Theorem grpsubpropd2 13887

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 1028	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ph )
2		simp2 1029	. . . . . . 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 1049	. . . . . . 7 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> B = ( Base ` G ) )
5	2, 4	eleqtrrd 2318	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> a e. B )
6		grpsubpropd2.3	. . . . . . . . 9 |- ( ph -> G e. Grp )
7	6	3ad2ant1 1049	. . . . . . . 8 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> G e. Grp )
8		simp3 1030	. . . . . . . 8 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> b e. ( Base ` G ) )
9		eqid 2238	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
10		eqid 2238	. . . . . . . . 9 |- ( invg ` G ) = ( invg ` G )
11	9, 10	grpinvcl 13830	. . . . . . . 8 |- ( ( G e. Grp /\ b e. ( Base ` G ) ) -> ( ( invg ` G ) ` b ) e. ( Base ` G ) )
12	7, 8, 11	syl2anc 415	. . . . . . 7 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ( ( invg ` G ) ` b ) e. ( Base ` G ) )
13	12, 4	eleqtrrd 2318	. . . . . 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 6102	. . . . . 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 1276	. . . . 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 2238	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
19	9, 18	grpidcl 13811	. . . . . . . . . . . 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 2318	. . . . . . . . . 10 |- ( ph -> ( 0g ` G ) e. B )
22	17, 21	basmexd 13391	. . . . . . . . 9 |- ( ph -> H e. _V )
23	3, 17, 6, 22, 14	grpinvpropdg 13857	. . . . . . . 8 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
24	23	fveq1d 5692	. . . . . . 7 |- ( ph -> ( ( invg ` G ) ` b ) = ( ( invg ` H ) ` b ) )
25	24	oveq2d 6091	. . . . . 6 |- ( ph -> ( a ( +g ` H ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
26	25	3ad2ant1 1049	. . . . 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 2271	. . . 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 6142	. . 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 2273	. . . 4 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
30		mpoeq12 6138	. . . 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 415	. . 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 2271	. 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 2238	. . . 4 |- ( +g ` G ) = ( +g ` G )
34		eqid 2238	. . . 4 |- ( -g ` G ) = ( -g ` G )
35	9, 33, 10, 34	grpsubfvalg 13827	. . 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 2238	. . . 4 |- ( Base ` H ) = ( Base ` H )
38		eqid 2238	. . . 4 |- ( +g ` H ) = ( +g ` H )
39		eqid 2238	. . . 4 |- ( invg ` H ) = ( invg ` H )
40		eqid 2238	. . . 4 |- ( -g ` H ) = ( -g ` H )
41	37, 38, 39, 40	grpsubfvalg 13827	. . 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 2281	1 |- ( ph -> ( -g ` G ) = ( -g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Grpcgrp 13782   invgcminusg 13783   -gcsg 13784

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-sbg 13787

This theorem is referenced by: (None)