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Theorem pcmptdvds 13265
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 1630 . . . . . . . . . 10  |-  F/ m  A  e.  NN0
3 nfcsb1v 3285 . . . . . . . . . . 11  |-  F/_ n [_ m  /  n ]_ A
43nfel1 2584 . . . . . . . . . 10  |-  F/ n [_ m  /  n ]_ A  e.  NN0
5 csbeq1a 3261 . . . . . . . . . . 11  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
65eleq1d 2504 . . . . . . . . . 10  |-  ( n  =  m  ->  ( A  e.  NN0  <->  [_ m  /  n ]_ A  e.  NN0 ) )
72, 4, 6cbvral 2930 . . . . . . . . 9  |-  ( A. n  e.  Prime  A  e. 
NN0 
<-> 
A. m  e.  Prime  [_ m  /  n ]_ A  e.  NN0 )
81, 7sylib 190 . . . . . . . 8  |-  ( ph  ->  A. m  e.  Prime  [_ m  /  n ]_ A  e.  NN0 )
9 csbeq1 3256 . . . . . . . . . 10  |-  ( m  =  p  ->  [_ m  /  n ]_ A  = 
[_ p  /  n ]_ A )
109eleq1d 2504 . . . . . . . . 9  |-  ( m  =  p  ->  ( [_ m  /  n ]_ A  e.  NN0  <->  [_ p  /  n ]_ A  e.  NN0 ) )
1110rspcv 3050 . . . . . . . 8  |-  ( p  e.  Prime  ->  ( A. m  e.  Prime  [_ m  /  n ]_ A  e. 
NN0  ->  [_ p  /  n ]_ A  e.  NN0 ) )
128, 11mpan9 457 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  [_ p  /  n ]_ A  e.  NN0 )
1312nn0ge0d 10279 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  0  <_  [_ p  /  n ]_ A )
14 0le0 10083 . . . . . 6  |-  0  <_  0
15 breq2 4218 . . . . . . 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 4218 . . . . . . 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 3772 . . . . . 6  |-  ( ( 0  <_  [_ p  /  n ]_ A  /\  0  <_  0 )  ->  0  <_  if ( ( p  <_  M  /\  -.  p  <_  N ) , 
[_ p  /  n ]_ A ,  0 ) )
1813, 14, 17sylancl 645 . . . . 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 2574 . . . . . . . 8  |-  F/_ m if ( n  e.  Prime ,  ( n ^ A
) ,  1 )
21 nfv 1630 . . . . . . . . 9  |-  F/ n  m  e.  Prime
22 nfcv 2574 . . . . . . . . . 10  |-  F/_ n m
23 nfcv 2574 . . . . . . . . . 10  |-  F/_ n ^
2422, 23, 3nfov 6106 . . . . . . . . 9  |-  F/_ n
( m ^ [_ m  /  n ]_ A
)
25 nfcv 2574 . . . . . . . . 9  |-  F/_ n
1
2621, 24, 25nfif 3765 . . . . . . . 8  |-  F/_ n if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A
) ,  1 )
27 eleq1 2498 . . . . . . . . 9  |-  ( n  =  m  ->  (
n  e.  Prime  <->  m  e.  Prime ) )
28 id 21 . . . . . . . . . 10  |-  ( n  =  m  ->  n  =  m )
2928, 5oveq12d 6101 . . . . . . . . 9  |-  ( n  =  m  ->  (
n ^ A )  =  ( m ^ [_ m  /  n ]_ A ) )
30 eqidd 2439 . . . . . . . . 9  |-  ( n  =  m  ->  1  =  1 )
3127, 29, 30ifbieq12d 3763 . . . . . . . 8  |-  ( n  =  m  ->  if ( n  e.  Prime ,  ( n ^ A
) ,  1 )  =  if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A ) ,  1 ) )
3220, 26, 31cbvmpt 4301 . . . . . . 7  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A ) ,  1 ) )  =  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A ) ,  1 ) )
3319, 32eqtri 2458 . . . . . 6  |-  F  =  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( m ^ [_ m  /  n ]_ A
) ,  1 ) )
348adantr 453 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  A. m  e.  Prime  [_ m  /  n ]_ A  e.  NN0 )
35 pcmpt.3 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
3635adantr 453 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  N  e.  NN )
