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Theorem pcmptdvds 14622
Description: The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)
Hypotheses
Ref Expression
pcmpt.1  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A
) ,  1 ) )
pcmpt.2  |-  ( ph  ->  A. n  e.  Prime  A  e.  NN0 )
pcmpt.3  |-  ( ph  ->  N  e.  NN )
pcmptdvds.3  |-  ( ph  ->  M  e.  ( ZZ>= `  N ) )
Assertion
Ref Expression
pcmptdvds  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 N )  ||  (  seq 1 (  x.  ,  F ) `  M ) )

Proof of Theorem pcmptdvds
Dummy variables  m  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcmpt.2 . . . . . . . . 9  |-  ( ph  ->  A. n  e.  Prime  A  e.  NN0 )
2 nfv 1728 . . . . . . . . . 10  |-  F/ m  A  e.  NN0
3 nfcsb1v 3389 . . . . . . . . . . 11  |-  F/_ n [_ m  /  n ]_ A
43nfel1 2580 . . . . . . . . . 10  |-  F/ n [_ m  /  n ]_ A  e.  NN0
5 csbeq1a 3382 . . . . . . . . . . 11  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
65eleq1d 2471 . . . . . . . . . 10  |-  ( n  =  m  ->  ( A  e.  NN0  <->  [_ m  /  n ]_ A  e.  NN0 ) )
72, 4, 6cbvral 3030 . . . . . . . . 9  |-  ( A. n  e.  Prime  A  e. 
NN0 
<-> 
A. m  e.  Prime  [_ m  /  n ]_ A  e.  NN0 )
81, 7sylib 196 . . . . . . . 8  |-  ( ph  ->  A. m  e.  Prime  [_ m  /  n ]_ A  e.  NN0 )
9 csbeq1 3376 . . . . . . . . . 10  |-  ( m  =  p  ->  [_ m  /  n ]_ A  = 
[_ p  /  n ]_ A )
109eleq1d 2471 . . . . . . . . 9  |-  ( m  =  p  ->  ( [_ m  /  n ]_ A  e.  NN0  <->  [_ p  /  n ]_ A  e.  NN0 ) )
1110rspcv 3156 . . . . . . . 8  |-  ( p  e.  Prime  ->  ( A. m  e.  Prime  [_ m  /  n ]_ A  e. 
NN0  ->  [_ p  /  n ]_ A  e.  NN0 ) )
128, 11mpan9 467 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  [_ p  /  n ]_ A  e.  NN0 )
1312nn0ge0d 10896 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  0  <_  [_ p  /  n ]_ A )
14 0le0 10666 . . . . . 6  |-  0  <_  0
15 breq2 4399 . . . . . . 7  |-  ( [_ p  /  n ]_ A  =  if ( ( p  <_  M  /\  -.  p  <_  N ) , 
[_ p  /  n ]_ A ,  0 )  ->  ( 0  <_  [_ p  /  n ]_ A  <->  0  <_  if ( ( p  <_  M  /\  -.  p  <_  N ) ,  [_ p  /  n ]_ A ,  0 ) ) )
16 breq2 4399 . . . . . . 7  |-  ( 0  =  if ( ( p  <_  M  /\  -.  p  <_  N ) ,  [_ p  /  n ]_ A ,  0 )  ->  ( 0  <_  0  <->  0  <_  if ( ( p  <_  M  /\  -.  p  <_  N ) ,  [_ p  /  n ]_ A ,  0 ) ) )
1715, 16ifboth 3921 . . . . . 6  |-  ( ( 0  <_  [_ p  /  n ]_ A  /\  0  <_  0 )  ->  0  <_  if ( ( p  <_  M  /\  -.  p  <_  N ) , 
[_ p  /  n ]_ A ,  0 ) )
1813, 14, 17sylancl 660 . . . . 5  |-  ( (
ph  /\  p  e.  Prime )  ->  0  <_  if ( ( p  <_  M  /\  -.  p  <_  N ) ,  [_ p  /  n ]_ A ,  0 ) )
19 pcmpt.1 . . . . . . 7  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A
) ,  1 ) )
