Theorem pcdvdsb 12266

Description: P ^ A divides N if and only if A is at most the count of P. (Contributed by Mario Carneiro, 3-Oct-2014.)

Assertion

Ref	Expression
pcdvdsb	|- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )

Proof of Theorem pcdvdsb

Step	Hyp	Ref	Expression
1		nn0re 9137	. . . . . . . . 9 |- ( A e. NN0 -> A e. RR )
2	1	3ad2ant3 1015	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A e. RR )
3	2	rexrd 7962	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A e. RR* )
4		pnfge 9739	. . . . . . 7 |- ( A e. RR* -> A <_ +oo )
5	3, 4	syl 14	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A <_ +oo )
6		pc0 12251	. . . . . . 7 |- ( P e. Prime -> ( P pCnt 0 ) = +oo )
7	6	3ad2ant1 1013	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P pCnt 0 ) = +oo )
8	5, 7	breqtrrd 4015	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A <_ ( P pCnt 0 ) )
9		prmnn 12057	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
10		nnexpcl 10482	. . . . . . . . 9 |- ( ( P e. NN /\ A e. NN0 ) -> ( P ^ A ) e. NN )
11	9, 10	sylan 281	. . . . . . . 8 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ A ) e. NN )
12	11	3adant2 1011	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. NN )
13	12	nnzd 9326	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. ZZ )
14		dvds0 11761	. . . . . 6 |- ( ( P ^ A ) e. ZZ -> ( P ^ A ) || 0 )
15	13, 14	syl 14	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) || 0 )
16	8, 15	2thd 174	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) )
17	16	adantr 274	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N = 0 ) -> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) )
18		oveq2 5859	. . . . . 6 |- ( N = 0 -> ( P pCnt N ) = ( P pCnt 0 ) )
19	18	breq2d 3999	. . . . 5 |- ( N = 0 -> ( A <_ ( P pCnt N ) <-> A <_ ( P pCnt 0 ) ) )
20		breq2 3991	. . . . 5 |- ( N = 0 -> ( ( P ^ A ) || N <-> ( P ^ A ) || 0 ) )
21	19, 20	bibi12d 234	. . . 4 |- ( N = 0 -> ( ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) <-> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) ) )
22	21	adantl 275	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N = 0 ) -> ( ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) <-> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) ) )
23	17, 22	mpbird 166	. 2 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N = 0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )
24		simpl3 997	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> A e. NN0 )
25	24	nn0zd 9325	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> A e. ZZ )
26		simpl1 995	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. Prime )
27		simpl2 996	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> N e. ZZ )
28		simpr 109	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> N =/= 0 )
29		pczcl 12245	. . . . . . . 8 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) e. NN0 )
30	26, 27, 28, 29	syl12anc 1231	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P pCnt N ) e. NN0 )
31	30	nn0zd 9325	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P pCnt N ) e. ZZ )
32		eluz 9493	. . . . . 6 |- ( ( A e. ZZ /\ ( P pCnt N ) e. ZZ ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) <-> A <_ ( P pCnt N ) ) )
33	25, 31, 32	syl2anc 409	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) <-> A <_ ( P pCnt N ) ) )
34	26, 9	syl 14	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. NN )
35	34	nnzd 9326	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. ZZ )
36		dvdsexp 11814	. . . . . . 7 |- ( ( P e. ZZ /\ A e. NN0 /\ ( P pCnt N ) e. ( ZZ>= ` A ) ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) )
37	36	3expia 1200	. . . . . 6 |- ( ( P e. ZZ /\ A e. NN0 ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) ) )
38	35, 24, 37	syl2anc 409	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) ) )
39	33, 38	sylbird 169	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A <_ ( P pCnt N ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) ) )
40		pczdvds 12260	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( P pCnt N ) ) || N )
41	26, 27, 28, 40	syl12anc 1231	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) || N )
42	13	adantr 274	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ A ) e. ZZ )
43	34, 30	nnexpcld 10624	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) e. NN )
44	43	nnzd 9326	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) e. ZZ )
45		dvdstr 11783	. . . . . 6 |- ( ( ( P ^ A ) e. ZZ /\ ( P ^ ( P pCnt N ) ) e. ZZ /\ N e. ZZ ) -> ( ( ( P ^ A ) || ( P ^ ( P pCnt N ) ) /\ ( P ^ ( P pCnt N ) ) || N ) -> ( P ^ A ) || N ) )
46	42, 44, 27, 45	syl3anc 1233	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( ( P ^ A ) || ( P ^ ( P pCnt N ) ) /\ ( P ^ ( P pCnt N ) ) || N ) -> ( P ^ A ) || N ) )
47	41, 46	mpan2d 426	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P ^ A ) || ( P ^ ( P pCnt N ) ) -> ( P ^ A ) || N ) )
48	39, 47	syld 45	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A <_ ( P pCnt N ) -> ( P ^ A ) || N ) )
49		zdcle 9281	. . . . 5 |- ( ( A e. ZZ /\ ( P pCnt N ) e. ZZ ) -> DECID A <_ ( P pCnt N ) )
50	25, 31, 49	syl2anc 409	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> DECID A <_ ( P pCnt N ) )
51		nn0z 9225	. . . . . . . 8 |- ( ( P pCnt N ) e. NN0 -> ( P pCnt N ) e. ZZ )
52		nn0z 9225	. . . . . . . 8 |- ( A e. NN0 -> A e. ZZ )
