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Theorem dvdscmulr 13559
Description: Cancellation law for the divides relation. (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 zmulcl 10691 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
213adant3 1008 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  ZZ )
3 zmulcl 10691 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
433adant2 1007 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N )  e.  ZZ )
52, 4jca 532 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
653coml 1194 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
763adant3r 1215 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
8 3simpa 985 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  e.  ZZ  /\  N  e.  ZZ ) )
9 simpr 461 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
10 zcn 10649 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
11 zcn 10649 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  CC )
1210, 11anim12i 566 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  e.  CC  /\  M  e.  CC ) )
13 zcn 10649 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
14 zcn 10649 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
1514anim1i 568 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( K  e.  CC  /\  K  =/=  0 ) )
16 mul12 9533 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  CC  /\  x  e.  CC  /\  M  e.  CC )  ->  ( K  x.  ( x  x.  M ) )  =  ( x  x.  ( K  x.  M )
) )
17163adant1r 1211 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  e.  CC  /\  K  =/=  0 )  /\  x  e.  CC  /\  M  e.  CC )  ->  ( K  x.  ( x  x.  M
) )  =  ( x  x.  ( K  x.  M ) ) )
18173expb 1188 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  CC  /\  K  =/=  0 )  /\  ( x  e.  CC  /\  M  e.  CC ) )  -> 
( K  x.  (
x  x.  M ) )  =  ( x  x.  ( K  x.  M ) ) )
1918ancoms 453 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  ( K  e.  CC  /\  K  =/=  0 ) )  -> 
( K  x.  (
x  x.  M ) )  =  ( x  x.  ( K  x.  M ) ) )
20193adant2 1007 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC  /\  ( K  e.  CC  /\  K  =/=  0 ) )  ->  ( K  x.  ( x  x.  M
) )  =  ( x  x.  ( K  x.  M ) ) )
2120eqeq1d 2449 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC  /\  ( K  e.  CC  /\  K  =/=  0 ) )  ->  ( ( K  x.  ( x  x.  M ) )  =  ( K  x.  N
)  <->  ( x  x.  ( K  x.  M
) )  =  ( K  x.  N ) ) )
22 mulcl 9364 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( x  x.  M
)  e.  CC )
23 mulcan 9971 . . . . . . . . . . . . 13  |-  ( ( ( x  x.  M
)  e.  CC  /\  N  e.  CC  /\  ( K  e.  CC  /\  K  =/=  0 ) )  -> 
( ( K  x.  ( x  x.  M
) )  =  ( K  x.  N )  <-> 
( x  x.  M
)  =  N ) )
2422, 23syl3an1 1251 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC  /\  ( K  e.  CC  /\  K  =/=  0 ) )  ->  ( ( K  x.  ( x  x.  M ) )  =  ( K  x.  N
)  <->  ( x  x.  M )  =  N ) )
2521, 24bitr3d 255 . . . . . . . . . . 11  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC  /\  ( K  e.  CC  /\  K  =/=  0 ) )  ->  ( (
x  x.  ( K  x.  M ) )  =  ( K  x.  N )  <->  ( x  x.  M )  =  N ) )
2612, 13, 15, 25syl3an 1260 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  M  e.  ZZ )  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  ( (
x  x.  ( K  x.  M ) )  =  ( K  x.  N )  <->  ( x  x.  M )  =  N ) )
27263expb 1188 . . . . . . . . 9  |-  ( ( ( x  e.  ZZ  /\  M  e.  ZZ )  /\  ( N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) ) )  ->  ( ( x  x.  ( K  x.  M ) )  =  ( K  x.  N
)  <->  ( x  x.  M )  =  N ) )
28273impa 1182 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  ( N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) ) )  ->  ( ( x  x.  ( K  x.  M ) )  =  ( K  x.  N
)  <->  ( x  x.  M )  =  N ) )
29283coml 1194 . . . . . . 7  |-  ( ( 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 ) )
30293expia 1189 . . . . . 6  |-  ( ( 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 ) ) )
31303impb 1183 . . . . 5  |-  ( ( 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 ) ) )
3231imp 429 . . . 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 ) )
3332biimpd 207 . . 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 ) )
347, 8, 9, 33dvds1lem 13542 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  ||  ( K  x.  N )  ->  M  ||  N ) )
35 dvdscmul 13557 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( K  x.  M )  ||  ( K  x.  N
) ) )
36353adant3r 1215 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  ||  N  ->  ( K  x.  M
)  ||  ( K  x.  N ) ) )
3734, 36impbid 191 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  ||  ( K  x.  N )  <->  M 
||  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   class class class wbr 4290  (class class class)co 6089   CCcc 9278   0cc0 9280    x. cmul 9285   ZZcz 10644    || cdivides 13533
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-dvds 13534
This theorem is referenced by:  bitsmod  13630  mulgcd  13728  pcpremul  13908  4sqlem17  14020  odmulg  16055  ablfacrp2  16566  ablfac1b  16569  pgpfac1lem3a  16575  znrrg  17996  fsumdvdsdiaglem  22521  oddpwdc  26735  jm2.20nn  29343
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