Theorem grpinvpropdg 13788

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 13587	. . . . . . . 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 2247	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( ( x ( +g ` K ) y ) = ( 0g ` K ) <-> ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
9	8	anass1rs 573	. . . . 5 |- ( ( ( ph /\ y e. B ) /\ x e. B ) -> ( ( x ( +g ` K ) y ) = ( 0g ` K ) <-> ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
10	9	riotabidva 6021	. . . 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 4200	. . 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 6004	. . . 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 4192	. . 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 6004	. . . 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 4192	. . 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 2273	. 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 2232	. . . 4 |- ( Base ` K ) = ( Base ` K )
18		eqid 2232	. . . 4 |- ( +g ` K ) = ( +g ` K )
19		eqid 2232	. . . 4 |- ( 0g ` K ) = ( 0g ` K )
20		eqid 2232	. . . 4 |- ( invg ` K ) = ( invg ` K )
21	17, 18, 19, 20	grpinvfvalg 13755	. . 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 2232	. . . 4 |- ( Base ` L ) = ( Base ` L )
24		eqid 2232	. . . 4 |- ( +g ` L ) = ( +g ` L )
25		eqid 2232	. . . 4 |- ( 0g ` L ) = ( 0g ` L )
26		eqid 2232	. . . 4 |- ( invg ` L ) = ( invg ` L )
27	23, 24, 25, 26	grpinvfvalg 13755	. . 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 2275	1 |- ( ph -> ( invg ` K ) = ( invg ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   |-> cmpt 4171   ` cfv 5352   iota_crio 6002  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   0gc0g 13469   invgcminusg 13714

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-0g 13471  df-minusg 13717

This theorem is referenced by:  grpsubpropdg  13817  grpsubpropd2  13818  mulgpropdg  13881  invrpropdg  14294  rlmvnegg  14613