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Theorem prmordvdslcmsOLDOLD 14999
Description: The primorial of a positive integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) Obsolete version of prmordvdslcmfOLD 14997 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
prmorlelcmsOLDOLD.f  |-  F  =  ( m  e.  NN  |->  if ( m  e.  Prime ,  m ,  1 ) )
prmorlelcmsOLDOLD.p  |-  P  =  ( n  e.  NN  |->  prod_ k  e.  ( 1 ... n ) ( F `  k ) )
prmorlelcmsOLDOLD.l  |-  L  =  ( x  e.  NN  |->  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... x
) m  ||  n } ,  RR ,  `'  <  ) )
Assertion
Ref Expression
prmordvdslcmsOLDOLD  |-  ( N  e.  NN  ->  ( P `  N )  ||  ( L `  N
) )
Distinct variable groups:    k, F, n, x    k, N, m, n, x
Allowed substitution hints:    P( x, k, m, n)    F( m)    L( x, k, m, n)

Proof of Theorem prmordvdslcmsOLDOLD
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 fz1ssnn 11824 . . . . 5  |-  ( 1 ... N )  C_  NN
21a1i 11 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  C_  NN )
3 fzfid 12179 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  e. 
Fin )
42, 3jca 534 . . 3  |-  ( N  e.  NN  ->  (
( 1 ... N
)  C_  NN  /\  (
1 ... N )  e. 
Fin ) )
5 ssrab2 3543 . . . 4  |-  { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n }  C_  NN
6 eqid 2420 . . . . . 6  |-  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  )  =  sup ( { n  e.  NN  |  A. m  e.  (
1 ... N ) m 
||  n } ,  RR ,  `'  <  )
76lcmscllemOLD 14560 . . . . 5  |-  ( ( ( 1 ... N
)  C_  NN  /\  (
1 ... N )  e. 
Fin )  ->  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  )  e.  { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n } )
84, 7syl 17 . . . 4  |-  ( N  e.  NN  ->  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  )  e.  { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n } )
95, 8sseldi 3459 . . 3  |-  ( N  e.  NN  ->  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  )  e.  NN )
10 simpr 462 . . . . 5  |-  ( ( N  e.  NN  /\  m  e.  NN )  ->  m  e.  NN )
11 1nn 10616 . . . . . 6  |-  1  e.  NN
1211a1i 11 . . . . 5  |-  ( ( N  e.  NN  /\  m  e.  NN )  ->  1  e.  NN )
1310, 12ifcld 3949 . . . 4  |-  ( ( N  e.  NN  /\  m  e.  NN )  ->  if ( m  e. 
Prime ,  m , 
1 )  e.  NN )
14 prmorlelcmsOLDOLD.f . . . 4  |-  F  =  ( m  e.  NN  |->  if ( m  e.  Prime ,  m ,  1 ) )
1513, 14fmptd 6053 . . 3  |-  ( N  e.  NN  ->  F : NN --> NN )
16 simpr 462 . . . . . . 7  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  ->  k  e.  ( 1 ... N ) )
1716adantr 466 . . . . . 6  |-  ( ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  /\  x  e.  ( ( 1 ... N
)  \  { k } ) )  -> 
k  e.  ( 1 ... N ) )
18 eldifi 3584 . . . . . . 7  |-  ( x  e.  ( ( 1 ... N )  \  { k } )  ->  x  e.  ( 1 ... N ) )
1918adantl 467 . . . . . 6  |-  ( ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  /\  x  e.  ( ( 1 ... N
)  \  { k } ) )  ->  x  e.  ( 1 ... N ) )
20 eldif 3443 . . . . . . . 8  |-  ( x  e.  ( ( 1 ... N )  \  { k } )  <-> 
( x  e.  ( 1 ... N )  /\  -.  x  e. 