37 simpr 449 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  p  e.  Prime )
38 pcmptdvds.3 . . . . . . 7  |-  ( ph  ->  M  e.  ( ZZ>= `  N ) )
3938adantr 453 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  M  e.  ( ZZ>= `  N )
)
4033, 34, 36, 37, 9, 39pcmpt2 13264 . . . . 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 ) )
4118, 40breqtrrd 4240 . . . 4  |-  ( (
ph  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  ( (  seq  1 (  x.  ,  F ) `  M
)  /  (  seq  1 (  x.  ,  F ) `  N
) ) ) )
4241ralrimiva 2791 . . 3  |-  ( ph  ->  A. p  e.  Prime  0  <_  ( p  pCnt  ( (  seq  1 (  x.  ,  F ) `
 M )  / 
(  seq  1 (  x.  ,  F ) `
 N ) ) ) )
4319, 1pcmptcl 13262 . . . . . . . 8  |-  ( ph  ->  ( F : NN --> NN  /\  seq  1 (  x.  ,  F ) : NN --> NN ) )
4443simprd 451 . . . . . . 7  |-  ( ph  ->  seq  1 (  x.  ,  F ) : NN --> NN )
45 nnuz 10523 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
4645uztrn2 10505 . . . . . . . 8  |-  ( ( N  e.  NN  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN )
4735, 38, 46syl2anc 644 . . . . . . 7  |-  ( ph  ->  M  e.  NN )
4844, 47ffvelrnd 5873 . . . . . 6  |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `
 M )  e.  NN )
4948nnzd 10376 . . . . 5  |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `
 M )  e.  ZZ )
5044, 35ffvelrnd 5873 . . . . 5  |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `
 N )  e.  NN )
51 znq 10580 . . . . 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 )
5249, 50, 51syl2anc 644 . . . 4  |-  ( ph  ->  ( (  seq  1
(  x.  ,  F
) `  M )  /  (  seq  1
(  x.  ,  F
) `  N )
)  e.  QQ )
53 pcz 13256 . . . 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
) ) ) ) )
5452, 53syl 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 ) ) ) ) )
5542, 54mpbird 225 . 2  |-  ( ph  ->  ( (  seq  1
(  x.  ,  F
) `  M )  /  (  seq  1
(  x.  ,  F
) `  N )
)  e.  ZZ )
5650nnzd 10376 . . 3  |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `
 N )  e.  ZZ )
5750nnne0d 10046 . . 3  |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `
 N )  =/=  0 )
58 dvdsval2 12857 . . 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 ) )
5956, 57, 49, 58syl3anc 1185 . 2  |-  ( ph  ->  ( (  seq  1
(  x.  ,  F
) `  N )  ||  (  seq  1
(  x.  ,  F
) `  M )  <->  ( (  seq  1 (  x.  ,  F ) `
 M )  / 
(  seq  1 (  x.  ,  F ) `
 N ) )  e.  ZZ ) )
6055, 59mpbird 225 1  |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `
 N )  ||  (  seq  1 (  x.  ,  F ) `  M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   [_csb 3253   ifcif 3741   class class class wbr 4214    e. cmpt 4268   -->wf 5452   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993    x. cmul 8997    <_ cle 9123    / cdiv 9679   NNcn 10002   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   QQcq 10576    seq cseq 11325   ^cexp 11384    || cdivides 12854   Primecprime 13081    pCnt cpc 13212
This theorem is referenced by:  bposlem6  21075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-fz 11046  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-gcd 13009  df-prm 13082  df-pc 13213
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