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Theorem dvdsmulcr 12558
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 10066 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  x.  K
)  e.  ZZ )
213adant2 974 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  x.  K )  e.  ZZ )
3 zmulcl 10066 . . . . . 6  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K
)  e.  ZZ )
433adant1 973 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K )  e.  ZZ )
52, 4jca 518 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M  x.  K
)  e.  ZZ  /\  ( N  x.  K
)  e.  ZZ ) )
653adant3r 1179 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( M  x.  K )  e.  ZZ  /\  ( N  x.  K
)  e.  ZZ ) )
7 3simpa 952 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  e.  ZZ  /\  N  e.  ZZ ) )
8 simpr 447 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
9 zcn 10029 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
10 zcn 10029 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  CC )
119, 10anim12i 549 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  e.  CC  /\  M  e.  CC ) )
12 zcn 10029 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
13 zcn 10029 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
1413anim1i 551 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( K  e.  CC  /\  K  =/=  0 ) )
15 mulass 8825 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  M  e.  CC  /\  K  e.  CC )  ->  (
( x  x.  M
)  x.  K )  =  ( x  x.  ( M  x.  K
) ) )
16153expa 1151 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  K  e.  CC )  ->  ( ( x  x.  M )  x.  K )  =  ( x  x.  ( M  x.  K ) ) )
1716adantrr 697 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  CC  /\  M  e.  CC )  /\  ( K  e.  CC  /\  K  =/=  0 ) )  -> 
( ( x  x.  M )  x.  K
)  =  ( x  x.  ( M  x.  K ) ) )
18173adant2 974 . . . . . . . . . . . . 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 2291 . . . . . . . . . . . 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 8821 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( x  x.  M
)  e.  CC )
21 mulcan2 9406 . . . . . . . . . . . . 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 1215 . . . . . . . . . . . 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 246 . . . . . . . . . . 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 1224 . . . . . . . . . 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 1152 . . . . . . . . 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 1146 . . . . . . . 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 1158 . . . . . . 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 1153 . . . . . 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 1147 . . . . 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 418 . . . 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 198 . . 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 12540 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( M  x.  K )  ||  ( N  x.  K )  ->  M  ||  N ) )
33 dvdsmulc 12556 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( M  x.  K )  ||  ( N  x.  K
) ) )
34333adant3r 1179 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( M  ||  N  ->  ( M  x.  K
)  ||  ( N  x.  K ) ) )
3532, 34impbid 183 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( M  x.  K )  ||  ( N  x.  K )  <->  M 
||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858   CCcc 8735   0cc0 8737    x. cmul 8742   ZZcz 10024    || cdivides 12531
This theorem is referenced by:  mulgcddvds  12783  prmpwdvds  12951  4sqlem10  12994  sylow3lem4  14941  odadd1  15140  odadd2  15141  ablfacrp2  15302  ablfac1eu  15308  fsumdvdsdiaglem  20423  nn0prpwlem  26238  jm2.20nn  27090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-dvds 12532
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