20 nfcv 2564 . . . . . . . 8  |-  F/_ m if ( n  e.  Prime ,  ( n ^ A
) ,  1 )
21 nfv 1728 . . . . . . . . 9  |-  F/ n  m  e.  Prime
22 nfcv 2564 . . . . . . . . . 10  |-  F/_ n m
23 nfcv 2564 . . . . . . . . . 10  |-  F/_ n ^
2422, 23, 3nfov 6304 . . . . . . . . 9  |-  F/_ n
( m ^ [_ m  /  n ]_ A
)
25 nfcv 2564 . . . . . . . . 9  |-  F/_ n
1
2621, 24, 25nfif 3914 . . . . . . . 8  |-  F/_ n if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A
) ,  1 )
27 eleq1 2474 . . . . . . . . 9  |-  ( n  =  m  ->  (
n  e.  Prime  <->  m  e.  Prime ) )
28 id 22 . . . . . . . . . 10  |-  ( n  =  m  ->  n  =  m )
2928, 5oveq12d 6296 . . . . . . . . 9  |-  ( n  =  m  ->  (
n ^ A )  =  ( m ^ [_ m  /  n ]_ A ) )
3027, 29ifbieq1d 3908 . . . . . . . 8  |-  ( n  =  m  ->  if ( n  e.  Prime ,  ( n ^ A
) ,  1 )  =  if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A ) ,  1 ) )
3120, 26, 30cbvmpt 4486 . . . . . . 7  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A ) ,  1 ) )  =  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A ) ,  1 ) )
3219, 31eqtri 2431 . . . . . 6  |-  F  =  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A
) ,  1 ) )
338adantr 463 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  A. m  e.  Prime  [_ m  /  n ]_ A  e.  NN0 )
34 pcmpt.3 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
3534adantr 463 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  N  e.  NN )
36 simpr 459 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  p  e.  Prime )
37 pcmptdvds.3 . . . . . . 7  |-  ( ph  ->  M  e.  ( ZZ>= `  N ) )
3837adantr 463 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  M  e.  ( ZZ>= `  N )
)
3932, 33, 35, 36, 9, 38pcmpt2 14621 . . . . 5  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  pCnt  ( (  seq 1
(  x.  ,  F
) `  M )  /  (  seq 1
(  x.  ,  F
) `  N )
) )  =  if ( ( p  <_  M  /\  -.  p  <_  N ) ,  [_ p  /  n ]_ A ,  0 ) )
4018, 39breqtrrd 4421 . . . 4  |-  ( (
ph  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  ( (  seq 1 (  x.  ,  F ) `  M
)  /  (  seq 1 (  x.  ,  F ) `  N
) ) ) )
4140ralrimiva 2818 . . 3  |-  ( ph  ->  A. p  e.  Prime  0  <_  ( p  pCnt  ( (  seq 1 (  x.  ,  F ) `
 M )  / 
(  seq 1 (  x.  ,  F ) `  N ) ) ) )
4219, 1pcmptcl 14619 . . . . . . . 8  |-  ( ph  ->  ( F : NN --> NN  /\  seq 1 (  x.  ,  F ) : NN --> NN ) )
4342simprd 461 . . . . . . 7  |-  ( ph  ->  seq 1 (  x.  ,  F ) : NN --> NN )
44 eluznn 11197 . . . . . . . 8  |-  ( ( N  e.  NN  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN )
4534, 37, 44syl2anc 659 . . . . . . 7  |-  ( ph  ->  M  e.  NN )
4643, 45ffvelrnd 6010 . . . . . 6  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 M )  e.  NN )
