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Theorem dvdslegcd 11982
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  f  g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll1 1037 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  e.  ZZ )
21zred 9392 . . . 4  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  e.  RR )
3 simpll2 1038 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  M  e.  ZZ )
4 simpll3 1039 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  N  e.  ZZ )
5 simplr 528 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  -.  ( M  =  0  /\  N  =  0 ) )
6 lttri3 8054 . . . . . . 7  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
76adantl 277 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  (
f  e.  RR  /\  g  e.  RR )
)  ->  ( f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
8 zssre 9277 . . . . . . 7  |-  ZZ  C_  RR
9 gcdsupex 11975 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  E. x  e.  ZZ  ( A. y  e.  {
n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) }  -.  x  < 
y  /\  A. y  e.  RR  ( y  < 
x  ->  E. z  e.  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } y  < 
z ) ) )
10 ssrexv 3234 . . . . . . 7  |-  ( ZZ  C_  RR  ->  ( E. x  e.  ZZ  ( A. y  e.  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) }  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } y  <  z ) )  ->  E. x  e.  RR  ( A. y  e.  {
n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) }  -.  x  < 
y  /\  A. y  e.  RR  ( y  < 
x  ->  E. z  e.  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } y  < 
z ) ) ) )
118, 9, 10mpsyl 65 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  E. x  e.  RR  ( A. y  e.  {
n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) }  -.  x  < 
y  /\  A. y  e.  RR  ( y  < 
x  ->  E. z  e.  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } y  < 
z ) ) )
127, 11supclti 7014 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  e.  RR )
133, 4, 5, 12syl21anc 1247 . . . 4  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  e.  RR )
14 simpr 110 . . . . . 6  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  -> 
( K  ||  M  /\  K  ||  N ) )
15 breq1 4020 . . . . . . . . 9  |-  ( n  =  K  ->  (
n  ||  M  <->  K  ||  M
) )
16 breq1 4020 . . . . . . . . 9  |-  ( n  =  K  ->  (
n  ||  N  <->  K  ||  N
) )
1715, 16anbi12d 473 . . . . . . . 8  |-  ( n  =  K  ->  (
( n  ||  M  /\  n  ||  N )  <-> 
( K  ||  M  /\  K  ||  N ) ) )
1817elrab3 2908 . . . . . . 7  |-  ( K  e.  ZZ  ->  ( K  e.  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) }  <->  ( K  ||  M  /\  K  ||  N ) ) )
191, 18syl 14 . . . . . 6  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  -> 
( K  e.  {
n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) }  <->  ( K  ||  M  /\  K  ||  N
) ) )
2014, 19mpbird 167 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  e.  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } )
217, 11supubti 7015 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( K  e. 
{ n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) }  ->  -.  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  < 
K ) )
223, 4, 5, 21syl21anc 1247 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  -> 
( K  e.  {
n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) }  ->  -.  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  )  <  K
) )
2320, 22mpd 13 . . . 4  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  -.  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  < 
K )
242, 13, 23nltled 8095 . . 3  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  <_  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  ) )
25 gcdn0val 11979 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )
263, 4, 5, 25syl21anc 1247 . . 3  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  -> 
( M  gcd  N
)  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )
2724, 26breqtrrd 4045 . 2  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  <_  ( M  gcd  N ) )
2827ex 115 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 104    <-> wb 105    /\ w3a 979    = wceq 1363    e. wcel 2159   A.wral 2467   E.wrex 2468   {crab 2471    C_ wss 3143   class class class wbr 4017  (class class class)co 5890   supcsup 6998   RRcr 7827   0cc0 7828    < clt 8009    <_ cle 8010   ZZcz 9270    || cdvds 11811    gcd cgcd 11960
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-sup 7000  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-q 9637  df-rp 9671  df-fz 10026  df-fzo 10160  df-fl 10287  df-mod 10340  df-seqfrec 10463  df-exp 10537  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025  df-dvds 11812  df-gcd 11961
This theorem is referenced by:  nndvdslegcd  11983  gcd0id  11997  gcdneg  12000  gcdaddm  12002  gcdzeq  12040  rpdvds  12116  coprm  12161  phimullem  12242  pockthlem  12371  2sqlem8a  14852  2sqlem8  14853
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