{ k } ) )
21 elsn 4007 . . . . . . . . . . . 12  |-  ( x  e.  { k }  <-> 
x  =  k )
2221biimpri 209 . . . . . . . . . . 11  |-  ( x  =  k  ->  x  e.  { k } )
2322equcoms 1844 . . . . . . . . . 10  |-  ( k  =  x  ->  x  e.  { k } )
2423necon3bi 2651 . . . . . . . . 9  |-  ( -.  x  e.  { k }  ->  k  =/=  x )
2524adantl 467 . . . . . . . 8  |-  ( ( x  e.  ( 1 ... N )  /\  -.  x  e.  { k } )  ->  k  =/=  x )
2620, 25sylbi 198 . . . . . . 7  |-  ( x  e.  ( ( 1 ... N )  \  { k } )  ->  k  =/=  x
)
2726adantl 467 . . . . . 6  |-  ( ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  /\  x  e.  ( ( 1 ... N
)  \  { k } ) )  -> 
k  =/=  x )
2814fvprmselgcd1 14981 . . . . . 6  |-  ( ( k  e.  ( 1 ... N )  /\  x  e.  ( 1 ... N )  /\  k  =/=  x )  -> 
( ( F `  k )  gcd  ( F `  x )
)  =  1 )
2917, 19, 27, 28syl3anc 1264 . . . . 5  |-  ( ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  /\  x  e.  ( ( 1 ... N
)  \  { k } ) )  -> 
( ( F `  k )  gcd  ( F `  x )
)  =  1 )
3029ralrimiva 2837 . . . 4  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  ->  A. x  e.  ( ( 1 ... N
)  \  { k } ) ( ( F `  k )  gcd  ( F `  x ) )  =  1 )
3130ralrimiva 2837 . . 3  |-  ( N  e.  NN  ->  A. k  e.  ( 1 ... N
) A. x  e.  ( ( 1 ... N )  \  {
k } ) ( ( F `  k
)  gcd  ( F `  x ) )  =  1 )
3214fvprmselelfz 14980 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  ->  ( F `  k )  e.  ( 1 ... N ) )
334adantr 466 . . . . . 6  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  ->  ( ( 1 ... N )  C_  NN  /\  ( 1 ... N )  e.  Fin ) )
34 breq2 4421 . . . . . . . . . . 11  |-  ( n  =  l  ->  (
m  ||  n  <->  m  ||  l
) )
3534ralbidv 2862 . . . . . . . . . 10  |-  ( n  =  l  ->  ( A. m  e.  (
1 ... N ) m 
||  n  <->  A. m  e.  ( 1 ... N
) m  ||  l
) )
3635cbvrabv 3077 . . . . . . . . 9  |-  { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n }  =  { l  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  l }
37 breq1 4420 . . . . . . . . . . . 12  |-  ( m  =  x  ->  (
m  ||  l  <->  x  ||  l
) )
3837cbvralv 3053 . . . . . . . . . . 11  |-  ( A. m  e.  ( 1 ... N ) m 
||  l  <->  A. x  e.  ( 1 ... N
) x  ||  l
)
3938a1i 11 . . . . . . . . . 10  |-  ( l  e.  NN  ->  ( A. m  e.  (
1 ... N ) m 
||  l  <->  A. x  e.  ( 1 ... N
) x  ||  l
) )
4039rabbiia 3067 . . . . . . . . 9  |-  { l  e.  NN  |  A. m  e.  ( 1 ... N ) m 
||  l }  =  { l  e.  NN  |  A. x  e.  ( 1 ... N ) x  ||  l }
4136, 40eqtri 2449 . . . . . . . 8  |-  { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n }  =  { l  e.  NN  |  A. x  e.  ( 1 ... N
) x  ||  l }
4241supeq1i 7959 . . . . . . 7  |-  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  )  =  sup ( { l  e.  NN  |  A. x  e.  (
1 ... N ) x 
||  l } ,  RR ,  `'  <  )
4342lcmsOLD 14562 . . . . . 6  |-  ( ( ( 1 ... N
)  C_  NN  /\  (
1 ... N )  e. 
Fin )  ->  A. x  e.  ( 1 ... N
) x  ||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  ) )
4433, 43syl 17 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  ->  A. x  e.  ( 1 ... N ) x  ||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  ) )
45 breq1 4420 . . . . . 6  |-  ( x  =  ( F `  k )  ->  (
x  ||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  )  <-> 
( F `  k
)  ||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  ) ) )
4645rspcv 3175 . . . . 5  |-  ( ( F `  k )  e.  ( 1 ... N )  ->  ( A. x  e.  (
1 ... N ) x 
||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n } ,  RR ,  `'  <  )  ->  ( F `  k )  ||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n } ,  RR ,  `'  <  ) ) )
4732, 44, 46sylc 62 . . . 4  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  ->  ( F `  k )  ||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  ) )
4847ralrimiva 2837 . . 3  |-  ( N  e.  NN  ->  A. k  e.  ( 1 ... N
) ( F `  k )  ||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  ) )
49 coprmproddvds 14658 . . 3  |-  ( ( ( ( 1 ... N )  C_  NN  /\  ( 1 ... N
