Theorem grpinvpropdg 13880

Description: If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.)

Hypotheses

Ref	Expression
grpinvpropd.1	|- ( ph -> B = ( Base ` K ) )
grpinvpropd.2	|- ( ph -> B = ( Base ` L ) )
grpinvpropdg.k	|- ( ph -> K e. V )
grpinvpropdg.l	|- ( ph -> L e. W )
grpinvpropd.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )

Assertion

Ref	Expression
grpinvpropdg	|- ( ph -> ( invg ` K ) = ( invg ` L ) )

Distinct variable groups:   x, y, B   x, K, y   x, L, y   ph, x, y

Allowed substitution hints:   V( x, y)   W( x, y)

Proof of Theorem grpinvpropdg

Step	Hyp	Ref	Expression
1		grpinvpropd.3	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
2		grpinvpropd.1	. . . . . . . . 9 |- ( ph -> B = ( Base ` K ) )
3		grpinvpropd.2	. . . . . . . . 9 |- ( ph -> B = ( Base ` L ) )
4		grpinvpropdg.k	. . . . . . . . 9 |- ( ph -> K e. V )
5		grpinvpropdg.l	. . . . . . . . 9 |- ( ph -> L e. W )
6	2, 3, 4, 5, 1	grpidpropdg 13694	. . . . . . . 8 |- ( ph -> ( 0g ` K ) = ( 0g ` L ) )
7	6	adantr 276	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( 0g ` K ) = ( 0g ` L ) )
8	1, 7	eqeq12d 2253	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( ( x ( +g ` K ) y ) = ( 0g ` K ) <-> ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
9	8	anass1rs 577	. . . . 5 |- ( ( ( ph /\ y e. B ) /\ x e. B ) -> ( ( x ( +g ` K ) y ) = ( 0g ` K ) <-> ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
10	9	riotabidva 6056	. . . 4 |- ( ( ph /\ y e. B ) -> ( iota_ x e. B ( x ( +g ` K ) y ) = ( 0g ` K ) ) = ( iota_ x e. B ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
11	10	mpteq2dva 4221	. . 3 |- ( ph -> ( y e. B |-> ( iota_ x e. B ( x ( +g ` K ) y ) = ( 0g ` K ) ) ) = ( y e. B |-> ( iota_ x e. B ( x ( +g ` L ) y ) = ( 0g ` L ) ) ) )
12	2	riotaeqdv 6039	. . . 4 |- ( ph -> ( iota_ x e. B ( x ( +g ` K ) y ) = ( 0g ` K ) ) = ( iota_ x e. ( Base ` K ) ( x ( +g ` K ) y ) = ( 0g ` K ) ) )
13	2, 12	mpteq12dv 4213	. . 3 |- ( ph -> ( y e. B |-> ( iota_ x e. B ( x ( +g ` K ) y ) = ( 0g ` K ) ) ) = ( y e. ( Base ` K ) |-> ( iota_ x e. ( Base ` K ) ( x ( +g ` K ) y ) = ( 0g ` K ) ) ) )
14	3	riotaeqdv 6039	. . . 4 |- ( ph -> ( iota_ x e. B ( x ( +g ` L ) y ) = ( 0g ` L ) ) = ( iota_ x e. ( Base ` L ) ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
15	3, 14	mpteq12dv 4213	. . 3 |- ( ph -> ( y e. B |-> ( iota_ x e. B ( x ( +g ` L ) y ) = ( 0g ` L ) ) ) = ( y e. ( Base ` L ) |-> ( iota_ x e. ( Base ` L ) ( x ( +g ` L ) y ) = ( 0g ` L ) ) ) )
16	11, 13, 15	3eqtr3d 2279	. 2 |- ( ph -> ( y e. ( Base ` K ) |-> ( iota_ x e. ( Base ` K ) ( x ( +g ` K ) y ) = ( 0g ` K ) ) ) = ( y e. ( Base ` L ) |-> ( iota_ x e. ( Base ` L ) ( x ( +g ` L ) y ) = ( 0g ` L ) ) ) )
17		eqid 2238	. . . 4 |- ( Base ` K ) = ( Base ` K )
18		eqid 2238	. . . 4 |- ( +g ` K ) = ( +g ` K )
19		eqid 2238	. . . 4 |- ( 0g ` K ) = ( 0g ` K )
20		eqid 2238	. . . 4 |- ( invg ` K ) = ( invg ` K )
21	17, 18, 19, 20	grpinvfvalg 13847	. . 3 |- ( K e. V -> ( invg ` K ) = ( y e. ( Base ` K ) |-> ( iota_ x e. ( Base ` K ) ( x ( +g ` K ) y ) = ( 0g ` K ) ) ) )
22	4, 21	syl 14	. 2 |- ( ph -> ( invg ` K ) = ( y e. ( Base ` K ) |-> ( iota_ x e. ( Base ` K ) ( x ( +g ` K ) y ) = ( 0g ` K ) ) ) )
23		eqid 2238	. . . 4 |- ( Base ` L ) = ( Base ` L )
24		eqid 2238	. . . 4 |- ( +g ` L ) = ( +g ` L )
25		eqid 2238	. . . 4 |- ( 0g ` L ) = ( 0g ` L )
26		eqid 2238	. . . 4 |- ( invg ` L ) = ( invg ` L )
27	23, 24, 25, 26	grpinvfvalg 13847	. . 3 |- ( L e. W -> ( invg ` L ) = ( y e. ( Base ` L ) |-> ( iota_ x e. ( Base ` L ) ( x ( +g ` L ) y ) = ( 0g ` L ) ) ) )
28	5, 27	syl 14	. 2 |- ( ph -> ( invg ` L ) = ( y e. ( Base ` L ) |-> ( iota_ x e. ( Base ` L ) ( x ( +g ` L ) y ) = ( 0g ` L ) ) ) )
29	16, 22, 28	3eqtr4d 2281	1 |- ( ph -> ( invg ` K ) = ( invg ` L ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   |-> cmpt 4192   ` cfv 5377   iota_crio 6037  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610   invgcminusg 13806

This proof depends on 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-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof 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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358  df-0g 13612  df-minusg 13809

This theorem is used by:  grpsubpropdg  13909  grpsubpropd2  13910  mulgpropdg  13967  invrpropdg  14456  rlmvnegg  14802