Theorem ablsubsub23 13877

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 1029	. . . 4 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> C e. V )
3		simpr2 1028	. . . 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 2229	. . . . 5 |- ( +g ` G ) = ( +g ` G )
6	4, 5	ablcom 13855	. . . 4 |- ( ( G e. Abel /\ C e. V /\ B e. V ) -> ( C ( +g ` G ) B ) = ( B ( +g ` G ) C ) )
7	1, 2, 3, 6	syl3anc 1271	. . 3 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( C ( +g ` G ) B ) = ( B ( +g ` G ) C ) )
8	7	eqeq1d 2238	. 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 13841	. . 3 |- ( G e. Abel -> G e. Grp )
10		ablsubsub23.m	. . . 4 |- .- = ( -g ` G )
11	4, 5, 10	grpsubadd 13636	. . 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 1010	. . . 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 13636	. . 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 1002   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6007   Basecbs 13047   +g cplusg 13125   Grpcgrp 13548   -gcsg 13550   Abelcabl 13837

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-inn 9122  df-2 9180  df-ndx 13050  df-slot 13051  df-base 13053  df-plusg 13138  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-minusg 13552  df-sbg 13553  df-cmn 13838  df-abl 13839

This theorem is referenced by: (None)