Theorem grpinvpropdg 13482

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 13281	. . . . . . . 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 2221	. . . . . 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 5929	. . . 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 4142	. . 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 5913	. . . 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 4134	. . 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 5913	. . . 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 4134	. . 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 2247	. 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 2206	. . . 4 |- ( Base ` K ) = ( Base ` K )
18		eqid 2206	. . . 4 |- ( +g ` K ) = ( +g ` K )
19		eqid 2206	. . . 4 |- ( 0g ` K ) = ( 0g ` K )
20		eqid 2206	. . . 4 |- ( invg ` K ) = ( invg ` K )
21	17, 18, 19, 20	grpinvfvalg 13449	. . 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 2206	. . . 4 |- ( Base ` L ) = ( Base ` L )
24		eqid 2206	. . . 4 |- ( +g ` L ) = ( +g ` L )
25		eqid 2206	. . . 4 |- ( 0g ` L ) = ( 0g ` L )
26		eqid 2206	. . . 4 |- ( invg ` L ) = ( invg ` L )
27	23, 24, 25, 26	grpinvfvalg 13449	. . 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 2249	1 |- ( ph -> ( invg ` K ) = ( invg ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   |-> cmpt 4113   ` cfv 5280   iota_crio 5911  (class class class)co 5957   Basecbs 12907   +g cplusg 12984   0gc0g 13163   invgcminusg 13408

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-inn 9057  df-ndx 12910  df-slot 12911  df-base 12913  df-0g 13165  df-minusg 13411

This theorem is referenced by:  grpsubpropdg  13511  grpsubpropd2  13512  mulgpropdg  13575  invrpropdg  13986  rlmvnegg  14302