MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pcbcctr Structured version   Unicode version

Theorem pcbcctr 23752
Description: Prime count of a central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.)
Assertion
Ref Expression
pcbcctr  |-  ( ( N  e.  NN  /\  P  e.  Prime )  -> 
( P  pCnt  (
( 2  x.  N
)  _C  N ) )  =  sum_ k  e.  ( 1 ... (
2  x.  N ) ) ( ( |_
`  ( ( 2  x.  N )  / 
( P ^ k
) ) )  -  ( 2  x.  ( |_ `  ( N  / 
( P ^ k
) ) ) ) ) )
Distinct variable groups:    k, N    P, k

Proof of Theorem pcbcctr
StepHypRef Expression
1 2nn 10689 . . . . 5  |-  2  e.  NN
2 nnmulcl 10554 . . . . 5  |-  ( ( 2  e.  NN  /\  N  e.  NN )  ->  ( 2  x.  N
)  e.  NN )
31, 2mpan 668 . . . 4  |-  ( N  e.  NN  ->  (
2  x.  N )  e.  NN )
43adantr 463 . . 3  |-  ( ( N  e.  NN  /\  P  e.  Prime )  -> 
( 2  x.  N
)  e.  NN )
5 nnnn0 10798 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
6 fzctr 11791 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  ( 0 ... (
2  x.  N ) ) )
75, 6syl 16 . . . 4  |-  ( N  e.  NN  ->  N  e.  ( 0 ... (
2  x.  N ) ) )
87adantr 463 . . 3  |-  ( ( N  e.  NN  /\  P  e.  Prime )  ->  N  e.  ( 0 ... ( 2  x.  N ) ) )
9 simpr 459 . . 3  |-  ( ( N  e.  NN  /\  P  e.  Prime )  ->  P  e.  Prime )
10 pcbc 14506 . . 3  |-  ( ( ( 2  x.  N
)  e.  NN  /\  N  e.  ( 0 ... ( 2  x.  N ) )  /\  P  e.  Prime )  -> 
( P  pCnt  (
( 2  x.  N
)  _C  N ) )  =  sum_ k  e.  ( 1 ... (
2  x.  N ) ) ( ( |_
`  ( ( 2  x.  N )  / 
( P ^ k
) ) )  -  ( ( |_ `  ( ( ( 2  x.  N )  -  N )  /  ( P ^ k ) ) )  +  ( |_
`  ( N  / 
( P ^ k
) ) ) ) ) )
114, 8, 9, 10syl3anc 1226 . 2  |-  ( ( N  e.  NN  /\  P  e.  Prime )  -> 
( P  pCnt  (
( 2  x.  N
)  _C  N ) )  =  sum_ k  e.  ( 1 ... (
2  x.  N ) ) ( ( |_
`  ( ( 2  x.  N )  / 
( P ^ k
) ) )  -  ( ( |_ `  ( ( ( 2  x.  N )  -  N )  /  ( P ^ k ) ) )  +  ( |_
`  ( N  / 
( P ^ k
) ) ) ) ) )
12 nncn 10539 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
13122timesd 10777 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
2  x.  N )  =  ( N  +  N ) )
1413oveq1d 6285 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( 2  x.  N
)  -  N )  =  ( ( N  +  N )  -  N ) )
1512, 12pncand 9923 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  +  N
)  -  N )  =  N )
1614, 15eqtrd 2495 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( 2  x.  N
)  -  N )  =  N )
1716oveq1d 6285 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( 2  x.  N )  -  N
)  /  ( P ^ k ) )  =  ( N  / 
( P ^ k
) ) )
1817fveq2d 5852 . . . . . . 7  |-  ( N  e.  NN  ->  ( |_ `  ( ( ( 2  x.  N )  -  N )  / 
( P ^ k
) ) )  =  ( |_ `  ( N  /  ( P ^
k ) ) ) )
1918oveq1d 6285 . . . . . 6  |-  ( N  e.  NN  ->  (
( |_ `  (
( ( 2  x.  N )  -  N
)  /  ( P ^ k ) ) )  +  ( |_
`  ( N  / 
( P ^ k
) ) ) )  =  ( ( |_
`  ( N  / 
( P ^ k
) ) )  +  ( |_ `  ( N  /  ( P ^
k ) ) ) ) )
2019ad2antrr 723 . . . . 5  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( ( |_
`  ( ( ( 2  x.  N )  -  N )  / 
( P ^ k
) ) )  +  ( |_ `  ( N  /  ( P ^
