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Theorem dvdslegcd 11968
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 1036 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  e.  ZZ )
21zred 9378 . . . 4  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  e.  RR )
3 simpll2 1037 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0
) )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  M  e.  ZZ )
4 simpll3 1038 . . . . 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 8040 . . . . . . 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 9263 . . . . . . 7  |-  ZZ  C_  RR
9 gcdsupex 11961 . . . . . . 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 3222 . . . . . . 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 7000 . . . . 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 1237 . . . 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 4008 . . . . . . . . 9  |-  ( n  =  K  ->  (
n  ||  M  <->  K  ||  M
) )
16 breq1 4008 . . . . . . . . 9  |-  ( n  =  K  ->  (
n  ||  N  <->  K  ||  N
) )
1715, 16anbi12d 473 . . . . . . . 8  |-  ( n  =  K  ->  (
( n  ||  M  /\  n  ||  N )  <-> 
( K  ||  M  /\  K  ||  N ) ) )
1817elrab3 2896 . . . . . . 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 7001 . . . . . 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 1237 . . . . 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 8081 . . 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 11965 . . . 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 1237 . . 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 4033 . 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 978    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459    C_ wss 3131   class class class wbr 4005  (class class class)co 5878   supcsup 6984   RRcr 7813   0cc0 7814    < clt 7995    <_ cle 7996   ZZcz 9256    || cdvds 11797    gcd cgcd 11946
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-sup 6986  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-fz 10012  df-fzo 10146  df-fl 10273  df-mod 10326  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-dvds 11798  df-gcd 11947
This theorem is referenced by:  nndvdslegcd  11969  gcd0id  11983  gcdneg  11986  gcdaddm  11988  gcdzeq  12026  rpdvds  12102  coprm  12147  phimullem  12228  pockthlem  12357  2sqlem8a  14657  2sqlem8  14658
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