Theorem ablsubsub23 13776

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 1008	. . . 4 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> C e. V )
3		simpr2 1007	. . . 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 2207	. . . . 5 |- ( +g ` G ) = ( +g ` G )
6	4, 5	ablcom 13754	. . . 4 |- ( ( G e. Abel /\ C e. V /\ B e. V ) -> ( C ( +g ` G ) B ) = ( B ( +g ` G ) C ) )
7	1, 2, 3, 6	syl3anc 1250	. . 3 |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( C ( +g ` G ) B ) = ( B ( +g ` G ) C ) )
8	7	eqeq1d 2216	. 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 13740	. . 3 |- ( G e. Abel -> G e. Grp )
10		ablsubsub23.m	. . . 4 |- .- = ( -g ` G )
11	4, 5, 10	grpsubadd 13535	. . 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 989	. . . 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 13535	. . 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 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   Grpcgrp 13447   -gcsg 13449   Abelcabl 13736

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-sbg 13452  df-cmn 13737  df-abl 13738

This theorem is referenced by: (None)