)  e.  Fin )  /\  ( sup ( { n  e.  NN  |  A. m  e.  (
1 ... N ) m 
||  n } ,  RR ,  `'  <  )  e.  NN  /\  F : NN --> NN )  /\  ( A. k  e.  ( 1 ... N ) A. x  e.  ( ( 1 ... N
)  \  { k } ) ( ( F `  k )  gcd  ( F `  x ) )  =  1  /\  A. k  e.  ( 1 ... N
) ( F `  k )  ||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  ) ) )  ->  prod_ k  e.  ( 1 ... N ) ( F `
 k )  ||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n } ,  RR ,  `'  <  ) )
504, 9, 15, 31, 48, 49syl122anc 1273 . 2  |-  ( N  e.  NN  ->  prod_ k  e.  ( 1 ... N ) ( F `
 k )  ||  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n } ,  RR ,  `'  <  ) )
51 oveq2 6305 . . . 4  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
5251prodeq1d 13953 . . 3  |-  ( n  =  N  ->  prod_ k  e.  ( 1 ... n ) ( F `
 k )  = 
prod_ k  e.  (
1 ... N ) ( F `  k ) )
53 prmorlelcmsOLDOLD.p . . 3  |-  P  =  ( n  e.  NN  |->  prod_ k  e.  ( 1 ... n ) ( F `  k ) )
54 prodex 13939 . . 3  |-  prod_ k  e.  ( 1 ... N
) ( F `  k )  e.  _V
5552, 53, 54fvmpt 5956 . 2  |-  ( N  e.  NN  ->  ( P `  N )  =  prod_ k  e.  ( 1 ... N ) ( F `  k
) )
56 oveq2 6305 . . . . . 6  |-  ( x  =  N  ->  (
1 ... x )  =  ( 1 ... N
) )
5756raleqdv 3029 . . . . 5  |-  ( x  =  N  ->  ( A. m  e.  (
1 ... x ) m 
||  n  <->  A. m  e.  ( 1 ... N
) m  ||  n
) )
5857rabbidv 3070 . . . 4  |-  ( x  =  N  ->  { n  e.  NN  |  A. m  e.  ( 1 ... x
) m  ||  n }  =  { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n } )
5958supeq1d 7958 . . 3  |-  ( x  =  N  ->  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... x ) m  ||  n } ,  RR ,  `'  <  )  =  sup ( { n  e.  NN  |  A. m  e.  (
1 ... N ) m 
||  n } ,  RR ,  `'  <  ) )
60 prmorlelcmsOLDOLD.l . . 3  |-  L  =  ( x  e.  NN  |->  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... x
) m  ||  n } ,  RR ,  `'  <  ) )
61 gtso 9711 . . . 4  |-  `'  <  Or  RR
6261supex 7975 . . 3  |-  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N ) m  ||  n } ,  RR ,  `'  <  )  e.  _V
6359, 60, 62fvmpt 5956 . 2  |-  ( N  e.  NN  ->  ( L `  N )  =  sup ( { n  e.  NN  |  A. m  e.  ( 1 ... N
) m  ||  n } ,  RR ,  `'  <  ) )
6450, 55, 633brtr4d 4448 1  |-  ( N  e.  NN  ->  ( P `  N )  ||  ( L `  N
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   {crab 2777    \ cdif 3430    C_ wss 3433   ifcif 3906   {csn 3993   class class class wbr 4417    |-> cmpt 4476   `'ccnv 4845   -->wf 5589   ` cfv 5593  (class class class)co 6297   Fincfn 7569   supcsup 7952   RRcr 9534   1c1 9536    < clt 9671   NNcn 10605   ...cfz 11778   prod_cprod 13937    || cdvds 14283    gcd cgcd 14446   Primecprime 14600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-inf2 8144  ax-cnex 9591  ax-resscn 9592  ax-1cn 9593  ax-icn 9594  ax-addcl 9595  ax-addrcl 9596  ax-mulcl 9597  ax-mulrcl 9598  ax-mulcom 9599  ax-addass 9600  ax-mulass 9601  ax-distr 9602  ax-i2m1 9603  ax-1ne0 9604  ax-1rid 9605  ax-rnegex 9606  ax-rrecex 9607  ax-cnre 9608  ax-pre-lttri 9609  ax-pre-lttrn 9610  ax-pre-ltadd 9611  ax-pre-mulgt0 9612  ax-pre-sup 9613
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-se 4806  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6259  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-om 6699  df-1st 6799  df-2nd 6800  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7954  df-inf 7955  df-oi 8023  df-card 8370  df-pnf 9673  df-mnf 9674  df-xr 9675  df-ltxr 9676  df-le 9677  df-sub 9858  df-neg 9859  df-div 10266  df-nn 10606  df-2 10664  df-3 10665  df-n0 10866  df-z 10934  df-uz 11156  df-rp 11299  df-fz 11779  df-fzo 11910  df-fl 12021  df-mod 12090  df-seq 12207  df-exp 12266  df-hash 12509  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-prod 13938  df-dvds 14284  df-gcd 14447  df-prm 14601
This theorem is referenced by:  prmorlelcmsOLDOLD  15000
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