ` G )
5	3, 4	eqtr4di 2247	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
6	5	eleq2d 2266	. . . . 5 |- ( g = G -> ( e e. ( Base ` g ) <-> e e. B ) )
7		fveq2 5561	. . . . . . . . . 10 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
8		grpidval.p	. . . . . . . . . 10 |- .+ = ( +g ` G )
9	7, 8	eqtr4di 2247	. . . . . . . . 9 |- ( g = G -> ( +g ` g ) = .+ )
10	9	oveqd 5942	. . . . . . . 8 |- ( g = G -> ( e ( +g ` g ) x ) = ( e .+ x ) )
11	10	eqeq1d 2205	. . . . . . 7 |- ( g = G -> ( ( e ( +g ` g ) x ) = x <-> ( e .+ x ) = x ) )
12	9	oveqd 5942	. . . . . . . 8 |- ( g = G -> ( x ( +g ` g ) e ) = ( x .+ e ) )
13	12	eqeq1d 2205	. . . . . . 7 |- ( g = G -> ( ( x ( +g ` g ) e ) = x <-> ( x .+ e ) = x ) )
14	11, 13	anbi12d 473	. . . . . 6 |- ( g = G -> ( ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) <-> ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
15	5, 14	raleqbidv 2709	. . . . 5 |- ( 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 ) ) )
16	6, 15	anbi12d 473	. . . 4 |- ( 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 ) ) ) )
17	16	iotabidv 5242	. . 3 |- ( 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 ) ) ) )
18		elex 2774	. . 3 |- ( G e. V -> G e. _V )
19		df-riota 5880	. . . 4 |- ( iota_ 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 ) ) )
20		basfn 12761	. . . . . . 7 |- Base Fn _V
21		funfvex 5578	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
22	21	funfni 5361	. . . . . . 7 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
23	20, 18, 22	sylancr 414	. . . . . 6 |- ( G e. V -> ( Base ` G ) e. _V )
24	4, 23	eqeltrid 2283	. . . . 5 |- ( G e. V -> B e. _V )
25		riotaexg 5884	. . . . 5 |- ( B e. _V -> ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) e. _V )
26	24, 25	syl 14	. . . 4 |- ( G e. V -> ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) e. _V )
27	19, 26	eqeltrrid 2284	. . 3 |- ( G e. V -> ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) e. _V )
28	2, 17, 18, 27	fvmptd3 5658	. 2 |- ( G e. V -> ( 0g ` G ) = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) )
29	1, 28	eqtrid 2241	1 |- ( G e. V -> .0. = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   iotacio 5218   Fn wfn 5254   ` cfv 5259   iota_crio 5879  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   0gc0g 12958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-0g 12960

This theorem is referenced by:  grpidpropdg  13076  0g0  13078  ismgmid  13079  sgrpidmndm  13122  dfur2g  13594  oppr0g  13713  oppr1g  13714