Theorem grpsubpropd2 13522

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 1000	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ph )
2		simp2 1001	. . . . . . 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 1021	. . . . . . 7 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> B = ( Base ` G ) )
5	2, 4	eleqtrrd 2286	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> a e. B )
6		grpsubpropd2.3	. . . . . . . . 9 |- ( ph -> G e. Grp )
7	6	3ad2ant1 1021	. . . . . . . 8 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> G e. Grp )
8		simp3 1002	. . . . . . . 8 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> b e. ( Base ` G ) )
9		eqid 2206	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
10		eqid 2206	. . . . . . . . 9 |- ( invg ` G ) = ( invg ` G )
11	9, 10	grpinvcl 13465	. . . . . . . 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 2286	. . . . . 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 5989	. . . . . 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 1248	. . . . 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 2206	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
19	9, 18	grpidcl 13446	. . . . . . . . . . . 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 2286	. . . . . . . . . 10 |- ( ph -> ( 0g ` G ) e. B )
22	17, 21	basmexd 12977	. . . . . . . . 9 |- ( ph -> H e. _V )
23	3, 17, 6, 22, 14	grpinvpropdg 13492	. . . . . . . 8 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
24	23	fveq1d 5596	. . . . . . 7 |- ( ph -> ( ( invg ` G ) ` b ) = ( ( invg ` H ) ` b ) )
25	24	oveq2d 5978	. . . . . 6 |- ( ph -> ( a ( +g ` H ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
26	25	3ad2ant1 1021	. . . . 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 2239	. . . 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 6027	. . 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 2241	. . . 4 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
30		mpoeq12 6023	. . . 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 2239	. 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 2206	. . . 4 |- ( +g ` G ) = ( +g ` G )
34		eqid 2206	. . . 4 |- ( -g ` G ) = ( -g ` G )
35	9, 33, 10, 34	grpsubfvalg 13462	. . 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 2206	. . . 4 |- ( Base ` H ) = ( Base ` H )
38		eqid 2206	. . . 4 |- ( +g ` H ) = ( +g ` H )
39		eqid 2206	. . . 4 |- ( invg ` H ) = ( invg ` H )
40		eqid 2206	. . . 4 |- ( -g ` H ) = ( -g ` H )
41	37, 38, 39, 40	grpsubfvalg 13462	. . 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 2249	1 |- ( ph -> ( -g ` G ) = ( -g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2177   _Vcvv 2773   ` cfv 5285  (class class class)co 5962   e. cmpo 5964   Basecbs 12917   +g cplusg 12994   0gc0g 13173   Grpcgrp 13417   invgcminusg 13418   -gcsg 13419

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 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-cnex 8046  ax-resscn 8047  ax-1re 8049  ax-addrcl 8052

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-rmo 2493  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 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-inn 9067  df-2 9125  df-ndx 12920  df-slot 12921  df-base 12923  df-plusg 13007  df-0g 13175  df-mgm 13273  df-sgrp 13319  df-mnd 13334  df-grp 13420  df-minusg 13421  df-sbg 13422

This theorem is referenced by: (None)