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Theorem dvdsgcd 10608
Description: An integer which divides each of two others also divides their gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 30-May-2014.)
Assertion
Ref Expression
dvdsgcd  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  gcd  N ) ) )

Proof of Theorem dvdsgcd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bezout 10607 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) ) )
213adant1 957 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) ) )
3 dvds2ln 10436 . . . . . . . . 9  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( K  ||  M  /\  K  ||  N
)  ->  K  ||  (
( x  x.  M
)  +  ( y  x.  N ) ) ) )
433impia 1136 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  ||  ( ( x  x.  M )  +  ( y  x.  N
) ) )
543coml 1146 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  K  ||  ( ( x  x.  M )  +  ( y  x.  N
) ) )
6 simp3l 967 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  x  e.  ZZ )
7 simp12 970 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  M  e.  ZZ )
8 zcn 8489 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 8489 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
10 mulcom 7216 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( x  x.  M
)  =  ( M  x.  x ) )
118, 9, 10syl2an 283 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  x.  M
)  =  ( M  x.  x ) )
126, 7, 11syl2anc 403 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( x  x.  M
)  =  ( M  x.  x ) )
13 simp3r 968 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
y  e.  ZZ )
14 simp13 971 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  N  e.  ZZ )
15 zcn 8489 . . . . . . . . . 10  |-  ( y  e.  ZZ  ->  y  e.  CC )
16 zcn 8489 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
17 mulcom 7216 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  N  e.  CC )  ->  ( y  x.  N
)  =  ( N  x.  y ) )
1815, 16, 17syl2an 283 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  N  e.  ZZ )  ->  ( y  x.  N
)  =  ( N  x.  y ) )
1913, 14, 18syl2anc 403 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( y  x.  N
)  =  ( N  x.  y ) )
2012, 19oveq12d 5581 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( ( x  x.  M )  +  ( y  x.  N ) )  =  ( ( M  x.  x )  +  ( N  x.  y ) ) )
215, 20breqtrd 3829 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  K  ||  ( ( M  x.  x )  +  ( N  x.  y
) ) )
22 breq2 3809 . . . . . 6  |-  ( ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y
) )  ->  ( K  ||  ( M  gcd  N )  <->  K  ||  ( ( M  x.  x )  +  ( N  x.  y ) ) ) )
2321, 22syl5ibrcom 155 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) )  ->  K  ||  ( M  gcd  N ) ) )
24233expia 1141 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  ( (
x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) )  ->  K  ||  ( M  gcd  N ) ) ) )
2524rexlimdvv 2488 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) )  ->  K  ||  ( M  gcd  N ) ) )
2625ex 113 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) )  ->  K  ||  ( M  gcd  N ) ) ) )
272, 26mpid 41 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  gcd  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434   E.wrex 2354   class class class wbr 3805  (class class class)co 5563   CCcc 7093    + caddc 7098    x. cmul 7100   ZZcz 8484    || cdvds 10403    gcd cgcd 10545
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 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
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 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546
This theorem is referenced by:  dvdsgcdb  10609  dfgcd2  10610  mulgcd  10612  ncoprmgcdne1b  10678  mulgcddvds  10683  rpmulgcd2  10684  rpexp  10739
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