Theorem grpidvalg 12627

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 12598	. . 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 5496	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpidval.b	. . . . . . 7 |- B = ( Base ` G )
5	3, 4	eqtr4di 2221	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
6	5	eleq2d 2240	. . . . 5 |- ( g = G -> ( e e. ( Base ` g ) <-> e e. B ) )
7		fveq2 5496	. . . . . . . . . 10 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
8		grpidval.p	. . . . . . . . . 10 |- .+ = ( +g ` G )
9	7, 8	eqtr4di 2221	. . . . . . . . 9 |- ( g = G -> ( +g ` g ) = .+ )
10	9	oveqd 5870	. . . . . . . 8 |- ( g = G -> ( e ( +g ` g ) x ) = ( e .+ x ) )
11	10	eqeq1d 2179	. . . . . . 7 |- ( g = G -> ( ( e ( +g ` g ) x ) = x <-> ( e .+ x ) = x ) )
12	9	oveqd 5870	. . . . . . . 8 |- ( g = G -> ( x ( +g ` g ) e ) = ( x .+ e ) )
13	12	eqeq1d 2179	. . . . . . 7 |- ( g = G -> ( ( x ( +g ` g ) e ) = x <-> ( x .+ e ) = x ) )
14	11, 13	anbi12d 470	. . . . . 6 |- ( g = G -> ( ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) <-> ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
15	5, 14	raleqbidv 2677	. . . . 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 470	. . . 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 5181	. . 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 2741	. . 3 |- ( G e. V -> G e. _V )
19		df-riota 5809	. . . 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 12473	. . . . . . 7 |- Base Fn _V
21		funfvex 5513	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
22	21	funfni 5298	. . . . . . 7 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
23	20, 18, 22	sylancr 412	. . . . . 6 |- ( G e. V -> ( Base ` G ) e. _V )
24	4, 23	eqeltrid 2257	. . . . 5 |- ( G e. V -> B e. _V )
25		riotaexg 5813	. . . . 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 2258	. . 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 5589	. 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 2215	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 103   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   iotacio 5158   Fn wfn 5193   ` cfv 5198   iota_crio 5808  (class class class)co 5853   Basecbs 12416   +g cplusg 12480   0gc0g 12596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-riota 5809  df-ov 5856  df-inn 8879  df-ndx 12419  df-slot 12420  df-base 12422  df-0g 12598

This theorem is referenced by:  grpidpropdg  12628  0g0  12630  ismgmid  12631  sgrpidmndm  12656