Theorem ablsubsub23 13975

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 1032	. . . 4 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> C e. V )
3		simpr2 1031	. . . 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 2231	. . . . 5 |- ( +g ` G ) = ( +g ` G )
6	4, 5	ablcom 13953	. . . 4 |- ( ( G e. Abel /\ C e. V /\ B e. V ) -> ( C ( +g ` G ) B ) = ( B ( +g ` G ) C ) )
7	1, 2, 3, 6	syl3anc 1274	. . 3 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( C ( +g ` G ) B ) = ( B ( +g ` G ) C ) )
8	7	eqeq1d 2240	. 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 13939	. . 3 |- ( G e. Abel -> G e. Grp )
10		ablsubsub23.m	. . . 4 |- .- = ( -g ` G )
11	4, 5, 10	grpsubadd 13734	. . 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 1013	. . . 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 13734	. . 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 1005   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13145   +g cplusg 13223   Grpcgrp 13646   -gcsg 13648   Abelcabl 13935

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-sbg 13651  df-cmn 13936  df-abl 13937

This theorem is referenced by: (None)