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Theorem dvdssub2 10382
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 8473 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N
)  e.  ZZ )
213adant1 957 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N )  e.  ZZ )
3 dvds2sub 10375 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  ( M  -  N )  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  ( M  -  N ) )  ->  K  ||  ( M  -  ( M  -  N ) ) ) )
42, 3syld3an3 1215 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  ( M  -  N ) )  ->  K  ||  ( M  -  ( M  -  N ) ) ) )
54ancomsd 265 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  ( M  -  N )  /\  K  ||  M )  ->  K  ||  ( M  -  ( M  -  N ) ) ) )
65imp 122 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  M ) )  ->  K  ||  ( M  -  ( M  -  N ) ) )
7 zcn 8437 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
8 zcn 8437 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
9 nncan 7404 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  -  ( M  -  N )
)  =  N )
107, 8, 9syl2an 283 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  ( M  -  N )
)  =  N )
11103adant1 957 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  ( M  -  N ) )  =  N )
1211adantr 270 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  M ) )  ->  ( M  -  ( M  -  N ) )  =  N )
136, 12breqtrd 3817 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  M ) )  ->  K  ||  N
)
1413expr 367 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  K  ||  ( M  -  N ) )  ->  ( K  ||  M  ->  K  ||  N
) )
15 dvds2add 10374 . . . . . 6  |-  ( ( K  e.  ZZ  /\  ( M  -  N
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  ( M  -  N
)  /\  K  ||  N
)  ->  K  ||  (
( M  -  N
)  +  N ) ) )
162, 15syld3an2 1217 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  ( M  -  N )  /\  K  ||  N )  ->  K  ||  (
( M  -  N
)  +  N ) ) )
1716imp 122 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  N ) )  ->  K  ||  (
( M  -  N
)  +  N ) )
18 npcan 7384 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  -  N )  +  N
)  =  M )
197, 8, 18syl2an 283 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  -  N )  +  N
)  =  M )
20193adant1 957 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  -  N
)  +  N )  =  M )
2120adantr 270 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  N ) )  ->  ( ( M  -  N )  +  N )  =  M )
2217, 21breqtrd 3817 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  ( M  -  N )  /\  K  ||  N ) )  ->  K  ||  M
)
2322expr 367 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  K  ||  ( M  -  N ) )  ->  ( K  ||  N  ->  K  ||  M
) )
2414, 23impbid 127 1  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  K  ||  ( M  -  N ) )  ->  ( K  ||  M 
<->  K  ||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   CCcc 7041    + caddc 7046    - cmin 7346   ZZcz 8432    || cdvds 10340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-dvds 10341
This theorem is referenced by:  dvdsadd  10383
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