Theorem grpsubpropdg 13407

Description: Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)

Hypotheses

Ref	Expression
grpsubpropd.b	|- ( ph -> ( Base ` G ) = ( Base ` H ) )
grpsubpropd.p	|- ( ph -> ( +g ` G ) = ( +g ` H ) )
grpsubpropdg.g	|- ( ph -> G e. V )
grpsubpropdg.h	|- ( ph -> H e. W )

Assertion

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

Proof of Theorem grpsubpropdg

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

Step	Hyp	Ref	Expression
1		grpsubpropd.b	. . 3 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
2		grpsubpropd.p	. . . 4 |- ( ph -> ( +g ` G ) = ( +g ` H ) )
3		eqidd 2205	. . . 4 |- ( ph -> a = a )
4		eqidd 2205	. . . . . 6 |- ( ph -> ( Base ` G ) = ( Base ` G ) )
5		grpsubpropdg.g	. . . . . 6 |- ( ph -> G e. V )
6		grpsubpropdg.h	. . . . . 6 |- ( ph -> H e. W )
7	2	oveqdr 5971	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
8	4, 1, 5, 6, 7	grpinvpropdg 13378	. . . . 5 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
9	8	fveq1d 5577	. . . 4 |- ( ph -> ( ( invg ` G ) ` b ) = ( ( invg ` H ) ` b ) )
10	2, 3, 9	oveq123d 5964	. . 3 |- ( ph -> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
11	1, 1, 10	mpoeq123dv 6006	. 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 ) ) ) )
12		eqid 2204	. . . 4 |- ( Base ` G ) = ( Base ` G )
13		eqid 2204	. . . 4 |- ( +g ` G ) = ( +g ` G )
14		eqid 2204	. . . 4 |- ( invg ` G ) = ( invg ` G )
15		eqid 2204	. . . 4 |- ( -g ` G ) = ( -g ` G )
16	12, 13, 14, 15	grpsubfvalg 13348	. . 3 |- ( G e. V -> ( -g ` G ) = ( a e. ( Base ` G ) , b e. ( Base ` G ) |-> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) ) )
17	5, 16	syl 14	. 2 |- ( ph -> ( -g ` G ) = ( a e. ( Base ` G ) , b e. ( Base ` G ) |-> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) ) )
18		eqid 2204	. . . 4 |- ( Base ` H ) = ( Base ` H )
19		eqid 2204	. . . 4 |- ( +g ` H ) = ( +g ` H )
20		eqid 2204	. . . 4 |- ( invg ` H ) = ( invg ` H )
21		eqid 2204	. . . 4 |- ( -g ` H ) = ( -g ` H )
22	18, 19, 20, 21	grpsubfvalg 13348	. . 3 |- ( H e. W -> ( -g ` H ) = ( a e. ( Base ` H ) , b e. ( Base ` H ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) ) )
23	6, 22	syl 14	. 2 |- ( ph -> ( -g ` H ) = ( a e. ( Base ` H ) , b e. ( Base ` H ) |-> ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) ) )
24	11, 17, 23	3eqtr4d 2247	1 |- ( ph -> ( -g ` G ) = ( -g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   ` cfv 5270  (class class class)co 5943   e. cmpo 5945   Basecbs 12803   +g cplusg 12880   invgcminusg 13304   -gcsg 13305

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-inn 9036  df-ndx 12806  df-slot 12807  df-base 12809  df-0g 13061  df-minusg 13307  df-sbg 13308

This theorem is referenced by:  rlmsubg  14191