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Theorem coprmdvds 12781
Description: If an integer divides the product of two integers and is coprime to one of them, then it divides the other. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
coprmdvds  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  ( M  x.  N )  /\  ( K  gcd  M
)  =  1 )  ->  K  ||  N
) )

Proof of Theorem coprmdvds
StepHypRef Expression
1 zcn 10029 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
2 zcn 10029 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
3 mulcom 8823 . . . . . . . . . 10  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
41, 2, 3syl2an 463 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
543adant1 973 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  =  ( N  x.  M ) )
65breq2d 4035 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  <->  K  ||  ( N  x.  M )
) )
7 dvdsmul2 12551 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  K  ||  ( N  x.  K ) )
87ancoms 439 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( N  x.  K ) )
983adant2 974 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( N  x.  K
) )
10 simp1 955 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
11 zmulcl 10066 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K
)  e.  ZZ )
1211ancoms 439 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  K
)  e.  ZZ )
13123adant2 974 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  K )  e.  ZZ )
14 zmulcl 10066 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  x.  M
)  e.  ZZ )
1514ancoms 439 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  M
)  e.  ZZ )
16153adant1 973 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  M )  e.  ZZ )
17 dvdsgcd 12722 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  ( N  x.  K
)  e.  ZZ  /\  ( N  x.  M
)  e.  ZZ )  ->  ( ( K 
||  ( N  x.  K )  /\  K  ||  ( N  x.  M
) )  ->  K  ||  ( ( N  x.  K )  gcd  ( N  x.  M )
) ) )
1810, 13, 16, 17syl3anc 1182 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  ( N  x.  K )  /\  K  ||  ( N  x.  M ) )  ->  K  ||  (
( N  x.  K
)  gcd  ( N  x.  M ) ) ) )
199, 18mpand 656 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( N  x.  M )  ->  K  ||  ( ( N  x.  K )  gcd  ( N  x.  M )
) ) )
206, 19sylbid 206 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  ->  K  ||  ( ( N  x.  K )  gcd  ( N  x.  M )
) ) )
2120adantr 451 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( M  x.  N
)  ->  K  ||  (
( N  x.  K
)  gcd  ( N  x.  M ) ) ) )
22 absmulgcd 12726 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
( N  x.  K
)  gcd  ( N  x.  M ) )  =  ( abs `  ( N  x.  ( K  gcd  M ) ) ) )
23223coml 1158 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( N  x.  K
)  gcd  ( N  x.  M ) )  =  ( abs `  ( N  x.  ( K  gcd  M ) ) ) )
2423adantr 451 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( ( N  x.  K )  gcd  ( N  x.  M
) )  =  ( abs `  ( N  x.  ( K  gcd  M ) ) ) )
25 oveq2 5866 . . . . . . . . . . 11  |-  ( ( K  gcd  M )  =  1  ->  ( N  x.  ( K  gcd  M ) )  =  ( N  x.  1 ) )
262mulid1d 8852 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N  x.  1 )  =  N )
2725, 26sylan9eqr 2337 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( K  gcd  M )  =  1 )  -> 
( N  x.  ( K  gcd  M ) )  =  N )
2827fveq2d 5529 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  ( K  gcd  M )  =  1 )  -> 
( abs `  ( N  x.  ( K  gcd  M ) ) )  =  ( abs `  N
) )
29283ad2antl3 1119 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( abs `  ( N  x.  ( K  gcd  M ) ) )  =  ( abs `  N
) )
3024, 29eqtrd 2315 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( ( N  x.  K )  gcd  ( N  x.  M
) )  =  ( abs `  N ) )
3130breq2d 4035 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( ( N  x.  K )  gcd  ( N  x.  M )
)  <->  K  ||  ( abs `  N ) ) )
32 dvdsabsb 12548 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  N  <->  K 
||  ( abs `  N
) ) )
33323adant2 974 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  N  <->  K  ||  ( abs `  N ) ) )
3433adantr 451 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  N 
<->  K  ||  ( abs `  N ) ) )
3531, 34bitr4d 247 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( ( N  x.  K )  gcd  ( N  x.  M )
)  <->  K  ||  N ) )
3621, 35sylibd 205 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( M  x.  N
)  ->  K  ||  N
) )
3736ex 423 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  =  1  -> 
( K  ||  ( M  x.  N )  ->  K  ||  N ) ) )
3837com23 72 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  ->  (
( K  gcd  M
)  =  1  ->  K  ||  N ) ) )
3938imp3a 420 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  ( M  x.  N )  /\  ( K  gcd  M
)  =  1 )  ->  K  ||  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    x. cmul 8742   ZZcz 10024   abscabs 11719    || cdivides 12531    gcd cgcd 12685
This theorem is referenced by:  coprmdvds2  12782  qredeq  12785  euclemma  12787  eulerthlem2  12850  prmdiveq  12854  prmpwdvds  12951  ablfacrp2  15302  dvdsmulf1o  20434  perfectlem1  20468  lgseisenlem1  20588  lgseisenlem2  20589  lgsquadlem2  20594  lgsquadlem3  20595  2sqlem8  20611  nn0prpwlem  26238  coprmdvdsb  27074  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-cnex 8793  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  ax-pre-sup 8815
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-rmo 2551  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-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686
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