Theorem grpsubpropdg 13817

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 2233	. . . 4 |- ( ph -> a = a )
4		eqidd 2233	. . . . . 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 6078	. . . . . 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 13788	. . . . 5 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
9	8	fveq1d 5672	. . . 4 |- ( ph -> ( ( invg ` G ) ` b ) = ( ( invg ` H ) ` b ) )
10	2, 3, 9	oveq123d 6071	. . 3 |- ( ph -> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
11	1, 1, 10	mpoeq123dv 6115	. 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 2232	. . . 4 |- ( Base ` G ) = ( Base ` G )
13		eqid 2232	. . . 4 |- ( +g ` G ) = ( +g ` G )
14		eqid 2232	. . . 4 |- ( invg ` G ) = ( invg ` G )
15		eqid 2232	. . . 4 |- ( -g ` G ) = ( -g ` G )
16	12, 13, 14, 15	grpsubfvalg 13758	. . 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 2232	. . . 4 |- ( Base ` H ) = ( Base ` H )
19		eqid 2232	. . . 4 |- ( +g ` H ) = ( +g ` H )
20		eqid 2232	. . . 4 |- ( invg ` H ) = ( invg ` H )
21		eqid 2232	. . . 4 |- ( -g ` H ) = ( -g ` H )
22	18, 19, 20, 21	grpsubfvalg 13758	. . 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 2275	1 |- ( ph -> ( -g ` G ) = ( -g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   e. cmpo 6052   Basecbs 13212   +g cplusg 13290   invgcminusg 13714   -gcsg 13715

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-0g 13471  df-minusg 13717  df-sbg 13718

This theorem is referenced by:  rlmsubg  14606