Theorem grpsubpropd2 13237

Description: Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
grpsubpropd2.1	|- ( ph -> B = ( Base ` G ) )
grpsubpropd2.2	|- ( ph -> B = ( Base ` H ) )
grpsubpropd2.3	|- ( ph -> G e. Grp )
grpsubpropd2.4	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )

Assertion

Ref	Expression
grpsubpropd2	|- ( ph -> ( -g ` G ) = ( -g ` H ) )

Distinct variable groups:   x, y, B   x, G, y   x, H, y   ph, x, y

Proof of Theorem grpsubpropd2

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 999	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ph )
2		simp2 1000	. . . . . . 7 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> a e. ( Base ` G ) )
3		grpsubpropd2.1	. . . . . . . 8 |- ( ph -> B = ( Base ` G ) )
4	3	3ad2ant1 1020	. . . . . . 7 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> B = ( Base ` G ) )
5	2, 4	eleqtrrd 2276	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> a e. B )
6		grpsubpropd2.3	. . . . . . . . 9 |- ( ph -> G e. Grp )
7	6	3ad2ant1 1020	. . . . . . . 8 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> G e. Grp )
8		simp3 1001	. . . . . . . 8 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> b e. ( Base ` G ) )
9		eqid 2196	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
10		eqid 2196	. . . . . . . . 9 |- ( invg ` G ) = ( invg ` G )
11	9, 10	grpinvcl 13180	. . . . . . . 8 |- ( ( G e. Grp /\ b e. ( Base ` G ) ) -> ( ( invg ` G ) ` b ) e. ( Base ` G ) )
12	7, 8, 11	syl2anc 411	. . . . . . 7 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ( ( invg ` G ) ` b ) e. ( Base ` G ) )
13	12, 4	eleqtrrd 2276	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ( ( invg ` G ) ` b ) e. B )
14		grpsubpropd2.4	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
15	14	oveqrspc2v 5949	. . . . . 6 |- ( ( ph /\ ( a e. B /\ ( ( invg ` G ) ` b ) e. B ) ) -> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` G ) ` b ) ) )
16	1, 5, 13, 15	syl12anc 1247	. . . . 5 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` G ) ` b ) ) )
17		grpsubpropd2.2	. . . . . . . . 9 |- ( ph -> B = ( Base ` H ) )
18		eqid 2196	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
19	9, 18	grpidcl 13161	. . . . . . . . . . . 12 |- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) )
20	6, 19	syl 14	. . . . . . . . . . 11 |- ( ph -> ( 0g ` G ) e. ( Base ` G ) )
21	20, 3	eleqtrrd 2276	. . . . . . . . . 10 |- ( ph -> ( 0g ` G ) e. B )
22	17, 21	basmexd 12738	. . . . . . . . 9 |- ( ph -> H e. _V )
23	3, 17, 6, 22, 14	grpinvpropdg 13207	. . . . . . . 8 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
24	23	fveq1d 5560	. . . . . . 7 |- ( ph -> ( ( invg ` G ) ` b ) = ( ( invg ` H ) ` b ) )
25	24	oveq2d 5938	. . . . . 6 |- ( ph -> ( a ( +g ` H ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
26	25	3ad2ant1 1020	. . . . 5 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ( a ( +g ` H ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
27	16, 26	eqtrd 2229	. . . 4 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
28	27	mpoeq3dva 5986	. . 3 |- ( ph -> ( a e. ( Base ` G ) , b e. ( Base ` G ) |-> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) ) = ( a e. ( Base ` G ) , b e. ( Base ` G ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) ) )
29	3, 17	eqtr3d 2231	. . . 4 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
30		mpoeq12 5982	. . . 4 |- ( ( ( Base ` G ) = ( Base ` H ) /\ ( Base ` G ) = ( Base ` H ) ) -> ( a e. ( Base ` G ) , b e. ( Base ` G ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) ) = ( a e. ( Base ` H ) , b e. ( Base ` H ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) ) )
31	29, 29, 30	syl2anc 411	. . 3 |- ( ph -> ( a e. ( Base ` G ) , b e. ( Base ` G ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) ) = ( a e. ( Base ` H ) , b e. ( Base ` H ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) ) )
32	28, 31	eqtrd 2229	. 2 |- ( ph -> ( a e. ( Base ` G ) , b e. ( Base ` G ) |-> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) ) = ( a e. ( Base ` H ) , b e. ( Base ` H ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) ) )
33		eqid 2196	. . . 4 |- ( +g ` G ) = ( +g ` G )
34		eqid 2196	. . . 4 |- ( -g ` G ) = ( -g ` G )
35	9, 33, 10, 34	grpsubfvalg 13177	. . 3 |- ( G e. Grp -> ( -g ` G ) = ( a e. ( Base ` G ) , b e. ( Base ` G ) |-> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) ) )
36	6, 35	syl 14	. 2 |- ( ph -> ( -g ` G ) = ( a e. ( Base ` G ) , b e. ( Base ` G ) |-> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) ) )
37		eqid 2196	. . . 4 |- ( Base ` H ) = ( Base ` H )
38		eqid 2196	. . . 4 |- ( +g ` H ) = ( +g ` H )
39		eqid 2196	. . . 4 |- ( invg ` H ) = ( invg ` H )
40		eqid 2196	. . . 4 |- ( -g ` H ) = ( -g ` H )
41	37, 38, 39, 40	grpsubfvalg 13177	. . 3 |- ( H e. _V -> ( -g ` H ) = ( a e. ( Base ` H ) , b e. ( Base ` H ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) ) )
42	22, 41	syl 14	. 2 |- ( ph -> ( -g ` H ) = ( a e. ( Base ` H ) , b e. ( Base ` H ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) ) )
43	32, 36, 42	3eqtr4d 2239	1 |- ( ph -> ( -g ` G ) = ( -g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   _Vcvv 2763   ` cfv 5258  (class class class)co 5922   e. cmpo 5924   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Grpcgrp 13132   invgcminusg 13133   -gcsg 13134

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-sbg 13137

This theorem is referenced by: (None)