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Theorem coprmdvds 14344
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 10830 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
2 zcn 10830 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
3 mulcom 9528 . . . . . . . . . 10  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
41, 2, 3syl2an 475 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
543adant1 1015 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  =  ( N  x.  M ) )
65breq2d 4406 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  <->  K  ||  ( N  x.  M )
) )
7 dvdsmul2 14107 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  K  ||  ( N  x.  K ) )
87ancoms 451 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( N  x.  K ) )
983adant2 1016 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( N  x.  K
) )
10 simp1 997 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
11 zmulcl 10873 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K
)  e.  ZZ )
1211ancoms 451 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  K
)  e.  ZZ )
13123adant2 1016 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  K )  e.  ZZ )
14 zmulcl 10873 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  x.  M
)  e.  ZZ )
1514ancoms 451 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  M
)  e.  ZZ )
16153adant1 1015 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  M )  e.  ZZ )
17 dvdsgcd 14282 . . . . . . . . 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 1230 . . . . . . . 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 673 . . . . . . 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 463 . . . . 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 14286 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
( N  x.  K
)  gcd  ( N  x.  M ) )  =  ( abs `  ( N  x.  ( K  gcd  M ) ) ) )
23223coml 1204 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( N  x.  K
)  gcd  ( N  x.  M ) )  =  ( abs `  ( N  x.  ( K  gcd  M ) ) ) )
2423adantr 463 . . . . . . . 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 6242 . . . . . . . . . . 11  |-  ( ( K  gcd  M )  =  1  ->  ( N  x.  ( K  gcd  M ) )  =  ( N  x.  1 ) )
262mulid1d 9563 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N  x.  1 )  =  N )
2725, 26sylan9eqr 2465 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( K  gcd  M )  =  1 )  -> 
( N  x.  ( K  gcd  M ) )  =  N )
2827fveq2d 5809 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  ( K  gcd  M )  =  1 )  -> 
( abs `  ( N  x.  ( K  gcd  M ) ) )  =  ( abs `  N
) )
29283ad2antl3 1161 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( abs `  ( N  x.  ( K  gcd  M ) ) )  =  ( abs `  N
) )
3024, 29eqtrd 2443 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( ( N  x.  K )  gcd  ( N  x.  M
) )  =  ( abs `  N ) )
3130breq2d 4406 . . . . . 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 14104 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  N  <->  K 
||  ( abs `  N
) ) )
33323adant2 1016 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  N  <->  K  ||  ( abs `  N ) ) )
3433adantr 463 . . . . . 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 432 . . 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 429 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   CCcc 9440   1c1 9443    x. cmul 9447   ZZcz 10825   abscabs 13123    || cdvds 14087    gcd cgcd 14245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-2nd 6739  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-sup 7855  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-rp 11184  df-fl 11879  df-mod 11948  df-seq 12062  df-exp 12121  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-dvds 14088  df-gcd 14246
This theorem is referenced by:  coprmdvds2  14345  qredeq  14348  euclemma  14350  eulerthlem2  14413  prmdiveq  14417  prmpwdvds  14523  ablfacrp2  17330  dvdsmulf1o  23743  perfectlem1  23777  lgseisenlem1  23897  lgseisenlem2  23898  lgsquadlem2  23903  lgsquadlem3  23904  2sqlem8  23920  2sqmod  27968  nn0prpwlem  30538  coprmdvdsb  35267  jm2.20nn  35282  perfectALTVlem1  37776
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