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Theorem dvdscmulr 10700
Description: Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvdscmulr  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  ||  ( K  x.  N )  <->  M 
||  N ) )

Proof of Theorem dvdscmulr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp3l 969 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  K  e.  ZZ )
2 simp1 941 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  M  e.  ZZ )
31, 2zmulcld 8807 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( K  x.  M
)  e.  ZZ )
4 simp2 942 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  N  e.  ZZ )
51, 4zmulcld 8807 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( K  x.  N
)  e.  ZZ )
63, 5jca 300 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
72, 4jca 300 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  e.  ZZ  /\  N  e.  ZZ ) )
8 simpr 108 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
91adantr 270 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  K  e.  ZZ )
109zcnd 8802 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  K  e.  CC )
118zcnd 8802 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  x  e.  CC )
122adantr 270 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  M  e.  ZZ )
1312zcnd 8802 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  M  e.  CC )
1410, 11, 13mul12d 7578 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  ( K  x.  ( x  x.  M ) )  =  ( x  x.  ( K  x.  M )
) )
1514eqeq1d 2093 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  (
( K  x.  (
x  x.  M ) )  =  ( K  x.  N )  <->  ( x  x.  ( K  x.  M
) )  =  ( K  x.  N ) ) )
1611, 13mulcld 7452 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  (
x  x.  M )  e.  CC )
174adantr 270 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  N  e.  ZZ )
1817zcnd 8802 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  N  e.  CC )
19 simpl3r 997 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  K  =/=  0 )
20 0z 8694 . . . . . . . 8  |-  0  e.  ZZ
21 zapne 8754 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  0  e.  ZZ )  ->  ( K #  0  <->  K  =/=  0 ) )
229, 20, 21sylancl 404 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  ( K #  0  <->  K  =/=  0
) )
2319, 22mpbird 165 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  K #  0 )
2416, 18, 10, 23mulcanapd 8069 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  (
( K  x.  (
x  x.  M ) )  =  ( K  x.  N )  <->  ( x  x.  M )  =  N ) )
2515, 24bitr3d 188 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  (
( x  x.  ( K  x.  M )
)  =  ( K  x.  N )  <->  ( x  x.  M )  =  N ) )
2625biimpd 142 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  (
( x  x.  ( K  x.  M )
)  =  ( K  x.  N )  -> 
( x  x.  M
)  =  N ) )
276, 7, 8, 26dvds1lem 10682 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  ||  ( K  x.  N )  ->  M  ||  N ) )
28 dvdscmul 10698 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( K  x.  M )  ||  ( K  x.  N
) ) )
29283adant3r 1169 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  ||  N  ->  ( K  x.  M
)  ||  ( K  x.  N ) ) )
3027, 29impbid 127 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  ||  ( K  x.  N )  <->  M 
||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 922    = wceq 1287    e. wcel 1436    =/= wne 2251   class class class wbr 3820  (class class class)co 5613   0cc0 7294    x. cmul 7299   # cap 7999   ZZcz 8683    || cdvds 10671
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-1cn 7382  ax-1re 7383  ax-icn 7384  ax-addcl 7385  ax-addrcl 7386  ax-mulcl 7387  ax-mulrcl 7388  ax-addcom 7389  ax-mulcom 7390  ax-addass 7391  ax-mulass 7392  ax-distr 7393  ax-i2m1 7394  ax-0lt1 7395  ax-1rid 7396  ax-0id 7397  ax-rnegex 7398  ax-precex 7399  ax-cnre 7400  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403  ax-pre-apti 7404  ax-pre-ltadd 7405  ax-pre-mulgt0 7406  ax-pre-mulext 7407
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-br 3821  df-opab 3875  df-id 4094  df-po 4097  df-iso 4098  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-iota 4946  df-fun 4983  df-fv 4989  df-riota 5569  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472  df-sub 7599  df-neg 7600  df-reap 7993  df-ap 8000  df-inn 8358  df-n0 8607  df-z 8684  df-dvds 10672
This theorem is referenced by:  modmulconst  10703  mulgcd  10880  oddpwdclemxy  11022  oddpwdclemodd  11025
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