Theorem grpsubpropdg 12979

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 2178	. . . 4 |- ( ph -> a = a )
4		eqidd 2178	. . . . . 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 5905	. . . . . 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 12950	. . . . 5 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
9	8	fveq1d 5519	. . . 4 |- ( ph -> ( ( invg ` G ) ` b ) = ( ( invg ` H ) ` b ) )
10	2, 3, 9	oveq123d 5898	. . 3 |- ( ph -> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
11	1, 1, 10	mpoeq123dv 5939	. 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 2177	. . . 4 |- ( Base ` G ) = ( Base ` G )
13		eqid 2177	. . . 4 |- ( +g ` G ) = ( +g ` G )
14		eqid 2177	. . . 4 |- ( invg ` G ) = ( invg ` G )
15		eqid 2177	. . . 4 |- ( -g ` G ) = ( -g ` G )
16	12, 13, 14, 15	grpsubfvalg 12923	. . 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 2177	. . . 4 |- ( Base ` H ) = ( Base ` H )
19		eqid 2177	. . . 4 |- ( +g ` H ) = ( +g ` H )
20		eqid 2177	. . . 4 |- ( invg ` H ) = ( invg ` H )
21		eqid 2177	. . . 4 |- ( -g ` H ) = ( -g ` H )
22	18, 19, 20, 21	grpsubfvalg 12923	. . 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 2220	1 |- ( ph -> ( -g ` G ) = ( -g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5877   e. cmpo 5879   Basecbs 12464   +g cplusg 12538   invgcminusg 12883   -gcsg 12884

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-inn 8922  df-ndx 12467  df-slot 12468  df-base 12470  df-0g 12712  df-minusg 12886  df-sbg 12887

This theorem is referenced by: (None)