Theorem ablpnpcan 14037

Description: Cancellation law for mixed addition and subtraction. (pnpcan 8512 analog.) (Contributed by NM, 29-May-2015.)

Hypotheses

Ref	Expression
ablsubadd.b	|- B = ( Base ` G )
ablsubadd.p	|- .+ = ( +g ` G )
ablsubadd.m	|- .- = ( -g ` G )
ablsubsub.g	|- ( ph -> G e. Abel )
ablsubsub.x	|- ( ph -> X e. B )
ablsubsub.y	|- ( ph -> Y e. B )
ablsubsub.z	|- ( ph -> Z e. B )
ablpnpcan.g	|- ( ph -> G e. Abel )
ablpnpcan.x	|- ( ph -> X e. B )
ablpnpcan.y	|- ( ph -> Y e. B )
ablpnpcan.z	|- ( ph -> Z e. B )

Assertion

Ref	Expression
ablpnpcan	|- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( Y .- Z ) )

Proof of Theorem ablpnpcan

Step	Hyp	Ref	Expression
1		ablsubsub.g	. . 3 |- ( ph -> G e. Abel )
2		ablsubsub.x	. . 3 |- ( ph -> X e. B )
3		ablsubsub.y	. . 3 |- ( ph -> Y e. B )
4		ablsubsub.z	. . 3 |- ( ph -> Z e. B )
5		ablsubadd.b	. . . 4 |- B = ( Base ` G )
6		ablsubadd.p	. . . 4 |- .+ = ( +g ` G )
7		ablsubadd.m	. . . 4 |- .- = ( -g ` G )
8	5, 6, 7	ablsub4 14030	. . 3 |- ( ( G e. Abel /\ ( X e. B /\ Y e. B ) /\ ( X e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( ( X .- X ) .+ ( Y .- Z ) ) )
9	1, 2, 3, 2, 4, 8	syl122anc 1283	. 2 |- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( ( X .- X ) .+ ( Y .- Z ) ) )
10		ablgrp 14006	. . . . 5 |- ( G e. Abel -> G e. Grp )
11	1, 10	syl 14	. . . 4 |- ( ph -> G e. Grp )
12		eqid 2232	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
13	5, 12, 7	grpsubid 13797	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = ( 0g ` G ) )
14	11, 2, 13	syl2anc 411	. . 3 |- ( ph -> ( X .- X ) = ( 0g ` G ) )
15	14	oveq1d 6065	. 2 |- ( ph -> ( ( X .- X ) .+ ( Y .- Z ) ) = ( ( 0g ` G ) .+ ( Y .- Z ) ) )
16	5, 7	grpsubcl 13793	. . . 4 |- ( ( G e. Grp /\ Y e. B /\ Z e. B ) -> ( Y .- Z ) e. B )
17	11, 3, 4, 16	syl3anc 1274	. . 3 |- ( ph -> ( Y .- Z ) e. B )
18	5, 6, 12	grplid 13744	. . 3 |- ( ( G e. Grp /\ ( Y .- Z ) e. B ) -> ( ( 0g ` G ) .+ ( Y .- Z ) ) = ( Y .- Z ) )
19	11, 17, 18	syl2anc 411	. 2 |- ( ph -> ( ( 0g ` G ) .+ ( Y .- Z ) ) = ( Y .- Z ) )
20	9, 15, 19	3eqtrd 2269	1 |- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( Y .- Z ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   0gc0g 13469   Grpcgrp 13713   -gcsg 13715   Abelcabl 14002

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-sbg 13718  df-cmn 14003  df-abl 14004

This theorem is referenced by: (None)