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Theorem dvdssub2 13687
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 10797 . . . . . . . 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 13682 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  ( M  -  N )  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  ( M  -  N ) )  ->  K  ||  ( M  -  ( M  -  N ) ) ) )
42, 3syld3an3 1264 . . . . . 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 10761 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 zcn 10761 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
9 nncan 9748 . . . . . . 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 4423 . . 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 13681 . . . . . 6  |-  ( ( K  e.  ZZ  /\  ( M  -  N
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  ( M  -  N
)  /\  K  ||  N
)  ->  K  ||  (
( M  -  N
)  +  N ) ) )
162, 15syld3an2 1266 . . . . 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 9729 . . . . . . 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 4423 . . 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 1370    e. wcel 1758   class class class wbr 4399  (class class class)co 6199   CCcc 9390    + caddc 9395    - cmin 9705   ZZcz 10756    || cdivides 13652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-n0 10690  df-z 10757  df-dvds 13653
This theorem is referenced by:  dvdsadd  13688  3dvds  13713  bitsmod  13749  bitsinv1lem  13754  sylow2blem3  16241  znunit  18120  perfectlem1  22700  lgsqr  22817  2sqlem8  22843  jm2.20nn  29493
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