MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvdscmulr Structured version   Unicode version

Theorem dvdscmulr 14309
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 10985 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
213adant3 1025 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  ZZ )
3 zmulcl 10985 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
433adant2 1024 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N )  e.  ZZ )
52, 4jca 534 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
653coml 1212 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
763adant3r 1261 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ ) )
8 3simpa 1002 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  e.  ZZ  /\  N  e.  ZZ ) )
9 simpr 462 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
10 zcn 10942 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
11 zcn 10942 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  CC )
1210, 11anim12i 568 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  e.  CC  /\  M  e.  CC ) )
13 zcn 10942 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
14 zcn 10942 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
1514anim1i 570 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( K  e.  CC  /\  K  =/=  0 ) )
16 mul12 9798 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  CC  /\  x  e.  CC  /\  M  e.  CC )  ->  ( K  x.  ( x  x.  M ) )  =  ( x  x.  ( K  x.  M )
) )
17163adant1r 1257 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  e.  CC  /\  K  =/=  0 )  /\  x  e.  CC  /\  M  e.  CC )  ->  ( K  x.  ( x  x.  M
) )  =  ( x  x.  ( K  x.  M ) ) )
18173expb 1206 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  CC  /\  K  =/=  0 )  /\  ( x  e.  CC  /\  M  e.  CC ) )  -> 
( K  x.  (
x  x.  M ) )  =  ( x  x.  ( K  x.  M ) ) )
1918ancoms 454 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  ( K  e.  CC  /\  K  =/=  0 ) )  -> 
( K  x.  (
x  x.  M ) )  =  ( x  x.  ( K  x.  M ) ) )
20193adant2 1024 . . . . . . . . . . . . 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 2431 . . . . . . . . . . . 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 9622 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( x  x.  M
)  e.  CC )
23 mulcan 10248 . . . . . . . . . . . . 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 1297 . . . . . . . . . . . 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 258 . . . . . . . . . . 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 1306 . . . . . . . . . 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 1206 . . . . . . . . 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 1200 . . . . . . . 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 1212 . . . . . . 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 1207 . . . . . 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 1201 . . . . 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 430 . . . 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 210 . . 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 14292 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  M )  ||  ( K  x.  N )  ->  M  ||  N ) )
35 dvdscmul 14307 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( K  x.  M )  ||  ( K  x.  N
) ) )
36353adant3r 1261 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  ||  N  ->  ( K  x.  M
)  ||  ( K  x.  N ) ) )
3734, 36impbid 193 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426  (class class class)co 6305   CCcc 9536   0cc0 9538    x. cmul 9543   ZZcz 10937    || cdvds 14283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-dvds 14284
This theorem is referenced by:  bitsmod  14384  mulgcd  14485  pcpremul  14756  4sqlem17OLD  14868  4sqlem17  14874  odmulg  17145  ablfacrp2  17635  ablfac1b  17638  pgpfac1lem3a  17644  znrrg  19067  fsumdvdsdiaglem  23975  oddpwdc  29013  jm2.20nn  35558
  Copyright terms: Public domain W3C validator