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Theorem dvdslegcd 11234
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 982 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  e.  ZZ )
21zred 8868 . . . 4  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  e.  RR )
3 simpll2 983 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  M  e.  ZZ )
4 simpll3 984 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  N  e.  ZZ )
5 simplr 497 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  -.  ( M  =  0  /\  N  =  0 ) )
6 lttri3 7565 . . . . . . 7  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
76adantl 271 . . . . . 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 8757 . . . . . . 7  |-  ZZ  C_  RR
9 gcdsupex 11227 . . . . . . 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 3086 . . . . . . 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 64 . . . . . 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 6693 . . . . 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 1173 . . . 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 108 . . . . . 6  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  -> 
( K  ||  M  /\  K  ||  N ) )
15 breq1 3848 . . . . . . . . 9  |-  ( n  =  K  ->  (
n  ||  M  <->  K  ||  M
) )
16 breq1 3848 . . . . . . . . 9  |-  ( n  =  K  ->  (
n  ||  N  <->  K  ||  N
) )
1715, 16anbi12d 457 . . . . . . . 8  |-  ( n  =  K  ->  (
( n  ||  M  /\  n  ||  N )  <-> 
( K  ||  M  /\  K  ||  N ) ) )
1817elrab3 2772 . . . . . . 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 165 . . . . 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 6694 . . . . . 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 1173 . . . . 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 7604 . . 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 11231 . . . 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 1173 . . 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 3871 . 2  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  <_  ( M  gcd  N ) )
2827ex 113 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363    C_ wss 2999   class class class wbr 3845  (class class class)co 5652   supcsup 6677   RRcr 7349   0cc0 7350    < clt 7522    <_ cle 7523   ZZcz 8750    || cdvds 11074    gcd cgcd 11216
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-sup 6679  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-fl 9677  df-mod 9730  df-iseq 9853  df-seq3 9854  df-exp 9955  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-dvds 11075  df-gcd 11217
This theorem is referenced by:  nndvdslegcd  11235  gcd0id  11248  gcdneg  11251  gcdaddm  11253  gcdzeq  11289  rpdvds  11359  coprm  11401  phimullem  11479
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