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Theorem lcmledvds 10659
Description: A positive integer which both operands of the lcm operator divide bounds it. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
Assertion
Ref Expression
lcmledvds  |-  ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  -> 
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  <_  K )
)

Proof of Theorem lcmledvds
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 lcmn0val 10655 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  = inf ( { n  e.  NN  | 
( M  ||  n  /\  N  ||  n ) } ,  RR ,  <  ) )
213adantl1 1095 . . . 4  |-  ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  -> 
( M lcm  N )  = inf ( { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } ,  RR ,  <  ) )
32adantr 270 . . 3  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  -> 
( M lcm  N )  = inf ( { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } ,  RR ,  <  ) )
4 1zzd 8511 . . . 4  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  -> 
1  e.  ZZ )
5 nnuz 8787 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
6 rabeq 2602 . . . . 5  |-  ( NN  =  ( ZZ>= `  1
)  ->  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  =  { n  e.  ( ZZ>=
`  1 )  |  ( M  ||  n  /\  N  ||  n ) } )
75, 6ax-mp 7 . . . 4  |-  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  =  { n  e.  ( ZZ>=
`  1 )  |  ( M  ||  n  /\  N  ||  n ) }
8 simpll1 978 . . . . 5  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  ->  K  e.  NN )
9 simpr 108 . . . . 5  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  -> 
( M  ||  K  /\  N  ||  K ) )
10 breq2 3809 . . . . . . 7  |-  ( n  =  K  ->  ( M  ||  n  <->  M  ||  K
) )
11 breq2 3809 . . . . . . 7  |-  ( n  =  K  ->  ( N  ||  n  <->  N  ||  K
) )
1210, 11anbi12d 457 . . . . . 6  |-  ( n  =  K  ->  (
( M  ||  n  /\  N  ||  n )  <-> 
( M  ||  K  /\  N  ||  K ) ) )
1312elrab 2757 . . . . 5  |-  ( K  e.  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  <->  ( K  e.  NN  /\  ( M 
||  K  /\  N  ||  K ) ) )
148, 9, 13sylanbrc 408 . . . 4  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  ->  K  e.  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } )
15 simpll2 979 . . . . . . 7  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  ->  M  e.  ZZ )
16 elfzelz 9173 . . . . . . . 8  |-  ( n  e.  ( 1 ... K )  ->  n  e.  ZZ )
1716adantl 271 . . . . . . 7  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  ->  n  e.  ZZ )
18 zdvdsdc 10424 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ )  -> DECID  M 
||  n )
1915, 17, 18syl2anc 403 . . . . . 6  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  -> DECID  M  ||  n )
20 simpll3 980 . . . . . . 7  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  ->  N  e.  ZZ )
21 zdvdsdc 10424 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  n  e.  ZZ )  -> DECID  N 
||  n )
2220, 17, 21syl2anc 403 . . . . . 6  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  -> DECID  N  ||  n )
23 dcan 876 . . . . . 6  |-  (DECID  M  ||  n  ->  (DECID  N  ||  n  -> DECID  ( M  ||  n  /\  N  ||  n ) ) )
2419, 22, 23sylc 61 . . . . 5  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  -> DECID  ( M  ||  n  /\  N  ||  n ) )
2524adantlr 461 . . . 4  |-  ( ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  /\  n  e.  ( 1 ... K ) )  -> DECID 
( M  ||  n  /\  N  ||  n ) )
264, 7, 14, 25infssuzledc 10553 . . 3  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  -> inf ( { n  e.  NN  |  ( M  ||  n  /\  N  ||  n
) } ,  RR ,  <  )  <_  K
)
273, 26eqbrtrd 3825 . 2  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  -> 
( M lcm  N )  <_  K )
2827ex 113 1  |-  ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  -> 
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  <_  K )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662  DECID wdc 776    /\ w3a 920    = wceq 1285    e. wcel 1434   {crab 2357   class class class wbr 3805   ` cfv 4952  (class class class)co 5563  infcinf 6490   RRcr 7094   0cc0 7095   1c1 7096    < clt 7267    <_ cle 7268   NNcn 8158   ZZcz 8484   ZZ>=cuz 8752   ...cfz 9157    || cdvds 10403   lcm clcm 10649
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-isom 4961  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-inf 6492  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-lcm 10650
This theorem is referenced by:  lcmneg  10663
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