Theorem grpidvalg 12821

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 12735	. . 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 5530	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpidval.b	. . . . . . 7 |- B = ( Base ` G )
5	3, 4	eqtr4di 2240	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
6	5	eleq2d 2259	. . . . 5 |- ( g = G -> ( e e. ( Base ` g ) <-> e e. B ) )
7		fveq2 5530	. . . . . . . . . 10 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
8		grpidval.p	. . . . . . . . . 10 |- .+ = ( +g ` G )
9	7, 8	eqtr4di 2240	. . . . . . . . 9 |- ( g = G -> ( +g ` g ) = .+ )
10	9	oveqd 5908	. . . . . . . 8 |- ( g = G -> ( e ( +g ` g ) x ) = ( e .+ x ) )
11	10	eqeq1d 2198	. . . . . . 7 |- ( g = G -> ( ( e ( +g ` g ) x ) = x <-> ( e .+ x ) = x ) )
12	9	oveqd 5908	. . . . . . . 8 |- ( g = G -> ( x ( +g ` g ) e ) = ( x .+ e ) )
13	12	eqeq1d 2198	. . . . . . 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 2698	. . . . 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 5214	. . 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 2763	. . 3 |- ( G e. V -> G e. _V )
19		df-riota 5847	. . . 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 12544	. . . . . . 7 |- Base Fn _V
21		funfvex 5547	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
22	21	funfni 5331	. . . . . . 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 2276	. . . . 5 |- ( G e. V -> B e. _V )
25		riotaexg 5851	. . . . 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 2277	. . 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 5625	. 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 2234	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 2160   A.wral 2468   _Vcvv 2752   iotacio 5191   Fn wfn 5226   ` cfv 5231   iota_crio 5846  (class class class)co 5891   Basecbs 12486   +g cplusg 12561   0gc0g 12733

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-riota 5847  df-ov 5894  df-inn 8939  df-ndx 12489  df-slot 12490  df-base 12492  df-0g 12735

This theorem is referenced by:  grpidpropdg  12822  0g0  12824  ismgmid  12825  sgrpidmndm  12853  dfur2g  13283  oppr0g  13398  oppr1g  13399