Theorem pcbcctr 16315

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

Step	Hyp	Ref	Expression
1		2nn 9478	. . . . 5 |- 2 e. NN
2		nnmulcl 9335	. . . . 5 |- ( ( 2 e. NN /\ N e. NN ) -> ( 2 x. N ) e. NN )
3	1, 2	mpan 428	. . . 4 |- ( N e. NN -> ( 2 x. N ) e. NN )
4	3	adantr 276	. . 3 |- ( ( N e. NN /\ P e. Prime ) -> ( 2 x. N ) e. NN )
5		nnnn0 9582	. . . . 5 |- ( N e. NN -> N e. NN0 )
6		fzctr 10558	. . . . 5 |- ( N e. NN0 -> N e. ( 0 ... ( 2 x. N ) ) )
7	5, 6	syl 14	. . . 4 |- ( N e. NN -> N e. ( 0 ... ( 2 x. N ) ) )
8	7	adantr 276	. . 3 |- ( ( N e. NN /\ P e. Prime ) -> N e. ( 0 ... ( 2 x. N ) ) )
9		simpr 110	. . 3 |- ( ( N e. NN /\ P e. Prime ) -> P e. Prime )
10		pcbc 13160	. . 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 ) ) ) ) ) )
11	4, 8, 9, 10	syl3anc 1278	. 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 9322	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
13	12	2timesd 9560	. . . . . . . . 9 |- ( N e. NN -> ( 2 x. N ) = ( N + N ) )
14	12, 12, 13	mvrladdd 8701	. . . . . . . 8 |- ( N e. NN -> ( ( 2 x. N ) - N ) = N )
15	14	fvoveq1d 6110	. . . . . . 7 |- ( N e. NN -> ( |_ ` ( ( ( 2 x. N ) - N ) / ( P ^ k ) ) ) = ( |_ ` ( N / ( P ^ k ) ) ) )
16	15	oveq1d 6103	. . . . . 6 |- ( N e. NN -> ( ( |_ ` ( ( ( 2 x. N ) - N ) / ( P ^ k ) ) ) + ( |_ ` ( N / ( P ^ k ) ) ) ) = ( ( |_ ` ( N / ( P ^ k ) ) ) + ( |_ ` ( N / ( P ^ k ) ) ) ) )
17	16	ad2antrr 492	. . . . 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 ) ) ) ) )
18		nnz 9675	. . . . . . . . . 10 |- ( N e. NN -> N e. ZZ )
19	18	ad2antrr 492	. . . . . . . . 9 |- ( ( ( N e. NN /\ P e. Prime ) /\ k e. ( 1 ... ( 2 x. N ) ) ) -> N e. ZZ )
20		prmnn 12914	. . . . . . . . . . 11 |- ( P e. Prime -> P e. NN )
21	20	adantl 277	. . . . . . . . . 10 |- ( ( N e. NN /\ P e. Prime ) -> P e. NN )
22		elfznn 10478	. . . . . . . . . . 11 |- ( k e. ( 1 ... ( 2 x. N ) ) -> k e. NN )
23	22	nnnn0d 9632	. . . . . . . . . 10 |- ( k e. ( 1 ... ( 2 x. N ) ) -> k e. NN0 )
24		nnexpcl 11011	. . . . . . . . . 10 |- ( ( P e. NN /\ k e. NN0 ) -> ( P ^ k ) e. NN )
25	21, 23, 24	syl2an 289	. . . . . . . . 9 |- ( ( ( N e. NN /\ P e. Prime ) /\ k e. ( 1 ... ( 2 x. N ) ) ) -> ( P ^ k ) e. NN )
26		znq 10041	. . . . . . . . 9 |- ( ( N e. ZZ /\ ( P ^ k ) e. NN ) -> ( N / ( P ^ k ) ) e. QQ )
27	19, 25, 26	syl2anc 415	. . . . . . . 8 |- ( ( ( N e. NN /\ P e. Prime ) /\ k e. ( 1 ... ( 2 x. N ) ) ) -> ( N / ( P ^ k ) ) e. QQ )
28	27	flqcld 10732	. . . . . . 7 |- ( ( ( N e. NN /\ P e. Prime ) /\ k e. ( 1 ... ( 2 x. N ) ) ) -> ( |_ ` ( N / ( P ^ k ) ) ) e. ZZ )
29	28	zcnd 9781	. . . . . 6 |- ( ( ( N e. NN /\ P e. Prime ) /\ k e. ( 1 ... ( 2 x. N ) ) ) -> ( |_ ` ( N / ( P ^ k ) ) ) e. CC )
30	29	2timesd 9560	. . . . 5 |- ( ( ( N e. NN /\ P e. Prime ) /\ k e. ( 1 ... ( 2 x. N ) ) ) -> ( 2 x. ( |_ ` ( N / ( P ^ k ) ) ) ) = ( ( |_ ` ( N / ( P ^ k ) ) ) + ( |_ ` ( N / ( P ^ k ) ) ) ) )
31	17, 30	eqtr4d 2274	. . . 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 ) ) ) ) )
32	31	oveq2d 6104	. . 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 ) ) ) ) ) )
33	32	sumeq2dv 12160	. 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 ) ) ) ) ) )
34	11, 33	eqtrd 2271	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 ) ) ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   0cc0 8186   1c1 8187   + caddc 8189   x. cmul 8191   - cmin 8505   / cdiv 9011   NNcn 9314   2c2 9365   NN0cn0 9575   ZZcz 9656   QQcq 10036   ...cfz 10429   |_cfl 10721   ^cexp 10997   _C cbc 11208   sum_csu 12145   Primecprime 12911   pCnt cpc 13093

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-fz 10430  df-fzo 10568  df-fl 10723  df-mod 10782  df-seqfrec 10907  df-exp 10998  df-fac 11187  df-bc 11209  df-ihash 11238  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-clim 12071  df-sumdc 12146  df-dvds 12581  df-gcd 12757  df-prm 12912  df-pc 13094

This theorem is used by:  bposlem1  16323  bposlem2  16324