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Theorem dvdsmulcr 13666
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 10796 . . . . . 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 10796 . . . . . 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 1216 . . 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 10754 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
10 zcn 10754 . . . . . . . . . . . 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 10754 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
13 zcn 10754 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
1413anim1i 568 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( K  e.  CC  /\  K  =/=  0 ) )
15 mulass 9473 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  M  e.  CC  /\  K  e.  CC )  ->  (
( x  x.  M
)  x.  K )  =  ( x  x.  ( M  x.  K
) ) )
16153expa 1188 . . . . . . . . . . . . . . 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 2453 . . . . . . . . . . . 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 9469 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( x  x.  M
)  e.  CC )
21 mulcan2 10077 . . . . . . . . . . . . 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 1252 . . . . . . . . . . . 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 1261 . . . . . . . . . 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 1189 . . . . . . . . 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 1183 . . . . . . . 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 1195 . . . . . . 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 1190 . . . . . 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 1184 . . . . 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 13648 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( M  x.  K )  ||  ( N  x.  K )  ->  M  ||  N ) )
33 dvdsmulc 13664 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( M  x.  K )  ||  ( N  x.  K
) ) )
34333adant3r 1216 . 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 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4392  (class class class)co 6192   CCcc 9383   0cc0 9385    x. cmul 9390   ZZcz 10749    || cdivides 13639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-dvds 13640
This theorem is referenced by:  mulgcddvds  13894  prmpwdvds  14069  4sqlem10  14112  sylow3lem4  16235  odadd1  16436  odadd2  16437  ablfacrp2  16675  ablfac1eu  16681  fsumdvdsdiaglem  22641  nn0prpwlem  28657  jm2.20nn  29486
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