k ) ) ) )  =  ( ( |_ `  ( N  /  ( P ^
k ) ) )  +  ( |_ `  ( N  /  ( P ^ k ) ) ) ) )
21 nnre 10538 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR )
2221ad2antrr 723 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  N  e.  RR )
23 prmnn 14307 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  NN )
2423adantl 464 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  P  e.  Prime )  ->  P  e.  NN )
25 elfznn 11717 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... ( 2  x.  N
) )  ->  k  e.  NN )
2625nnnn0d 10848 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( 2  x.  N
) )  ->  k  e.  NN0 )
27 nnexpcl 12164 . . . . . . . . . 10  |-  ( ( P  e.  NN  /\  k  e.  NN0 )  -> 
( P ^ k
)  e.  NN )
2824, 26, 27syl2an 475 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( P ^
k )  e.  NN )
2922, 28nndivred 10580 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( N  / 
( P ^ k
) )  e.  RR )
3029flcld 11916 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( |_ `  ( N  /  ( P ^ k ) ) )  e.  ZZ )
3130zcnd 10966 . . . . . 6  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( |_ `  ( N  /  ( P ^ k ) ) )  e.  CC )
32312timesd 10777 . . . . 5  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( 2  x.  ( |_ `  ( N  /  ( P ^
k ) ) ) )  =  ( ( |_ `  ( N  /  ( P ^
k ) ) )  +  ( |_ `  ( N  /  ( P ^ k ) ) ) ) )
3320, 32eqtr4d 2498 . . . 4  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( ( |_
`  ( ( ( 2  x.  N )  -  N )  / 
( P ^ k
) ) )  +  ( |_ `  ( N  /  ( P ^
k ) ) ) )  =  ( 2  x.  ( |_ `  ( N  /  ( P ^ k ) ) ) ) )
3433oveq2d 6286 . . 3  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( ( |_
`  ( ( 2  x.  N )  / 
( P ^ k
) ) )  -  ( ( |_ `  ( ( ( 2  x.  N )  -  N )  /  ( P ^ k ) ) )  +  ( |_
`  ( N  / 
( P ^ k
) ) ) ) )  =  ( ( |_ `  ( ( 2  x.  N )  /  ( P ^
k ) ) )  -  ( 2  x.  ( |_ `  ( N  /  ( P ^
k ) ) ) ) ) )
3534sumeq2dv 13610 . 2  |-  ( ( N  e.  NN  /\  P  e.  Prime )  ->  sum_ k  e.  ( 1 ... ( 2  x.  N ) ) ( ( |_ `  (
( 2  x.  N
)  /  ( P ^ k ) ) )  -  ( ( |_ `  ( ( ( 2  x.  N
)  -  N )  /  ( P ^
k ) ) )  +  ( |_ `  ( N  /  ( P ^ k ) ) ) ) )  = 
sum_ k  e.  ( 1 ... ( 2  x.  N ) ) ( ( |_ `  ( ( 2  x.  N )  /  ( P ^ k ) ) )  -  ( 2  x.  ( |_ `  ( N  /  ( P ^ k ) ) ) ) ) )
3611, 35eqtrd 2495 1  |-  ( ( N  e.  NN  /\  P  e.  Prime )  -> 
( P  pCnt  (
( 2  x.  N
)  _C  N ) )  =  sum_ k  e.  ( 1 ... (
2  x.  N ) ) ( ( |_
`  ( ( 2  x.  N )  / 
( P ^ k
) ) )  -  ( 2  x.  ( |_ `  ( N  / 
( P ^ k
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796    / cdiv 10202   NNcn 10531   2c2 10581   NN0cn0 10791   ...cfz 11675   |_cfl 11908   ^cexp 12151    _C cbc 12365   sum_csu 13593   Primecprime 14304    pCnt cpc 14447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-fac 12339  df-bc 12366  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-sum 13594  df-dvds 14074  df-gcd 14232  df-prm 14305  df-pc 14448
This theorem is referenced by:  bposlem1  23760  bposlem2  23761
  Copyright terms: Public domain W3C validator