Theorem ablsubsub23 13632

Description: Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.)

Hypotheses

Ref	Expression
ablsubsub23.v	|- V = ( Base ` G )
ablsubsub23.m	|- .- = ( -g ` G )

Assertion

Ref	Expression
ablsubsub23	|- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) = C <-> ( A .- C ) = B ) )

Proof of Theorem ablsubsub23

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> G e. Abel )
2		simpr3 1007	. . . 4 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> C e. V )
3		simpr2 1006	. . . 4 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> B e. V )
4		ablsubsub23.v	. . . . 5 |- V = ( Base ` G )
5		eqid 2204	. . . . 5 |- ( +g ` G ) = ( +g ` G )
6	4, 5	ablcom 13610	. . . 4 |- ( ( G e. Abel /\ C e. V /\ B e. V ) -> ( C ( +g ` G ) B ) = ( B ( +g ` G ) C ) )
7	1, 2, 3, 6	syl3anc 1249	. . 3 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( C ( +g ` G ) B ) = ( B ( +g ` G ) C ) )
8	7	eqeq1d 2213	. 2 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( C ( +g ` G ) B ) = A <-> ( B ( +g ` G ) C ) = A ) )
9		ablgrp 13596	. . 3 |- ( G e. Abel -> G e. Grp )
10		ablsubsub23.m	. . . 4 |- .- = ( -g ` G )
11	4, 5, 10	grpsubadd 13391	. . 3 |- ( ( G e. Grp /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) = C <-> ( C ( +g ` G ) B ) = A ) )
12	9, 11	sylan 283	. 2 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) = C <-> ( C ( +g ` G ) B ) = A ) )
13		3ancomb 988	. . . 4 |- ( ( A e. V /\ B e. V /\ C e. V ) <-> ( A e. V /\ C e. V /\ B e. V ) )
14	13	biimpi 120	. . 3 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A e. V /\ C e. V /\ B e. V ) )
15	4, 5, 10	grpsubadd 13391	. . 3 |- ( ( G e. Grp /\ ( A e. V /\ C e. V /\ B e. V ) ) -> ( ( A .- C ) = B <-> ( B ( +g ` G ) C ) = A ) )
16	9, 14, 15	syl2an 289	. 2 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- C ) = B <-> ( B ( +g ` G ) C ) = A ) )
17	8, 12, 16	3bitr4d 220	1 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) = C <-> ( A .- C ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1372   e. wcel 2175   ` cfv 5270  (class class class)co 5943   Basecbs 12803   +g cplusg 12880   Grpcgrp 13303   -gcsg 13305   Abelcabl 13592

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-in1 615  ax-in2 616  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-setind 4584  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-fal 1378  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-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  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-2 9094  df-ndx 12806  df-slot 12807  df-base 12809  df-plusg 12893  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-grp 13306  df-minusg 13307  df-sbg 13308  df-cmn 13593  df-abl 13594

This theorem is referenced by: (None)