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Theorem coprmdvds 10854
Description: Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.)
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 8651 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  CC )
2 zcn 8651 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
3 mulcom 7374 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
41, 2, 3syl2an 283 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
54breq2d 3823 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  <->  K 
||  ( N  x.  M ) ) )
6 dvdsmulgcd 10794 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  ||  ( N  x.  M )  <->  K 
||  ( N  x.  ( M  gcd  K ) ) ) )
76ancoms 264 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( N  x.  M )  <->  K 
||  ( N  x.  ( M  gcd  K ) ) ) )
85, 7bitrd 186 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  <->  K 
||  ( N  x.  ( M  gcd  K ) ) ) )
983adant1 957 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  <->  K  ||  ( N  x.  ( M  gcd  K ) ) ) )
109adantr 270 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( M  x.  N
)  <->  K  ||  ( N  x.  ( M  gcd  K ) ) ) )
11 gcdcom 10745 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  gcd  M
)  =  ( M  gcd  K ) )
12113adant3 959 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  =  ( M  gcd  K
) )
1312eqeq1d 2091 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  =  1  <->  ( M  gcd  K )  =  1 ) )
14 oveq2 5599 . . . . . . . . . 10  |-  ( ( M  gcd  K )  =  1  ->  ( N  x.  ( M  gcd  K ) )  =  ( N  x.  1 ) )
1513, 14syl6bi 161 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  =  1  -> 
( N  x.  ( M  gcd  K ) )  =  ( N  x.  1 ) ) )
1615imp 122 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( N  x.  ( M  gcd  K ) )  =  ( N  x.  1 ) )
172mulid1d 7408 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( N  x.  1 )  =  N )
18173ad2ant3 962 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  1 )  =  N )
1918adantr 270 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( N  x.  1 )  =  N )
2016, 19eqtrd 2115 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( N  x.  ( M  gcd  K ) )  =  N )
2120breq2d 3823 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( N  x.  ( M  gcd  K ) )  <-> 
K  ||  N )
)
2210, 21bitrd 186 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( M  x.  N
)  <->  K  ||  N ) )
2322biimpd 142 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( M  x.  N
)  ->  K  ||  N
) )
2423ex 113 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  =  1  -> 
( K  ||  ( M  x.  N )  ->  K  ||  N ) ) )
2524com23 77 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  ->  (
( K  gcd  M
)  =  1  ->  K  ||  N ) ) )
2625impd 251 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    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3811  (class class class)co 5591   CCcc 7251   1c1 7254    x. cmul 7258   ZZcz 8646    || cdvds 10576    gcd cgcd 10718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-mulrcl 7347  ax-addcom 7348  ax-mulcom 7349  ax-addass 7350  ax-mulass 7351  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-1rid 7355  ax-0id 7356  ax-rnegex 7357  ax-precex 7358  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-apti 7363  ax-pre-ltadd 7364  ax-pre-mulgt0 7365  ax-pre-mulext 7366  ax-arch 7367  ax-caucvg 7368
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-ilim 4160  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-frec 6088  df-sup 6586  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-reap 7952  df-ap 7959  df-div 8038  df-inn 8317  df-2 8375  df-3 8376  df-4 8377  df-n0 8566  df-z 8647  df-uz 8915  df-q 9000  df-rp 9030  df-fz 9320  df-fzo 9444  df-fl 9566  df-mod 9619  df-iseq 9741  df-iexp 9792  df-cj 10103  df-re 10104  df-im 10105  df-rsqrt 10258  df-abs 10259  df-dvds 10577  df-gcd 10719
This theorem is referenced by:  coprmdvds2  10855  qredeq  10858  cncongr1  10865  euclemma  10905
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