Theorem grpsubpropdg 12830

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 2174	. . . 4 |- ( ph -> a = a )
4		eqidd 2174	. . . . . 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 5890	. . . . . 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 12801	. . . . 5 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
9	8	fveq1d 5506	. . . 4 |- ( ph -> ( ( invg ` G ) ` b ) = ( ( invg ` H ) ` b ) )
10	2, 3, 9	oveq123d 5883	. . 3 |- ( ph -> ( a ( +g ` G ) ( ( invg ` G ) ` b ) ) = ( a ( +g ` H ) ( ( invg ` H ) ` b ) ) )
11	1, 1, 10	mpoeq123dv 5924	. 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 2173	. . . 4 |- ( Base ` G ) = ( Base ` G )
13		eqid 2173	. . . 4 |- ( +g ` G ) = ( +g ` G )
14		eqid 2173	. . . 4 |- ( invg ` G ) = ( invg ` G )
15		eqid 2173	. . . 4 |- ( -g ` G ) = ( -g ` G )
16	12, 13, 14, 15	grpsubfvalg 12775	. . 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 2173	. . . 4 |- ( Base ` H ) = ( Base ` H )
19		eqid 2173	. . . 4 |- ( +g ` H ) = ( +g ` H )
20		eqid 2173	. . . 4 |- ( invg ` H ) = ( invg ` H )
21		eqid 2173	. . . 4 |- ( -g ` H ) = ( -g ` H )
22	18, 19, 20, 21	grpsubfvalg 12775	. . 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 2216	1 |- ( ph -> ( -g ` G ) = ( -g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1351   e. wcel 2144   ` cfv 5205  (class class class)co 5862   e. cmpo 5864   Basecbs 12425   +g cplusg 12489   invgcminusg 12736   -gcsg 12737

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 707  ax-5 1443  ax-7 1444  ax-gen 1445  ax-ie1 1489  ax-ie2 1490  ax-8 1500  ax-10 1501  ax-11 1502  ax-i12 1503  ax-bndl 1505  ax-4 1506  ax-17 1522  ax-i9 1526  ax-ial 1530  ax-i5r 1531  ax-13 2146  ax-14 2147  ax-ext 2155  ax-coll 4110  ax-sep 4113  ax-pow 4166  ax-pr 4200  ax-un 4424  ax-cnex 7874  ax-resscn 7875  ax-1re 7877  ax-addrcl 7880

This theorem depends on definitions:  df-bi 117  df-3an 978  df-tru 1354  df-nf 1457  df-sb 1759  df-eu 2025  df-mo 2026  df-clab 2160  df-cleq 2166  df-clel 2169  df-nfc 2304  df-ral 2456  df-rex 2457  df-reu 2458  df-rab 2460  df-v 2735  df-sbc 2959  df-csb 3053  df-un 3128  df-in 3130  df-ss 3137  df-pw 3571  df-sn 3592  df-pr 3593  df-op 3595  df-uni 3803  df-int 3838  df-iun 3881  df-br 3996  df-opab 4057  df-mpt 4058  df-id 4284  df-xp 4623  df-rel 4624  df-cnv 4625  df-co 4626  df-dm 4627  df-rn 4628  df-res 4629  df-ima 4630  df-iota 5167  df-fun 5207  df-fn 5208  df-f 5209  df-f1 5210  df-fo 5211  df-f1o 5212  df-fv 5213  df-riota 5818  df-ov 5865  df-oprab 5866  df-mpo 5867  df-1st 6128  df-2nd 6129  df-inn 8888  df-ndx 12428  df-slot 12429  df-base 12431  df-0g 12625  df-minusg 12739  df-sbg 12740

This theorem is referenced by: (None)