Theorem grpidvalg 13670

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 13589	. . 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 5690	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpidval.b	. . . . . . 7 |- B = ( Base ` G )
5	3, 4	eqtr4di 2289	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
6	5	eleq2d 2308	. . . . 5 |- ( g = G -> ( e e. ( Base ` g ) <-> e e. B ) )
7		fveq2 5690	. . . . . . . . . 10 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
8		grpidval.p	. . . . . . . . . 10 |- .+ = ( +g ` G )
9	7, 8	eqtr4di 2289	. . . . . . . . 9 |- ( g = G -> ( +g ` g ) = .+ )
10	9	oveqd 6092	. . . . . . . 8 |- ( g = G -> ( e ( +g ` g ) x ) = ( e .+ x ) )
11	10	eqeq1d 2247	. . . . . . 7 |- ( g = G -> ( ( e ( +g ` g ) x ) = x <-> ( e .+ x ) = x ) )
12	9	oveqd 6092	. . . . . . . 8 |- ( g = G -> ( x ( +g ` g ) e ) = ( x .+ e ) )
13	12	eqeq1d 2247	. . . . . . 7 |- ( g = G -> ( ( x ( +g ` g ) e ) = x <-> ( x .+ e ) = x ) )
14	11, 13	anbi12d 477	. . . . . 6 |- ( g = G -> ( ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) <-> ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
15	5, 14	raleqbidv 2765	. . . . 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 477	. . . 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 5355	. . 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 2833	. . 3 |- ( G e. V -> G e. _V )
19		df-riota 6028	. . . 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 13389	. . . . . . 7 |- Base Fn _V
21		funfvex 5707	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
22	21	funfni 5478	. . . . . . 7 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
23	20, 18, 22	sylancr 418	. . . . . 6 |- ( G e. V -> ( Base ` G ) e. _V )
24	4, 23	eqeltrid 2325	. . . . 5 |- ( G e. V -> B e. _V )
25		riotaexg 6032	. . . . 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 2326	. . 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 5793	. 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 2283	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 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   iotacio 5330   Fn wfn 5367   ` cfv 5372   iota_crio 6027  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-0g 13589

This theorem is referenced by:  grpidpropdg  13671  0g0  13673  ismgmid  13674  sgrpidmndm  13710  dfur2g  14240  oppr0g  14360  oppr1g  14361