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Theorem dvdscmulr 12573
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 10082 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
213adant3 975 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  ZZ )
3 zmulcl 10082 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
433adant2 974 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N )  e.  ZZ )
52, 4jca 518 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
653coml 1158 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
763adant3r 1179 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
8 3simpa 952 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  e.  ZZ  /\  N  e.  ZZ ) )
9 simpr 447 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
10 zcn 10045 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
11 zcn 10045 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  CC )
1210, 11anim12i 549 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  e.  CC  /\  M  e.  CC ) )
13 zcn 10045 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
14 zcn 10045 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
1514anim1i 551 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( K  e.  CC  /\  K  =/=  0 ) )
16 mul12 8994 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  CC  /\  x  e.  CC  /\  M  e.  CC )  ->  ( K  x.  ( x  x.  M ) )  =  ( x  x.  ( K  x.  M )
) )
17163adant1r 1175 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  e.  CC  /\  K  =/=  0 )  /\  x  e.  CC  /\  M  e.  CC )  ->  ( K  x.  ( x  x.  M
) )  =  ( x  x.  ( K  x.  M ) ) )
18173expb 1152 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  CC  /\  K  =/=  0 )  /\  ( x  e.  CC  /\  M  e.  CC ) )  -> 
( K  x.  (
x  x.  M ) )  =  ( x  x.  ( K  x.  M ) ) )
1918ancoms 439 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  ( K  e.  CC  /\  K  =/=  0 ) )  -> 
( K  x.  (
x  x.  M ) )  =  ( x  x.  ( K  x.  M ) ) )
20193adant2 974 . . . . . . . . . . . . 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 2304 . . . . . . . . . . . 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 8837 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( x  x.  M
)  e.  CC )
23 mulcan 9421 . . . . . . . . . . . . 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 1215 . . . . . . . . . . . 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 246 . . . . . . . . . . 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 1224 . . . . . . . . . 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 1152 . . . . . . . . 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 1146 . . . . . . . 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 1158 . . . . . . 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 1153 . . . . . 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 1147 . . . . 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 418 . . . 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 198 . . 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 12556 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  ||  ( K  x.  N )  ->  M  ||  N ) )
35 dvdscmul 12571 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( K  x.  M )  ||  ( K  x.  N
) ) )
36353adant3r 1179 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  ||  N  ->  ( K  x.  M
)  ||  ( K  x.  N ) ) )
3734, 36impbid 183 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039  (class class class)co 5874   CCcc 8751   0cc0 8753    x. cmul 8758   ZZcz 10040    || cdivides 12547
This theorem is referenced by:  bitsmod  12643  mulgcd  12741  pcpremul  12912  4sqlem17  13024  odmulg  14885  ablfacrp2  15318  ablfac1b  15321  pgpfac1lem3a  15327  znrrg  16535  fsumdvdsdiaglem  20439  jm2.20nn  27193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-dvds 12548
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