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Theorem dvdssub2 13562
Description: If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)
Assertion
Ref Expression
dvdssub2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  K  ||  ( M  -  N ) )  ->  ( K  ||  M 
<->  K  ||  N ) )

Proof of Theorem dvdssub2
StepHypRef Expression
1 zsubcl 10679 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N
)  e.  ZZ )
213adant1 1006 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N )  e.  ZZ )
3 dvds2sub 13557 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  ( M  -  N )  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  ( M  -  N ) )  ->  K  ||  ( M  -  ( M  -  N ) ) ) )
42, 3syld3an3 1263 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  ( M  -  N ) )  ->  K  ||  ( M  -  ( M  -  N ) ) ) )
54ancomsd 454 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  ( M  -  N )  /\  K  ||  M )  ->  K  ||  ( M  -  ( M  -  N ) ) ) )
65imp 429 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  M ) )  ->  K  ||  ( M  -  ( M  -  N ) ) )
7 zcn 10643 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 zcn 10643 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
9 nncan 9630 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  -  ( M  -  N )
)  =  N )
107, 8, 9syl2an 477 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  ( M  -  N )
)  =  N )
11103adant1 1006 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  ( M  -  N ) )  =  N )
1211adantr 465 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  M ) )  ->  ( M  -  ( M  -  N ) )  =  N )
136, 12breqtrd 4311 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  M ) )  ->  K  ||  N
)
1413expr 615 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  K  ||  ( M  -  N ) )  ->  ( K  ||  M  ->  K  ||  N
) )
15 dvds2add 13556 . . . . . 6  |-  ( ( K  e.  ZZ  /\  ( M  -  N
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  ( M  -  N
)  /\  K  ||  N
)  ->  K  ||  (
( M  -  N
)  +  N ) ) )
162, 15syld3an2 1265 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  ( M  -  N )  /\  K  ||  N )  ->  K  ||  (
( M  -  N
)  +  N ) ) )
1716imp 429 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  N ) )  ->  K  ||  (
( M  -  N
)  +  N ) )
18 npcan 9611 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  -  N )  +  N
)  =  M )
197, 8, 18syl2an 477 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  -  N )  +  N
)  =  M )
20193adant1 1006 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  -  N
)  +  N )  =  M )
2120adantr 465 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  N ) )  ->  ( ( M  -  N )  +  N )  =  M )
2217, 21breqtrd 4311 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  N ) )  ->  K  ||  M
)
2322expr 615 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  K  ||  ( M  -  N ) )  ->  ( K  ||  N  ->  K  ||  M
) )
2414, 23impbid 191 1  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  K  ||  ( M  -  N ) )  ->  ( K  ||  M 
<->  K  ||  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4287  (class class class)co 6086   CCcc 9272    + caddc 9277    - cmin 9587   ZZcz 10638    || cdivides 13527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-dvds 13528
This theorem is referenced by:  dvdsadd  13563  3dvds  13588  bitsmod  13624  bitsinv1lem  13629  sylow2blem3  16112  znunit  17971  perfectlem1  22543  lgsqr  22660  2sqlem8  22686  jm2.20nn  29299
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