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Theorem gcddvds 13007
Description: The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
gcddvds  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )

Proof of Theorem gcddvds
Dummy variables  n  K  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0z 10285 . . . . . 6  |-  0  e.  ZZ
2 dvds0 12857 . . . . . 6  |-  ( 0  e.  ZZ  ->  0  ||  0 )
31, 2ax-mp 8 . . . . 5  |-  0  ||  0
4 breq2 4208 . . . . . . 7  |-  ( M  =  0  ->  (
0  ||  M  <->  0  ||  0 ) )
5 breq2 4208 . . . . . . 7  |-  ( N  =  0  ->  (
0  ||  N  <->  0  ||  0 ) )
64, 5bi2anan9 844 . . . . . 6  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( 0 
||  M  /\  0  ||  N )  <->  ( 0 
||  0  /\  0  ||  0 ) ) )
7 anidm 626 . . . . . 6  |-  ( ( 0  ||  0  /\  0  ||  0 )  <->  0  ||  0 )
86, 7syl6bb 253 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( 0 
||  M  /\  0  ||  N )  <->  0  ||  0 ) )
93, 8mpbiri 225 . . . 4  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( 0  ||  M  /\  0  ||  N
) )
10 oveq12 6082 . . . . . . 7  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( 0  gcd  0 ) )
11 gcd0val 13001 . . . . . . 7  |-  ( 0  gcd  0 )  =  0
1210, 11syl6eq 2483 . . . . . 6  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  0 )
1312breq1d 4214 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  M 
<->  0  ||  M ) )
1412breq1d 4214 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  N 
<->  0  ||  N ) )
1513, 14anbi12d 692 . . . 4  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( ( M  gcd  N ) 
||  M  /\  ( M  gcd  N )  ||  N )  <->  ( 0 
||  M  /\  0  ||  N ) ) )
169, 15mpbird 224 . . 3  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) )
1716adantl 453 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
18 eqid 2435 . . . . 5  |-  { n  e.  ZZ  |  A. z  e.  { M ,  N } n  ||  z }  =  { n  e.  ZZ  |  A. z  e.  { M ,  N } n  ||  z }
19 eqid 2435 . . . . 5  |-  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) }
2018, 19gcdcllem3 13005 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  )  e.  NN  /\  ( sup ( { n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) } ,  RR ,  <  )  ||  M  /\  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  ||  N )  /\  (
( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N )  ->  K  <_  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  ) ) ) )
2120simp2d 970 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  )  ||  M  /\  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  ||  N ) )
22 gcdn0val 13002 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )
2322breq1d 4214 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( M  gcd  N )  ||  M 
<->  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  ||  M ) )
2422breq1d 4214 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( M  gcd  N )  ||  N 
<->  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  ||  N ) )
2523, 24anbi12d 692 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( ( M  gcd  N ) 
||  M  /\  ( M  gcd  N )  ||  N )  <->  ( sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  )  ||  M  /\  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  ||  N ) ) )
2621, 25mpbird 224 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) )
2717, 26pm2.61dan 767 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   {cpr 3807   class class class wbr 4204  (class class class)co 6073   supcsup 7437   RRcr 8981   0cc0 8982    < clt 9112    <_ cle 9113   NNcn 9992   ZZcz 10274    || cdivides 12844    gcd cgcd 12998
This theorem is referenced by:  gcd0id  13015  gcdneg  13018  gcdaddmlem  13020  gcd1  13024  bezoutlem4  13033  dvdsgcdb  13036  mulgcd  13038  gcdeq  13044  dvdsmulgcd  13046  sqgcd  13050  dvdssqlem  13051  coprm  13092  mulgcddvds  13096  rpmulgcd2  13097  qredeu  13099  divgcdodd  13111  rpexp  13112  rpdvds  13116  divnumden  13132  phimullem  13160  pythagtriplem4  13185  pythagtriplem19  13199  pcgcd1  13242  pc2dvds  13244  pockthlem  13265  odmulg  15184  odadd1  15455  odadd2  15456  znunit  16836  znrrg  16838  dvdsmulf1o  20971  2sqlem8  21148  qqhval2lem  24357  bezoutr  27041  hashgcdlem  27484  hashgcdeq  27485  phisum  27486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999
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