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

Theorem dvdsmulcr 14222
Description: Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvdsmulcr  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( M  x.  K )  ||  ( N  x.  K )  <->  M 
||  N ) )

Proof of Theorem dvdsmulcr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 zmulcl 10953 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  x.  K
)  e.  ZZ )
213adant2 1016 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  x.  K )  e.  ZZ )
3 zmulcl 10953 . . . . . 6  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K
)  e.  ZZ )
433adant1 1015 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K )  e.  ZZ )
52, 4jca 530 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M  x.  K
)  e.  ZZ  /\  ( N  x.  K
)  e.  ZZ ) )
653adant3r 1227 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( M  x.  K )  e.  ZZ  /\  ( N  x.  K
)  e.  ZZ ) )
7 3simpa 994 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  e.  ZZ  /\  N  e.  ZZ ) )
8 simpr 459 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
9 zcn 10910 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
10 zcn 10910 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  CC )
119, 10anim12i 564 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  e.  CC  /\  M  e.  CC ) )
12 zcn 10910 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
13 zcn 10910 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
1413anim1i 566 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( K  e.  CC  /\  K  =/=  0 ) )
15 mulass 9610 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  M  e.  CC  /\  K  e.  CC )  ->  (
( x  x.  M
)  x.  K )  =  ( x  x.  ( M  x.  K
) ) )
16153expa 1197 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  K  e.  CC )  ->  ( ( x  x.  M )  x.  K )  =  ( x  x.  ( M  x.  K ) ) )
1716adantrr 715 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  ( K  e.  CC  /\  K  =/=  0 ) )  -> 
( ( x  x.  M )  x.  K
)  =  ( x  x.  ( M  x.  K ) ) )
18173adant2 1016 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC  /\  ( K  e.  CC  /\  K  =/=  0 ) )  ->  ( (
x  x.  M )  x.  K )  =  ( x  x.  ( M  x.  K )
) )
1918eqeq1d 2404 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC  /\  ( K  e.  CC  /\  K  =/=  0 ) )  ->  ( (
( x  x.  M
)  x.  K )  =  ( N  x.  K )  <->  ( x  x.  ( M  x.  K
) )  =  ( N  x.  K ) ) )
20 mulcl 9606 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( x  x.  M
)  e.  CC )
21 mulcan2 10228 . . . . . . . . . . . . 13  |-  ( ( ( x  x.  M
)  e.  CC  /\  N  e.  CC  /\  ( K  e.  CC  /\  K  =/=  0 ) )  -> 
( ( ( x  x.  M )  x.  K )  =  ( N  x.  K )  <-> 
( x  x.  M
)  =  N ) )
2220, 21syl3an1 1263 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC  /\  ( K  e.  CC  /\  K  =/=  0 ) )  ->  ( (
( x  x.  M
)  x.  K )  =  ( N  x.  K )  <->  ( x  x.  M )  =  N ) )
2319, 22bitr3d 255 . . . . . . . . . . 11  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  N  e.  CC  /\  ( K  e.  CC  /\  K  =/=  0 ) )  ->  ( (
x  x.  ( M  x.  K ) )  =  ( N  x.  K )  <->  ( x  x.  M )  =  N ) )
2411, 12, 14, 23syl3an 1272 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  M  e.  ZZ )  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  ( (
x  x.  ( M  x.  K ) )  =  ( N  x.  K )  <->  ( x  x.  M )  =  N ) )
25243expb 1198 . . . . . . . . 9  |-  ( ( ( x  e.  ZZ  /\  M  e.  ZZ )  /\  ( N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) ) )  ->  ( ( x  x.  ( M  x.  K ) )  =  ( N  x.  K
)  <->  ( x  x.  M )  =  N ) )
26253impa 1192 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  ( N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) ) )  ->  ( ( x  x.  ( M  x.  K ) )  =  ( N  x.  K
)  <->  ( x  x.  M )  =  N ) )
27263coml 1204 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  (
( x  x.  ( M  x.  K )
)  =  ( N  x.  K )  <->  ( x  x.  M )  =  N ) )
28273expia 1199 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) ) )  ->  (
x  e.  ZZ  ->  ( ( x  x.  ( M  x.  K )
)  =  ( N  x.  K )  <->  ( x  x.  M )  =  N ) ) )
29283impb 1193 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( x  e.  ZZ  ->  ( ( x  x.  ( M  x.  K
) )  =  ( N  x.  K )  <-> 
( x  x.  M
)  =  N ) ) )
3029imp 427 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  (
( x  x.  ( M  x.  K )
)  =  ( N  x.  K )  <->  ( x  x.  M )  =  N ) )
3130biimpd 207 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  (
( x  x.  ( M  x.  K )
)  =  ( N  x.  K )  -> 
( x  x.  M
)  =  N ) )
326, 7, 8, 31dvds1lem 14204 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( M  x.  K )  ||  ( N  x.  K )  ->  M  ||  N ) )
33 dvdsmulc 14220 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( M  x.  K )  ||  ( N  x.  K
) ) )
34333adant3r 1227 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  ||  N  ->  ( M  x.  K
)  ||  ( N  x.  K ) ) )
3532, 34impbid 190 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( M  x.  K )  ||  ( N  x.  K )  <->  M 
||  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4395  (class class class)co 6278   CCcc 9520   0cc0 9522    x. cmul 9527   ZZcz 10905    || cdvds 14195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-n0 10837  df-z 10906  df-dvds 14196
This theorem is referenced by:  mulgcddvds  14454  prmpwdvds  14631  4sqlem10  14674  sylow3lem4  16974  odadd1  17178  odadd2  17179  ablfacrp2  17438  ablfac1eu  17444  fsumdvdsdiaglem  23840  nn0prpwlem  30550  jm2.20nn  35301  etransclem38  37423
  Copyright terms: Public domain W3C validator