Theorem grpinvpropdg 12945

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 12793	. . . . . . . 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 2192	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( ( x ( +g ` K ) y ) = ( 0g ` K ) <-> ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
9	8	anass1rs 571	. . . . 5 |- ( ( ( ph /\ y e. B ) /\ x e. B ) -> ( ( x ( +g ` K ) y ) = ( 0g ` K ) <-> ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
10	9	riotabidva 5847	. . . 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 4094	. . 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 5832	. . . 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 4086	. . 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 5832	. . . 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 4086	. . 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 2218	. 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 2177	. . . 4 |- ( Base ` K ) = ( Base ` K )
18		eqid 2177	. . . 4 |- ( +g ` K ) = ( +g ` K )
19		eqid 2177	. . . 4 |- ( 0g ` K ) = ( 0g ` K )
20		eqid 2177	. . . 4 |- ( invg ` K ) = ( invg ` K )
21	17, 18, 19, 20	grpinvfvalg 12915	. . 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 2177	. . . 4 |- ( Base ` L ) = ( Base ` L )
24		eqid 2177	. . . 4 |- ( +g ` L ) = ( +g ` L )
25		eqid 2177	. . . 4 |- ( 0g ` L ) = ( 0g ` L )
26		eqid 2177	. . . 4 |- ( invg ` L ) = ( invg ` L )
27	23, 24, 25, 26	grpinvfvalg 12915	. . 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 2220	1 |- ( ph -> ( invg ` K ) = ( invg ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   |-> cmpt 4065   ` cfv 5217   iota_crio 5830  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   0gc0g 12705   invgcminusg 12878

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-inn 8920  df-ndx 12465  df-slot 12466  df-base 12468  df-0g 12707  df-minusg 12881

This theorem is referenced by:  grpsubpropdg  12974  grpsubpropd2  12975  mulgpropdg  13025  invrpropdg  13318