Theorem grpsubpropdg 13436

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 2206	. . . 4 |- ( ph -> a = a )
4		eqidd 2206	. . . . . 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 5972	. . . . . 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 13407	. . . . 5 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
9	8	fveq1d 5578	. . . 4 |- ( ph -> ( ( invg ` G ) ` b ) = ( ( invg ` H ) ` b ) )
10	2, 3, 9	oveq123d 5965	. . 3 |- ( ph -> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
11	1, 1, 10	mpoeq123dv 6007	. 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 2205	. . . 4 |- ( Base ` G ) = ( Base ` G )
13		eqid 2205	. . . 4 |- ( +g ` G ) = ( +g ` G )
14		eqid 2205	. . . 4 |- ( invg ` G ) = ( invg ` G )
15		eqid 2205	. . . 4 |- ( -g ` G ) = ( -g ` G )
16	12, 13, 14, 15	grpsubfvalg 13377	. . 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 2205	. . . 4 |- ( Base ` H ) = ( Base ` H )
19		eqid 2205	. . . 4 |- ( +g ` H ) = ( +g ` H )
20		eqid 2205	. . . 4 |- ( invg ` H ) = ( invg ` H )
21		eqid 2205	. . . 4 |- ( -g ` H ) = ( -g ` H )
22	18, 19, 20, 21	grpsubfvalg 13377	. . 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 2248	1 |- ( ph -> ( -g ` G ) = ( -g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   ` cfv 5271  (class class class)co 5944   e. cmpo 5946   Basecbs 12832   +g cplusg 12909   invgcminusg 13333   -gcsg 13334

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-0g 13090  df-minusg 13336  df-sbg 13337

This theorem is referenced by:  rlmsubg  14220