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Theorem coprmdvds 13809
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 10672 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
2 zcn 10672 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
3 mulcom 9389 . . . . . . . . . 10  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
41, 2, 3syl2an 477 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
543adant1 1006 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  =  ( N  x.  M ) )
65breq2d 4325 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  <->  K  ||  ( N  x.  M )
) )
7 dvdsmul2 13576 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  K  ||  ( N  x.  K ) )
87ancoms 453 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( N  x.  K ) )
983adant2 1007 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( N  x.  K
) )
10 simp1 988 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
11 zmulcl 10714 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K
)  e.  ZZ )
1211ancoms 453 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  K
)  e.  ZZ )
13123adant2 1007 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  K )  e.  ZZ )
14 zmulcl 10714 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  x.  M
)  e.  ZZ )
1514ancoms 453 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  M
)  e.  ZZ )
16153adant1 1006 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  M )  e.  ZZ )
17 dvdsgcd 13748 . . . . . . . . 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 1218 . . . . . . . 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 675 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( N  x.  M )  ->  K  ||  ( ( N  x.  K )  gcd  ( N  x.  M )
) ) )
206, 19sylbid 215 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  ->  K  ||  ( ( N  x.  K )  gcd  ( N  x.  M )
) ) )
2120adantr 465 . . . . 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 13752 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
( N  x.  K
)  gcd  ( N  x.  M ) )  =  ( abs `  ( N  x.  ( K  gcd  M ) ) ) )
23223coml 1194 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( N  x.  K
)  gcd  ( N  x.  M ) )  =  ( abs `  ( N  x.  ( K  gcd  M ) ) ) )
2423adantr 465 . . . . . . . 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 6120 . . . . . . . . . . 11  |-  ( ( K  gcd  M )  =  1  ->  ( N  x.  ( K  gcd  M ) )  =  ( N  x.  1 ) )
262mulid1d 9424 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N  x.  1 )  =  N )
2725, 26sylan9eqr 2497 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( K  gcd  M )  =  1 )  -> 
( N  x.  ( K  gcd  M ) )  =  N )
2827fveq2d 5716 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  ( K  gcd  M )  =  1 )  -> 
( abs `  ( N  x.  ( K  gcd  M ) ) )  =  ( abs `  N
) )
29283ad2antl3 1152 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( abs `  ( N  x.  ( K  gcd  M ) ) )  =  ( abs `  N
) )
3024, 29eqtrd 2475 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( ( N  x.  K )  gcd  ( N  x.  M
) )  =  ( abs `  N ) )
3130breq2d 4325 . . . . . 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 13573 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  N  <->  K 
||  ( abs `  N
) ) )
33323adant2 1007 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  N  <->  K  ||  ( abs `  N ) ) )
3433adantr 465 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  N 
<->  K  ||  ( abs `  N ) ) )
3531, 34bitr4d 256 . . . . 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 214 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( M  x.  N
)  ->  K  ||  N
) )
3736ex 434 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  =  1  -> 
( K  ||  ( M  x.  N )  ->  K  ||  N ) ) )
3837com23 78 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  ->  (
( K  gcd  M
)  =  1  ->  K  ||  N ) ) )
3938impd 431 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  ( M  x.  N )  /\  ( K  gcd  M
)  =  1 )  ->  K  ||  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   CCcc 9301   1c1 9304    x. cmul 9308   ZZcz 10667   abscabs 12744    || cdivides 13556    gcd cgcd 13711
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 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-dvds 13557  df-gcd 13712
This theorem is referenced by:  coprmdvds2  13810  qredeq  13813  euclemma  13815  eulerthlem2  13878  prmdiveq  13882  prmpwdvds  13986  ablfacrp2  16590  dvdsmulf1o  22556  perfectlem1  22590  lgseisenlem1  22710  lgseisenlem2  22711  lgsquadlem2  22716  lgsquadlem3  22717  2sqlem8  22733  nn0prpwlem  28543  coprmdvdsb  29356  jm2.20nn  29372
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