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Theorem dvdsmulcr 13554
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 10685 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  x.  K
)  e.  ZZ )
213adant2 1007 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  x.  K )  e.  ZZ )
3 zmulcl 10685 . . . . . 6  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K
)  e.  ZZ )
433adant1 1006 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K )  e.  ZZ )
52, 4jca 532 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M  x.  K
)  e.  ZZ  /\  ( N  x.  K
)  e.  ZZ ) )
653adant3r 1215 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( M  x.  K )  e.  ZZ  /\  ( N  x.  K
)  e.  ZZ ) )
7 3simpa 985 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  e.  ZZ  /\  N  e.  ZZ ) )
8 simpr 461 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
9 zcn 10643 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
10 zcn 10643 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  CC )
119, 10anim12i 566 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  e.  CC  /\  M  e.  CC ) )
12 zcn 10643 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
13 zcn 10643 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
1413anim1i 568 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( K  e.  CC  /\  K  =/=  0 ) )
15 mulass 9362 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  M  e.  CC  /\  K  e.  CC )  ->  (
( x  x.  M
)  x.  K )  =  ( x  x.  ( M  x.  K
) ) )
16153expa 1187 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  K  e.  CC )  ->  ( ( x  x.  M )  x.  K )  =  ( x  x.  ( M  x.  K ) ) )
1716adantrr 716 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  ( K  e.  CC  /\  K  =/=  0 ) )  -> 
( ( x  x.  M )  x.  K
)  =  ( x  x.  ( M  x.  K ) ) )
18173adant2 1007 . . . . . . . . . . . . 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 2446 . . . . . . . . . . . 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 9358 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( x  x.  M
)  e.  CC )
21 mulcan2 9966 . . . . . . . . . . . . 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 1251 . . . . . . . . . . . 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 1260 . . . . . . . . . 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 1188 . . . . . . . . 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 1182 . . . . . . . 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 1194 . . . . . . 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 1189 . . . . . 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 1183 . . . . 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 429 . . . 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 13536 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( M  x.  K )  ||  ( N  x.  K )  ->  M  ||  N ) )
33 dvdsmulc 13552 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( M  x.  K )  ||  ( N  x.  K
) ) )
34333adant3r 1215 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  ||  N  ->  ( M  x.  K
)  ||  ( N  x.  K ) ) )
3532, 34impbid 191 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   class class class wbr 4287  (class class class)co 6086   CCcc 9272   0cc0 9274    x. cmul 9279   ZZcz 10638    || cdivides 13527
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-dvds 13528
This theorem is referenced by:  mulgcddvds  13782  prmpwdvds  13957  4sqlem10  14000  sylow3lem4  16120  odadd1  16321  odadd2  16322  ablfacrp2  16556  ablfac1eu  16562  fsumdvdsdiaglem  22498  nn0prpwlem  28470  jm2.20nn  29299
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