Theorem pcdvdsb 12514

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 9275	. . . . . . . . 9 |- ( A e. NN0 -> A e. RR )
2	1	3ad2ant3 1022	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A e. RR )
3	2	rexrd 8093	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A e. RR* )
4		pnfge 9881	. . . . . . 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 12498	. . . . . . 7 |- ( P e. Prime -> ( P pCnt 0 ) = +oo )
7	6	3ad2ant1 1020	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P pCnt 0 ) = +oo )
8	5, 7	breqtrrd 4062	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A <_ ( P pCnt 0 ) )
9		prmnn 12303	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
10		nnexpcl 10661	. . . . . . . . 9 |- ( ( P e. NN /\ A e. NN0 ) -> ( P ^ A ) e. NN )
11	9, 10	sylan 283	. . . . . . . 8 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ A ) e. NN )
12	11	3adant2 1018	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. NN )
13	12	nnzd 9464	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. ZZ )
14		dvds0 11988	. . . . . 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 175	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) )
17	16	adantr 276	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N = 0 ) -> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) )
18		oveq2 5933	. . . . . 6 |- ( N = 0 -> ( P pCnt N ) = ( P pCnt 0 ) )
19	18	breq2d 4046	. . . . 5 |- ( N = 0 -> ( A <_ ( P pCnt N ) <-> A <_ ( P pCnt 0 ) ) )
20		breq2 4038	. . . . 5 |- ( N = 0 -> ( ( P ^ A ) || N <-> ( P ^ A ) || 0 ) )
21	19, 20	bibi12d 235	. . . 4 |- ( N = 0 -> ( ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) <-> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) ) )
22	21	adantl 277	. . 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 167	. 2 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N = 0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )
24		simpl3 1004	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> A e. NN0 )
25	24	nn0zd 9463	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> A e. ZZ )
26		simpl1 1002	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. Prime )
27		simpl2 1003	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> N e. ZZ )
28		simpr 110	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> N =/= 0 )
29		pczcl 12492	. . . . . . . 8 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) e. NN0 )
30	26, 27, 28, 29	syl12anc 1247	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P pCnt N ) e. NN0 )
31	30	nn0zd 9463	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P pCnt N ) e. ZZ )
32		eluz 9631	. . . . . 6 |- ( ( A e. ZZ /\ ( P pCnt N ) e. ZZ ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) <-> A <_ ( P pCnt N ) ) )
33	25, 31, 32	syl2anc 411	. . . . 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 9464	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. ZZ )
36		dvdsexp 12043	. . . . . . 7 |- ( ( P e. ZZ /\ A e. NN0 /\ ( P pCnt N ) e. ( ZZ>= ` A ) ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) )
37	36	3expia 1207	. . . . . 6 |- ( ( P e. ZZ /\ A e. NN0 ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) ) )
38	35, 24, 37	syl2anc 411	. . . . 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 170	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A <_ ( P pCnt N ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) ) )
40		pczdvds 12508	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( P pCnt N ) ) || N )
41	26, 27, 28, 40	syl12anc 1247	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) || N )
42	13	adantr 276	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ A ) e. ZZ )
43	34, 30	nnexpcld 10804	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) e. NN )
44	43	nnzd 9464	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) e. ZZ )
45		dvdstr 12010	. . . . . 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 1249	. . . . 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 428	. . . 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 9419	. . . . 5 |- ( ( A e. ZZ /\ ( P pCnt N ) e. ZZ ) -> DECID A <_ ( P pCnt N ) )
50	25, 31, 49	syl2anc 411	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> DECID A <_ ( P pCnt N ) )
51		nn0z 9363	. . . . . . . 8 |- ( ( P pCnt N ) e. NN0 -> ( P pCnt N ) e. ZZ )
52		nn0z 9363	. . . . . . . 8 |- ( A e. NN0 -> A e. ZZ )
53		zltnle 9389	. . . . . . . 8 |- ( ( ( P pCnt N ) e. ZZ /\ A e. ZZ ) -> ( ( P pCnt N ) < A <-> -. A <_ ( P pCnt N ) ) )
54	51, 52, 53	syl2an 289	. . . . . . 7 |- ( ( ( P pCnt N ) e. NN0 /\ A e. NN0 ) -> ( ( P pCnt N ) < A <-> -. A <_ ( P pCnt N ) ) )
55		nn0ltp1le 9405	. . . . . . 7 |- ( ( ( P pCnt N ) e. NN0 /\ A e. NN0 ) -> ( ( P pCnt N ) < A <-> ( ( P pCnt N ) + 1 ) <_ A ) )
56	54, 55	bitr3d 190	. . . . . 6 |- ( ( ( P pCnt N ) e. NN0 /\ A e. NN0 ) -> ( -. A <_ ( P pCnt N ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
57	30, 24, 56	syl2anc 411	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( -. A <_ ( P pCnt N ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
58		peano2nn0 9306	. . . . . . . . . 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 9463	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) + 1 ) e. ZZ )
61		eluz 9631	. . . . . . . 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 411	. . . . . . 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 12043	. . . . . . . . 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 1207	. . . . . . . 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 411	. . . . . . 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 170	. . . . . 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 12510	. . . . . . . . 9 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
68	26, 27, 28, 67	syl12anc 1247	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
69	34, 59	nnexpcld 10804	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) e. NN )
70	69	nnzd 9464	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) e. ZZ )
71		dvdstr 12010	. . . . . . . . 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 1249	. . . . . . . 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 664	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> -. ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) /\ ( P ^ A ) || N ) )
74		imnan 691	. . . . . . 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 134	. . . . . 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 150	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( -. A <_ ( P pCnt N ) -> -. ( P ^ A ) || N ) )
78		condc 854	. . . 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 129	. 2 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )
81		simp2 1000	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> N e. ZZ )
82		0zd 9355	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> 0 e. ZZ )
83		zdceq 9418	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
84	81, 82, 83	syl2anc 411	. . 3 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> DECID N = 0 )
85		dcne 2378	. . 3 |- (DECID N = 0 <-> ( N = 0 \/ N =/= 0 ) )
86	84, 85	sylib 122	. 2 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( N = 0 \/ N =/= 0 ) )
87	23, 80, 86	mpjaodan 799	1 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   RRcr 7895   0cc0 7896   1c1 7897   + caddc 7899   +oocpnf 8075   RR*cxr 8077   < clt 8078   <_ cle 8079   NNcn 9007   NN0cn0 9266   ZZcz 9343   ZZ>=cuz 9618   ^cexp 10647   || cdvds 11969   Primecprime 12300   pCnt cpc 12478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-er 6601  df-en 6809  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-gcd 12146  df-prm 12301  df-pc 12479

This theorem is referenced by:  pcelnn  12515  pcidlem  12517  pcdvdstr  12521  pcgcd1  12522  pcfac  12544  pockthlem  12550  pockthg  12551