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Theorem dvdslegcd 12695
Description: An integer which divides both operands of the  gcd operator is bounded by it. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvdslegcd  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  -> 
( ( K  ||  M  /\  K  ||  N
)  ->  K  <_  ( M  gcd  N ) ) )

Proof of Theorem dvdslegcd
Dummy variables  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . . . . . . . . 12  |-  { n  e.  ZZ  |  A. z  e.  { M ,  N } n  ||  z }  =  { n  e.  ZZ  |  A. z  e.  { M ,  N } n  ||  z }
2 eqid 2283 . . . . . . . . . . . 12  |-  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) }  =  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) }
31, 2gcdcllem3 12692 . . . . . . . . . . 11  |-  ( ( ( 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 ,  <  ) ) ) )
43simp3d 969 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N )  ->  K  <_  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  ) ) )
5 gcdn0val 12689 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )
65breq2d 4035 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( K  <_ 
( M  gcd  N
)  <->  K  <_  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) ) )
74, 6sylibrd 225 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N )  ->  K  <_  ( M  gcd  N
) ) )
87com12 27 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N )  ->  (
( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  ->  K  <_  ( M  gcd  N
) ) )
983expb 1152 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  ( K  ||  M  /\  K  ||  N ) )  ->  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  K  <_  ( M  gcd  N ) ) )
109com12 27 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( K  e.  ZZ  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  <_  ( M  gcd  N ) ) )
1110exp4b 590 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  -> 
( K  e.  ZZ  ->  ( ( K  ||  M  /\  K  ||  N
)  ->  K  <_  ( M  gcd  N ) ) ) ) )
1211com23 72 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  ->  ( -.  ( M  =  0  /\  N  =  0 )  -> 
( ( K  ||  M  /\  K  ||  N
)  ->  K  <_  ( M  gcd  N ) ) ) ) )
1312impcom 419 . . 3  |-  ( ( K  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( -.  ( M  =  0  /\  N  =  0
)  ->  ( ( K  ||  M  /\  K  ||  N )  ->  K  <_  ( M  gcd  N
) ) ) )
14133impb 1147 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( ( K  ||  M  /\  K  ||  N )  ->  K  <_  ( M  gcd  N
) ) ) )
1514imp 418 1  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  -> 
( ( K  ||  M  /\  K  ||  N
)  ->  K  <_  ( M  gcd  N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   {cpr 3641   class class class wbr 4023  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737    < clt 8867    <_ cle 8868   NNcn 9746   ZZcz 10024    || cdivides 12531    gcd cgcd 12685
This theorem is referenced by:  gcd0id  12702  gcdneg  12705  gcdaddmlem  12707  bezoutlem4  12720  gcdeq  12731  coprm  12779  rpdvds  12803  phimullem  12847  pockthlem  12952  2sqlem8a  20610  2sqlem8  20611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686
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