Theorem pcbcctr 16269

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 9471	. . . . 5 |- 2 e. NN
2		nnmulcl 9328	. . . . 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 9575	. . . . 5 |- ( N e. NN -> N e. NN0 )
6		fzctr 10551	. . . . 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 13153	. . 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 9315	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
13	12	2timesd 9553	. . . . . . . . 9 |- ( N e. NN -> ( 2 x. N ) = ( N + N ) )
14	12, 12, 13	mvrladdd 8695	. . . . . . . 8 |- ( N e. NN -> ( ( 2 x. N ) - N ) = N )
15	14	fvoveq1d 6107	. . . . . . 7 |- ( N e. NN -> ( |_ ` ( ( ( 2 x. N ) - N ) / ( P ^ k ) ) ) = ( |_ ` ( N / ( P ^ k ) ) ) )
16	15	oveq1d 6100	. . . . . 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 9668	. . . . . . . . . 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 12907	. . . . . . . . . . 11 |- ( P e. Prime -> P e. NN )
21	20	adantl 277	. . . . . . . . . 10 |- ( ( N e. NN /\ P e. Prime ) -> P e. NN )
22		elfznn 10471	. . . . . . . . . . 11 |- ( k e. ( 1 ... ( 2 x. N ) ) -> k e. NN )
23	22	nnnn0d 9625	. . . . . . . . . 10 |- ( k e. ( 1 ... ( 2 x. N ) ) -> k e. NN0 )
24		nnexpcl 11004	. . . . . . . . . 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 10034	. . . . . . . . 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 10725	. . . . . . 7 |- ( ( ( N e. NN /\ P e. Prime ) /\ k e. ( 1 ... ( 2 x. N ) ) ) -> ( |_ ` ( N / ( P ^ k ) ) ) e. ZZ )
29	28	zcnd 9774	. . . . . 6 |- ( ( ( N e. NN /\ P e. Prime ) /\ k e. ( 1 ... ( 2 x. N ) ) ) -> ( |_ ` ( N / ( P ^ k ) ) ) e. CC )
30	29	2timesd 9553	. . . . 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 6101	. . 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 12153	. 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 5377  (class class class)co 6085   0cc0 8180   1c1 8181   + caddc 8183   x. cmul 8185   - cmin 8499   / cdiv 9005   NNcn 9307   2c2 9358   NN0cn0 9568   ZZcz 9649   QQcq 10029   ...cfz 10422   |_cfl 10714   ^cexp 10990   _C cbc 11201   sum_csu 12138   Primecprime 12904   pCnt cpc 13086

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7325  df-inf 7326  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-fz 10423  df-fzo 10561  df-fl 10716  df-mod 10775  df-seqfrec 10900  df-exp 10991  df-fac 11180  df-bc 11202  df-ihash 11231  df-cj 11623  df-re 11624  df-im 11625  df-rsqrt 11780  df-abs 11781  df-clim 12064  df-sumdc 12139  df-dvds 12574  df-gcd 12750  df-prm 12905  df-pc 13087

This theorem is used by:  bposlem1  16277  bposlem2  16278