Theorem grpsubpropd2 13675

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 1021	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ph )
2		simp2 1022	. . . . . . 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 1042	. . . . . . 7 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> B = ( Base ` G ) )
5	2, 4	eleqtrrd 2309	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> a e. B )
6		grpsubpropd2.3	. . . . . . . . 9 |- ( ph -> G e. Grp )
7	6	3ad2ant1 1042	. . . . . . . 8 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> G e. Grp )
8		simp3 1023	. . . . . . . 8 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> b e. ( Base ` G ) )
9		eqid 2229	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
10		eqid 2229	. . . . . . . . 9 |- ( invg ` G ) = ( invg ` G )
11	9, 10	grpinvcl 13618	. . . . . . . 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 2309	. . . . . 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 6038	. . . . . 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 1269	. . . . 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 2229	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
19	9, 18	grpidcl 13599	. . . . . . . . . . . 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 2309	. . . . . . . . . 10 |- ( ph -> ( 0g ` G ) e. B )
22	17, 21	basmexd 13130	. . . . . . . . 9 |- ( ph -> H e. _V )
23	3, 17, 6, 22, 14	grpinvpropdg 13645	. . . . . . . 8 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
24	23	fveq1d 5635	. . . . . . 7 |- ( ph -> ( ( invg ` G ) ` b ) = ( ( invg ` H ) ` b ) )
25	24	oveq2d 6027	. . . . . 6 |- ( ph -> ( a ( +g ` H ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
26	25	3ad2ant1 1042	. . . . 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 2262	. . . 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 6078	. . 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 2264	. . . 4 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
30		mpoeq12 6074	. . . 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 2262	. 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 2229	. . . 4 |- ( +g ` G ) = ( +g ` G )
34		eqid 2229	. . . 4 |- ( -g ` G ) = ( -g ` G )
35	9, 33, 10, 34	grpsubfvalg 13615	. . 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 2229	. . . 4 |- ( Base ` H ) = ( Base ` H )
38		eqid 2229	. . . 4 |- ( +g ` H ) = ( +g ` H )
39		eqid 2229	. . . 4 |- ( invg ` H ) = ( invg ` H )
40		eqid 2229	. . . 4 |- ( -g ` H ) = ( -g ` H )
41	37, 38, 39, 40	grpsubfvalg 13615	. . 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 2272	1 |- ( ph -> ( -g ` G ) = ( -g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   _Vcvv 2800   ` cfv 5322  (class class class)co 6011   e. cmpo 6013   Basecbs 13069   +g cplusg 13147   0gc0g 13326   Grpcgrp 13570   invgcminusg 13571   -gcsg 13572

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-cnex 8111  ax-resscn 8112  ax-1re 8114  ax-addrcl 8117

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-inn 9132  df-2 9190  df-ndx 13072  df-slot 13073  df-base 13075  df-plusg 13160  df-0g 13328  df-mgm 13426  df-sgrp 13472  df-mnd 13487  df-grp 13573  df-minusg 13574  df-sbg 13575

This theorem is referenced by: (None)