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Theorem pcmptdvds 14261
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 1678 . . . . . . . . . 10  |-  F/ m  A  e.  NN0
3 nfcsb1v 3444 . . . . . . . . . . 11  |-  F/_ n [_ m  /  n ]_ A
43nfel1 2638 . . . . . . . . . 10  |-  F/ n [_ m  /  n ]_ A  e.  NN0
5 csbeq1a 3437 . . . . . . . . . . 11  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
65eleq1d 2529 . . . . . . . . . 10  |-  ( n  =  m  ->  ( A  e.  NN0  <->  [_ m  /  n ]_ A  e.  NN0 ) )
72, 4, 6cbvral 3077 . . . . . . . . 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 3431 . . . . . . . . . 10  |-  ( m  =  p  ->  [_ m  /  n ]_ A  = 
[_ p  /  n ]_ A )
109eleq1d 2529 . . . . . . . . 9  |-  ( m  =  p  ->  ( [_ m  /  n ]_ A  e.  NN0  <->  [_ p  /  n ]_ A  e.  NN0 ) )
1110rspcv 3203 . . . . . . . 8  |-  ( p  e.  Prime  ->  ( A. m  e.  Prime  [_ m  /  n ]_ A  e. 
NN0  ->  [_ p  /  n ]_ A  e.  NN0 ) )
128, 11mpan9 469 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  [_ p  /  n ]_ A  e.  NN0 )
1312nn0ge0d 10844 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  0  <_  [_ p  /  n ]_ A )
14 0le0 10614 . . . . . 6  |-  0  <_  0
15 breq2 4444 . . . . . . 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 4444 . . . . . . 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 3968 . . . . . 6  |-  ( ( 0  <_  [_ p  /  n ]_ A  /\  0  <_  0 )  ->  0  <_  if ( ( p  <_  M  /\  -.  p  <_  N ) , 
[_ p  /  n ]_ A ,  0 ) )
1813, 14, 17sylancl 662 . . . . 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 2622 . . . . . . . 8  |-  F/_ m if ( n  e.  Prime ,  ( n ^ A
) ,  1 )
21 nfv 1678 . . . . . . . . 9  |-  F/ n  m  e.  Prime
22 nfcv 2622 . . . . . . . . . 10  |-  F/_ n m
23 nfcv 2622 . . . . . . . . . 10  |-  F/_ n ^
2422, 23, 3nfov 6298 . . . . . . . . 9  |-  F/_ n
( m ^ [_ m  /  n ]_ A
)
25 nfcv 2622 . . . . . . . . 9  |-  F/_ n
1
2621, 24, 25nfif 3961 . . . . . . . 8  |-  F/_ n if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A
) ,  1 )
27 eleq1 2532 . . . . . . . . 9  |-  ( n  =  m  ->  (
n  e.  Prime  <->  m  e.  Prime ) )
28 id 22 . . . . . . . . . 10  |-  ( n  =  m  ->  n  =  m )
2928, 5oveq12d 6293 . . . . . . . . 9  |-  ( n  =  m  ->  (
n ^ A )  =  ( m ^ [_ m  /  n ]_ A ) )
3027, 29ifbieq1d 3955 . . . . . . . 8  |-  ( n  =  m  ->  if ( n  e.  Prime ,  ( n ^ A
) ,  1 )  =  if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A ) ,  1 ) )
3120, 26, 30cbvmpt 4530 . . . . . . 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 2489 . . . . . 6  |-  F  =  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A
) ,  1 ) )
338adantr 465 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  A. m  e.  Prime  [_ m  /  n ]_ A  e.  NN0 )
34 pcmpt.3 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
3534adantr 465 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  N  e.  NN )
36 simpr 461 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  p  e.  Prime )
37 pcmptdvds.3 . . . . . . 7  |-  ( ph  ->  M  e.  ( ZZ>= `  N ) )
3837adantr 465 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  M  e.  ( ZZ>= `  N )
)
3932, 33, 35, 36, 9, 38pcmpt2 14260 . . . . 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 4466 . . . 4  |-  ( (
ph  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  ( (  seq 1 (  x.  ,  F ) `  M
)  /  (  seq 1 (  x.  ,  F ) `  N
) ) ) )
4140ralrimiva 2871 . . 3  |-  ( ph  ->  A. p  e.  Prime  0  <_  ( p  pCnt  ( (  seq 1 (  x.  ,  F ) `
 M )  / 
(  seq 1 (  x.  ,  F ) `  N ) ) ) )
4219, 1pcmptcl 14258 . . . . . . . 8  |-  ( ph  ->  ( F : NN --> NN  /\  seq 1 (  x.  ,  F ) : NN --> NN ) )
4342simprd 463 . . . . . . 7  |-  ( ph  ->  seq 1 (  x.  ,  F ) : NN --> NN )
44 eluznn 11141 . . . . . . . 8  |-  ( ( N  e.  NN  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN )
4534, 37, 44syl2anc 661 . . . . . . 7  |-  ( ph  ->  M  e.  NN )
4643, 45ffvelrnd 6013 . . . . . 6  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 M )  e.  NN )
4746nnzd 10954 . . . . 5  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 M )  e.  ZZ )
4843, 34ffvelrnd 6013 . . . . 5  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 N )  e.  NN )
49 znq 11175 . . . . 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 661 . . . 4  |-  ( ph  ->  ( (  seq 1
(  x.  ,  F
) `  M )  /  (  seq 1
(  x.  ,  F
) `  N )
)  e.  QQ )
51 pcz 14252 . . . 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 16 . . 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 10954 . . 3  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 N )  e.  ZZ )
5548nnne0d 10569 . . 3  |-  ( ph  ->  (  seq 1 (  x.  ,  F ) `
 N )  =/=  0 )
56 dvdsval2 13839 . . 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 1223 . 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 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   [_csb 3428   ifcif 3932   class class class wbr 4440    |-> cmpt 4498   -->wf 5575   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    x. cmul 9486    <_ cle 9618    / cdiv 10195   NNcn 10525   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   QQcq 11171    seqcseq 12063   ^cexp 12122    || cdivides 13836   Primecprime 14065    pCnt cpc 14208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-fz 11662  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-dvds 13837  df-gcd 13993  df-prm 14066  df-pc 14209
This theorem is referenced by:  bposlem6  23285
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