Theorem ablsubsub23 12924

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 1005	. . . 4 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> C e. V )
3		simpr2 1004	. . . 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 2175	. . . . 5 |- ( +g ` G ) = ( +g ` G )
6	4, 5	ablcom 12902	. . . 4 |- ( ( G e. Abel /\ C e. V /\ B e. V ) -> ( C ( +g ` G ) B ) = ( B ( +g ` G ) C ) )
7	1, 2, 3, 6	syl3anc 1238	. . 3 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( C ( +g ` G ) B ) = ( B ( +g ` G ) C ) )
8	7	eqeq1d 2184	. 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 12889	. . 3 |- ( G e. Abel -> G e. Grp )
10		ablsubsub23.m	. . . 4 |- .- = ( -g ` G )
11	4, 5, 10	grpsubadd 12817	. . 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 986	. . . 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 12817	. . 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 978   = wceq 1353   e. wcel 2146   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   Grpcgrp 12738   -gcsg 12740   Abelcabl 12885

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-inn 8891  df-2 8949  df-ndx 12431  df-slot 12432  df-base 12434  df-plusg 12505  df-0g 12628  df-mgm 12640  df-sgrp 12673  df-mnd 12683  df-grp 12741  df-minusg 12742  df-sbg 12743  df-cmn 12886  df-abl 12887

This theorem is referenced by: (None)