53		zltnle 9251	. . . . . . . 8 |- ( ( ( P pCnt N ) e. ZZ /\ A e. ZZ ) -> ( ( P pCnt N ) < A <-> -. A <_ ( P pCnt N ) ) )
54	51, 52, 53	syl2an 287	. . . . . . 7 |- ( ( ( P pCnt N ) e. NN0 /\ A e. NN0 ) -> ( ( P pCnt N ) < A <-> -. A <_ ( P pCnt N ) ) )
55		nn0ltp1le 9267	. . . . . . 7 |- ( ( ( P pCnt N ) e. NN0 /\ A e. NN0 ) -> ( ( P pCnt N ) < A <-> ( ( P pCnt N ) + 1 ) <_ A ) )
56	54, 55	bitr3d 189	. . . . . 6 |- ( ( ( P pCnt N ) e. NN0 /\ A e. NN0 ) -> ( -. A <_ ( P pCnt N ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
57	30, 24, 56	syl2anc 409	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( -. A <_ ( P pCnt N ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
58		peano2nn0 9168	. . . . . . . . . 10 |- ( ( P pCnt N ) e. NN0 -> ( ( P pCnt N ) + 1 ) e. NN0 )
59	30, 58	syl 14	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) + 1 ) e. NN0 )
60	59	nn0zd 9325	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) + 1 ) e. ZZ )
61		eluz 9493	. . . . . . . 8 |- ( ( ( ( P pCnt N ) + 1 ) e. ZZ /\ A e. ZZ ) -> ( A e. ( ZZ>= ` ( ( P pCnt N ) + 1 ) ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
62	60, 25, 61	syl2anc 409	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A e. ( ZZ>= ` ( ( P pCnt N ) + 1 ) ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
63		dvdsexp 11814	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( ( P pCnt N ) + 1 ) e. NN0 /\ A e. ( ZZ>= ` ( ( P pCnt N ) + 1 ) ) ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) )
64	63	3expia 1200	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( P pCnt N ) + 1 ) e. NN0 ) -> ( A e. ( ZZ>= ` ( ( P pCnt N ) + 1 ) ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) ) )
65	35, 59, 64	syl2anc 409	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A e. ( ZZ>= ` ( ( P pCnt N ) + 1 ) ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) ) )
66	62, 65	sylbird 169	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( ( P pCnt N ) + 1 ) <_ A -> ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) ) )
67		pczndvds 12262	. . . . . . . . 9 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
68	26, 27, 28, 67	syl12anc 1231	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
69	34, 59	nnexpcld 10624	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) e. NN )
70	69	nnzd 9326	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) e. ZZ )
71		dvdstr 11783	. . . . . . . . 9 |- ( ( ( P ^ ( ( P pCnt N ) + 1 ) ) e. ZZ /\ ( P ^ A ) e. ZZ /\ N e. ZZ ) -> ( ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) /\ ( P ^ A ) || N ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) || N ) )
72	70, 42, 27, 71	syl3anc 1233	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) /\ ( P ^ A ) || N ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) || N ) )
73	68, 72	mtod 658	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> -. ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) /\ ( P ^ A ) || N ) )
74		imnan 685	. . . . . . 7 |- ( ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) -> -. ( P ^ A ) || N ) <-> -. ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) /\ ( P ^ A ) || N ) )
75	73, 74	sylibr 133	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) -> -. ( P ^ A ) || N ) )
76	66, 75	syld 45	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( ( P pCnt N ) + 1 ) <_ A -> -. ( P ^ A ) || N ) )
77	57, 76	sylbid 149	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( -. A <_ ( P pCnt N ) -> -. ( P ^ A ) || N ) )
78		condc 848	. . . 4 |- (DECID A <_ ( P pCnt N ) -> ( ( -. A <_ ( P pCnt N ) -> -. ( P ^ A ) || N ) -> ( ( P ^ A ) || N -> A <_ ( P pCnt N ) ) ) )
79	50, 77, 78	sylc 62	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P ^ A ) || N -> A <_ ( P pCnt N ) ) )
80	48, 79	impbid 128	. 2 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )
81		simp2 993	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> N e. ZZ )
82		0zd 9217	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> 0 e. ZZ )
83		zdceq 9280	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
84	81, 82, 83	syl2anc 409	. . 3 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> DECID N = 0 )
85		dcne 2351	. . 3 |- (DECID N = 0 <-> ( N = 0 \/ N =/= 0 ) )
86	84, 85	sylib 121	. 2 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( N = 0 \/ N =/= 0 ) )
87	23, 80, 86	mpjaodan 793	1 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3987   ` cfv 5196  (class class class)co 5851   RRcr 7766   0cc0 7767   1c1 7768   + caddc 7770   +oocpnf 7944   RR*cxr 7946   < clt 7947   <_ cle 7948   NNcn 8871   NN0cn0 9128   ZZcz 9205   ZZ>=cuz 9480   ^cexp 10468   || cdvds 11742   Primecprime 12054   pCnt cpc 12231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885  ax-arch 7886  ax-caucvg 7887

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-1o 6393  df-2o 6394  df-er 6511  df-en 6717  df-sup 6959  df-inf 6960  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583  df-inn 8872  df-2 8930  df-3 8931  df-4 8932  df-n0 9129  df-z 9206  df-uz 9481  df-q 9572  df-rp 9604  df-fz 9959  df-fzo 10092  df-fl 10219  df-mod 10272  df-seqfrec 10395  df-exp 10469  df-cj 10799  df-re 10800  df-im 10801  df-rsqrt 10955  df-abs 10956  df-dvds 11743  df-gcd 11891  df-prm 12055  df-pc 12232

This theorem is referenced by:  pcelnn  12267  pcidlem  12269  pcdvdstr  12273  pcgcd1  12274  pcfac  12295  pockthlem  12301  pockthg  12302