Theorem grpidvalg 13455

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 13340	. . 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 5639	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpidval.b	. . . . . . 7 |- B = ( Base ` G )
5	3, 4	eqtr4di 2282	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
6	5	eleq2d 2301	. . . . 5 |- ( g = G -> ( e e. ( Base ` g ) <-> e e. B ) )
7		fveq2 5639	. . . . . . . . . 10 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
8		grpidval.p	. . . . . . . . . 10 |- .+ = ( +g ` G )
9	7, 8	eqtr4di 2282	. . . . . . . . 9 |- ( g = G -> ( +g ` g ) = .+ )
10	9	oveqd 6034	. . . . . . . 8 |- ( g = G -> ( e ( +g ` g ) x ) = ( e .+ x ) )
11	10	eqeq1d 2240	. . . . . . 7 |- ( g = G -> ( ( e ( +g ` g ) x ) = x <-> ( e .+ x ) = x ) )
12	9	oveqd 6034	. . . . . . . 8 |- ( g = G -> ( x ( +g ` g ) e ) = ( x .+ e ) )
13	12	eqeq1d 2240	. . . . . . 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 2746	. . . . 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 5309	. . 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 2814	. . 3 |- ( G e. V -> G e. _V )
19		df-riota 5970	. . . 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 13140	. . . . . . 7 |- Base Fn _V
21		funfvex 5656	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
22	21	funfni 5432	. . . . . . 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 2318	. . . . 5 |- ( G e. V -> B e. _V )
25		riotaexg 5974	. . . . 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 2319	. . 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 5740	. 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 2276	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 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   iotacio 5284   Fn wfn 5321   ` cfv 5326   iota_crio 5969  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-0g 13340

This theorem is referenced by:  grpidpropdg  13456  0g0  13458  ismgmid  13459  sgrpidmndm  13502  dfur2g  13974  oppr0g  14093  oppr1g  14094