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Theorem coprmdvds 12111
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 9277 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  CC )
2 zcn 9277 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
3 mulcom 7959 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
41, 2, 3syl2an 289 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  =  ( N  x.  M ) )
54breq2d 4030 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  <->  K 
||  ( N  x.  M ) ) )
6 dvdsmulgcd 12045 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  ||  ( N  x.  M )  <->  K 
||  ( N  x.  ( M  gcd  K ) ) ) )
76ancoms 268 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( N  x.  M )  <->  K 
||  ( N  x.  ( M  gcd  K ) ) ) )
85, 7bitrd 188 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  <->  K 
||  ( N  x.  ( M  gcd  K ) ) ) )
983adant1 1017 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  <->  K  ||  ( N  x.  ( M  gcd  K ) ) ) )
109adantr 276 . . . . . 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 11993 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  gcd  M
)  =  ( M  gcd  K ) )
12113adant3 1019 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  =  ( M  gcd  K
) )
1312eqeq1d 2198 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  =  1  <->  ( M  gcd  K )  =  1 ) )
14 oveq2 5899 . . . . . . . . . 10  |-  ( ( M  gcd  K )  =  1  ->  ( N  x.  ( M  gcd  K ) )  =  ( N  x.  1 ) )
1513, 14biimtrdi 163 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  =  1  -> 
( N  x.  ( M  gcd  K ) )  =  ( N  x.  1 ) ) )
1615imp 124 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( N  x.  ( M  gcd  K ) )  =  ( N  x.  1 ) )
172mulridd 7993 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( N  x.  1 )  =  N )
18173ad2ant3 1022 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  x.  1 )  =  N )
1918adantr 276 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( N  x.  1 )  =  N )
2016, 19eqtrd 2222 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( N  x.  ( M  gcd  K ) )  =  N )
2120breq2d 4030 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( N  x.  ( M  gcd  K ) )  <-> 
K  ||  N )
)
2210, 21bitrd 188 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( M  x.  N
)  <->  K  ||  N ) )
2322biimpd 144 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  ||  ( M  x.  N
)  ->  K  ||  N
) )
2423ex 115 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  =  1  -> 
( K  ||  ( M  x.  N )  ->  K  ||  N ) ) )
2524com23 78 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( M  x.  N )  ->  (
( K  gcd  M
)  =  1  ->  K  ||  N ) ) )
2625impd 254 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 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5891   CCcc 7828   1c1 7831    x. cmul 7835   ZZcz 9272    || cdvds 11813    gcd cgcd 11962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-sup 7002  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-n0 9196  df-z 9273  df-uz 9548  df-q 9639  df-rp 9673  df-fz 10028  df-fzo 10162  df-fl 10289  df-mod 10342  df-seqfrec 10465  df-exp 10539  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027  df-dvds 11814  df-gcd 11963
This theorem is referenced by:  coprmdvds2  12112  qredeq  12115  cncongr1  12122  euclemma  12165  eulerthlemh  12250  eulerthlemth  12251  prmdiveq  12255  prmpwdvds  12372  lgseisenlem1  14853  lgseisenlem2  14854  2sqlem8  14873
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