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Theorem lcmledvds 11678
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 11674 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  = inf ( { n  e.  NN  | 
( M  ||  n  /\  N  ||  n ) } ,  RR ,  <  ) )
213adantl1 1122 . . . 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 274 . . 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 9049 . . . 4  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  -> 
1  e.  ZZ )
5 nnuz 9329 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
6 rabeq 2652 . . . . 5  |-  ( NN  =  ( ZZ>= `  1
)  ->  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  =  { n  e.  ( ZZ>=
`  1 )  |  ( M  ||  n  /\  N  ||  n ) } )
75, 6ax-mp 5 . . . 4  |-  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  =  { n  e.  ( ZZ>=
`  1 )  |  ( M  ||  n  /\  N  ||  n ) }
8 simpll1 1005 . . . . 5  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  ->  K  e.  NN )
9 simpr 109 . . . . 5  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  -> 
( M  ||  K  /\  N  ||  K ) )
10 breq2 3903 . . . . . . 7  |-  ( n  =  K  ->  ( M  ||  n  <->  M  ||  K
) )
11 breq2 3903 . . . . . . 7  |-  ( n  =  K  ->  ( N  ||  n  <->  N  ||  K
) )
1210, 11anbi12d 464 . . . . . 6  |-  ( n  =  K  ->  (
( M  ||  n  /\  N  ||  n )  <-> 
( M  ||  K  /\  N  ||  K ) ) )
1312elrab 2813 . . . . 5  |-  ( K  e.  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  <->  ( K  e.  NN  /\  ( M 
||  K  /\  N  ||  K ) ) )
148, 9, 13sylanbrc 413 . . . 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 1006 . . . . . . 7  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  ->  M  e.  ZZ )
16 elfzelz 9774 . . . . . . . 8  |-  ( n  e.  ( 1 ... K )  ->  n  e.  ZZ )
1716adantl 275 . . . . . . 7  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  ->  n  e.  ZZ )
18 zdvdsdc 11441 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ )  -> DECID  M 
||  n )
1915, 17, 18syl2anc 408 . . . . . 6  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  -> DECID  M  ||  n )
20 simpll3 1007 . . . . . . 7  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  ->  N  e.  ZZ )
21 zdvdsdc 11441 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  n  e.  ZZ )  -> DECID  N 
||  n )
2220, 17, 21syl2anc 408 . . . . . 6  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  -> DECID  N  ||  n )
23 dcan 903 . . . . . 6  |-  (DECID  M  ||  n  ->  (DECID  N  ||  n  -> DECID  ( M  ||  n  /\  N  ||  n ) ) )
2419, 22, 23sylc 62 . . . . 5  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... K
) )  -> DECID  ( M  ||  n  /\  N  ||  n ) )
2524adantlr 468 . . . 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 11570 . . 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 3920 . 2  |-  ( ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  ( M  ||  K  /\  N  ||  K ) )  -> 
( M lcm  N )  <_  K )
2827ex 114 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 103    \/ wo 682  DECID wdc 804    /\ w3a 947    = wceq 1316    e. wcel 1465   {crab 2397   class class class wbr 3899   ` cfv 5093  (class class class)co 5742  infcinf 6838   RRcr 7587   0cc0 7588   1c1 7589    < clt 7768    <_ cle 7769   NNcn 8688   ZZcz 9022   ZZ>=cuz 9294   ...cfz 9758    || cdvds 11420   lcm clcm 11668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-sup 6839  df-inf 6840  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421  df-lcm 11669
This theorem is referenced by:  lcmneg  11682
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