Theorem grpsubpropd2 17521

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 1061	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ph )
2		simp2 1062	. . . . . . 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 1082	. . . . . . 7 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> B = ( Base ` G ) )
5	2, 4	eleqtrrd 2704	. . . . . 6 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> a e. B )
6		grpsubpropd2.3	. . . . . . . . 9 |- ( ph -> G e. Grp )
7	6	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> G e. Grp )
8		simp3 1063	. . . . . . . 8 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> b e. ( Base ` G ) )
9		eqid 2622	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
10		eqid 2622	. . . . . . . . 9 |- ( invg ` G ) = ( invg ` G )
11	9, 10	grpinvcl 17467	. . . . . . . 8 |- ( ( G e. Grp /\ b e. ( Base ` G ) ) -> ( ( invg ` G ) ` b ) e. ( Base ` G ) )
12	7, 8, 11	syl2anc 693	. . . . . . 7 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ( ( invg ` G ) ` b ) e. ( Base ` G ) )
13	12, 4	eleqtrrd 2704	. . . . . 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 6673	. . . . . 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 1324	. . . . 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	3, 17, 14	grpinvpropd 17490	. . . . . . . 8 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
19	18	fveq1d 6193	. . . . . . 7 |- ( ph -> ( ( invg ` G ) ` b ) = ( ( invg ` H ) ` b ) )
20	19	oveq2d 6666	. . . . . 6 |- ( ph -> ( a ( +g ` H ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
21	20	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ( a ( +g ` H ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
22	16, 21	eqtrd 2656	. . . 4 |- ( ( ph /\ a e. ( Base ` G ) /\ b e. ( Base ` G ) ) -> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
23	22	mpt2eq3dva 6719	. . 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 ) ) ) )
24	3, 17	eqtr3d 2658	. . . 4 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
25		mpt2eq12 6715	. . . 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 ) ) ) )
26	24, 24, 25	syl2anc 693	. . 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 ) ) ) )
27	23, 26	eqtrd 2656	. 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 ) ) ) )
28		eqid 2622	. . 3 |- ( +g ` G ) = ( +g ` G )
29		eqid 2622	. . 3 |- ( -g ` G ) = ( -g ` G )
30	9, 28, 10, 29	grpsubfval 17464	. 2 |- ( -g ` G ) = ( a e. ( Base ` G ) , b e. ( Base ` G ) |-> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) )
31		eqid 2622	. . 3 |- ( Base ` H ) = ( Base ` H )
32		eqid 2622	. . 3 |- ( +g ` H ) = ( +g ` H )
33		eqid 2622	. . 3 |- ( invg ` H ) = ( invg ` H )
34		eqid 2622	. . 3 |- ( -g ` H ) = ( -g ` H )
35	31, 32, 33, 34	grpsubfval 17464	. 2 |- ( -g ` H ) = ( a e. ( Base ` H ) , b e. ( Base ` H ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
36	27, 30, 35	3eqtr4g 2681	1 |- ( ph -> ( -g ` G ) = ( -g ` H ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427

This theorem is referenced by:  ngppropd  22441