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Theorem coprmdvds 14091
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 10858 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
2 zcn 10858 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
3 mulcom 9567 . . . . . . . . . 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 1009 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  =  ( N  x.  M ) )
65breq2d 4452 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  <->  K  ||  ( N  x.  M )
) )
7 dvdsmul2 13856 . . . . . . . . . 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 1010 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( N  x.  K
) )
10 simp1 991 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
11 zmulcl 10900 . . . . . . . . . . 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 1010 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  K )  e.  ZZ )
14 zmulcl 10900 . . . . . . . . . . 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 1009 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  M )  e.  ZZ )
17 dvdsgcd 14029 . . . . . . . . 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 1223 . . . . . . . 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 14033 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
( N  x.  K
)  gcd  ( N  x.  M ) )  =  ( abs `  ( N  x.  ( K  gcd  M ) ) ) )
23223coml 1198 . . . . . . . . 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 6283 . . . . . . . . . . 11  |-  ( ( K  gcd  M )  =  1  ->  ( N  x.  ( K  gcd  M ) )  =  ( N  x.  1 ) )
262mulid1d 9602 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N  x.  1 )  =  N )
2725, 26sylan9eqr 2523 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( K  gcd  M )  =  1 )  -> 
( N  x.  ( K  gcd  M ) )  =  N )
2827fveq2d 5861 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  ( K  gcd  M )  =  1 )  -> 
( abs `  ( N  x.  ( K  gcd  M ) ) )  =  ( abs `  N
) )
29283ad2antl3 1155 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( abs `  ( N  x.  ( K  gcd  M ) ) )  =  ( abs `  N
) )
3024, 29eqtrd 2501 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( ( N  x.  K )  gcd  ( N  x.  M
) )  =  ( abs `  N ) )
3130breq2d 4452 . . . . . 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 13853 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  N  <->  K 
||  ( abs `  N
) ) )
33323adant2 1010 . . . . . . 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 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   1c1 9482    x. cmul 9486   ZZcz 10853   abscabs 13017    || cdivides 13836    gcd cgcd 13992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-dvds 13837  df-gcd 13993
This theorem is referenced by:  coprmdvds2  14092  qredeq  14095  euclemma  14097  eulerthlem2  14160  prmdiveq  14164  prmpwdvds  14270  ablfacrp2  16901  dvdsmulf1o  23191  perfectlem1  23225  lgseisenlem1  23345  lgseisenlem2  23346  lgsquadlem2  23351  lgsquadlem3  23352  2sqlem8  23368  nn0prpwlem  29704  coprmdvdsb  30516  jm2.20nn  30532
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