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Theorem pcbcctr 21052
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 10125 . . . . 5  |-  2  e.  NN
2 nnmulcl 10015 . . . . 5  |-  ( ( 2  e.  NN  /\  N  e.  NN )  ->  ( 2  x.  N
)  e.  NN )
31, 2mpan 652 . . . 4  |-  ( N  e.  NN  ->  (
2  x.  N )  e.  NN )
43adantr 452 . . 3  |-  ( ( N  e.  NN  /\  P  e.  Prime )  -> 
( 2  x.  N
)  e.  NN )
5 nnnn0 10220 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
6 fzctr 11109 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  ( 0 ... (
2  x.  N ) ) )
75, 6syl 16 . . . 4  |-  ( N  e.  NN  ->  N  e.  ( 0 ... (
2  x.  N ) ) )
87adantr 452 . . 3  |-  ( ( N  e.  NN  /\  P  e.  Prime )  ->  N  e.  ( 0 ... ( 2  x.  N ) ) )
9 simpr 448 . . 3  |-  ( ( N  e.  NN  /\  P  e.  Prime )  ->  P  e.  Prime )
10 pcbc 13261 . . 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 1184 . 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 10000 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
13122timesd 10202 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
2  x.  N )  =  ( N  +  N ) )
1413oveq1d 6088 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( 2  x.  N
)  -  N )  =  ( ( N  +  N )  -  N ) )
1512, 12pncand 9404 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  +  N
)  -  N )  =  N )
1614, 15eqtrd 2467 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( 2  x.  N
)  -  N )  =  N )
1716oveq1d 6088 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( 2  x.  N )  -  N
)  /  ( P ^ k ) )  =  ( N  / 
( P ^ k
) ) )
1817fveq2d 5724 . . . . . . 7  |-  ( N  e.  NN  ->  ( |_ `  ( ( ( 2  x.  N )  -  N )  / 
( P ^ k
) ) )  =  ( |_ `  ( N  /  ( P ^
k ) ) ) )
1918oveq1d 6088 . . . . . 6  |-  ( N  e.  NN  ->  (
( |_ `  (
( ( 2  x.  N )  -  N
)  /  ( P ^ k ) ) )  +  ( |_
`  ( N  / 
( P ^ k
) ) ) )  =  ( ( |_
`  ( N  / 
( P ^ k
) ) )  +  ( |_ `  ( N  /  ( P ^
k ) ) ) ) )
2019ad2antrr 707 . . . . 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 9999 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR )
2221ad2antrr 707 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  N  e.  RR )
23 prmnn 13074 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  NN )
2423adantl 453 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  P  e.  Prime )  ->  P  e.  NN )
25 elfznn 11072 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... ( 2  x.  N
) )  ->  k  e.  NN )
2625nnnn0d 10266 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( 2  x.  N
) )  ->  k  e.  NN0 )
27 nnexpcl 11386 . . . . . . . . . 10  |-  ( ( P  e.  NN  /\  k  e.  NN0 )  -> 
( P ^ k
)  e.  NN )
2824, 26, 27syl2an 464 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( P ^
k )  e.  NN )
2922, 28nndivred 10040 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( N  / 
( P ^ k
) )  e.  RR )
3029flcld 11199 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( |_ `  ( N  /  ( P ^ k ) ) )  e.  ZZ )
3130zcnd 10368 . . . . . 6  |-  ( ( ( N  e.  NN  /\  P  e.  Prime )  /\  k  e.  (
1 ... ( 2  x.  N ) ) )  ->  ( |_ `  ( N  /  ( P ^ k ) ) )  e.  CC )
32312timesd 10202 . . . . 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 2470 . . . 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 6089 . . 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 12489 . 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 2467 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   NN0cn0 10213   ...cfz 11035   |_cfl 11193   ^cexp 11374    _C cbc 11585   sum_csu 12471   Primecprime 13071    pCnt cpc 13202
This theorem is referenced by:  bposlem1  21060  bposlem2  21061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203
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