4746nnzd 11007 . . . . 5  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 M )  e.  ZZ )
4843, 34ffvelrnd 6010 . . . . 5  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 N )  e.  NN )
49 znq 11231 . . . . 5  |-  ( ( (  seq 1 (  x.  ,  F ) `
 M )  e.  ZZ  /\  (  seq 1 (  x.  ,  F ) `  N
)  e.  NN )  ->  ( (  seq 1 (  x.  ,  F ) `  M
)  /  (  seq 1 (  x.  ,  F ) `  N
) )  e.  QQ )
5047, 48, 49syl2anc 659 . . . 4  |-  ( ph  ->  ( (  seq 1
(  x.  ,  F
) `  M )  /  (  seq 1
(  x.  ,  F
) `  N )
)  e.  QQ )
51 pcz 14613 . . . 4  |-  ( ( (  seq 1 (  x.  ,  F ) `
 M )  / 
(  seq 1 (  x.  ,  F ) `  N ) )  e.  QQ  ->  ( (
(  seq 1 (  x.  ,  F ) `  M )  /  (  seq 1 (  x.  ,  F ) `  N
) )  e.  ZZ  <->  A. p  e.  Prime  0  <_  ( p  pCnt  (
(  seq 1 (  x.  ,  F ) `  M )  /  (  seq 1 (  x.  ,  F ) `  N
) ) ) ) )
5250, 51syl 17 . . 3  |-  ( ph  ->  ( ( (  seq 1 (  x.  ,  F ) `  M
)  /  (  seq 1 (  x.  ,  F ) `  N
) )  e.  ZZ  <->  A. p  e.  Prime  0  <_  ( p  pCnt  (
(  seq 1 (  x.  ,  F ) `  M )  /  (  seq 1 (  x.  ,  F ) `  N
) ) ) ) )
5341, 52mpbird 232 . 2  |-  ( ph  ->  ( (  seq 1
(  x.  ,  F
) `  M )  /  (  seq 1
(  x.  ,  F
) `  N )
)  e.  ZZ )
5448nnzd 11007 . . 3  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 N )  e.  ZZ )
5548nnne0d 10621 . . 3  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 N )  =/=  0 )
56 dvdsval2 14198 . . 3  |-  ( ( (  seq 1 (  x.  ,  F ) `
 N )  e.  ZZ  /\  (  seq 1 (  x.  ,  F ) `  N
)  =/=  0  /\  (  seq 1 (  x.  ,  F ) `
 M )  e.  ZZ )  ->  (
(  seq 1 (  x.  ,  F ) `  N )  ||  (  seq 1 (  x.  ,  F ) `  M
)  <->  ( (  seq 1 (  x.  ,  F ) `  M
)  /  (  seq 1 (  x.  ,  F ) `  N
) )  e.  ZZ ) )
5754, 55, 47, 56syl3anc 1230 . 2  |-  ( ph  ->  ( (  seq 1
(  x.  ,  F
) `  N )  ||  (  seq 1
(  x.  ,  F
) `  M )  <->  ( (  seq 1 (  x.  ,  F ) `
 M )  / 
(  seq 1 (  x.  ,  F ) `  N ) )  e.  ZZ ) )
5853, 57mpbird 232 1  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 N )  ||  (  seq 1 (  x.  ,  F ) `  M ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   [_csb 3373   ifcif 3885   class class class wbr 4395    |-> cmpt 4453   -->wf 5565   ` cfv 5569  (class class class)co 6278   0cc0 9522   1c1 9523    x. cmul 9527    <_ cle 9659    / cdiv 10247   NNcn 10576   NN0cn0 10836   ZZcz 10905   ZZ>=cuz 11127   QQcq 11227    seqcseq 12151   ^cexp 12210    || cdvds 14195   Primecprime 14426    pCnt cpc 14569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-fz 11727  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-dvds 14196  df-gcd 14354  df-prm 14427  df-pc 14570
This theorem is referenced by:  bposlem6  23945
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