Theorem pcdvdsb 13099

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 9572	. . . . . . . . 9 |- ( A e. NN0 -> A e. RR )
2	1	3ad2ant3 1051	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A e. RR )
3	2	rexrd 8375	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A e. RR* )
4		pnfge 10191	. . . . . . 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 13083	. . . . . . 7 |- ( P e. Prime -> ( P pCnt 0 ) = +oo )
7	6	3ad2ant1 1049	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P pCnt 0 ) = +oo )
8	5, 7	breqtrrd 4158	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A <_ ( P pCnt 0 ) )
9		prmnn 12888	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
10		nnexpcl 10989	. . . . . . . . 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 1047	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. NN )
13	12	nnzd 9767	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. ZZ )
14		dvds0 12573	. . . . . 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 6093	. . . . . 6 |- ( N = 0 -> ( P pCnt N ) = ( P pCnt 0 ) )
19	18	breq2d 4142	. . . . 5 |- ( N = 0 -> ( A <_ ( P pCnt N ) <-> A <_ ( P pCnt 0 ) ) )
20		breq2 4134	. . . . 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 1033	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> A e. NN0 )
25	24	nn0zd 9766	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> A e. ZZ )
26		simpl1 1031	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. Prime )
27		simpl2 1032	. . . . . . . 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 13077	. . . . . . . 8 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) e. NN0 )
30	26, 27, 28, 29	syl12anc 1276	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P pCnt N ) e. NN0 )
31	30	nn0zd 9766	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P pCnt N ) e. ZZ )
32		eluz 9935	. . . . . 6 |- ( ( A e. ZZ /\ ( P pCnt N ) e. ZZ ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) <-> A <_ ( P pCnt N ) ) )
33	25, 31, 32	syl2anc 415	. . . . 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 9767	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. ZZ )
36		dvdsexp 12628	. . . . . . 7 |- ( ( P e. ZZ /\ A e. NN0 /\ ( P pCnt N ) e. ( ZZ>= ` A ) ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) )
37	36	3expia 1236	. . . . . 6 |- ( ( P e. ZZ /\ A e. NN0 ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) ) )
38	35, 24, 37	syl2anc 415	. . . . 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 13093	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( P pCnt N ) ) || N )
41	26, 27, 28, 40	syl12anc 1276	. . . . 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 11133	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) e. NN )
44	43	nnzd 9767	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) e. ZZ )
45		dvdstr 12595	. . . . . 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 1278	. . . . 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 432	. . . 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 9721	. . . . 5 |- ( ( A e. ZZ /\ ( P pCnt N ) e. ZZ ) -> DECID A <_ ( P pCnt N ) )
50	25, 31, 49	syl2anc 415	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> DECID A <_ ( P pCnt N ) )
51		nn0z 9664	. . . . . . . 8 |- ( ( P pCnt N ) e. NN0 -> ( P pCnt N ) e. ZZ )
52		nn0z 9664	. . . . . . . 8 |- ( A e. NN0 -> A e. ZZ )
53		zltnle 9690	. . . . . . . 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 9707	. . . . . . 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 415	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( -. A <_ ( P pCnt N ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
58		peano2nn0 9603	. . . . . . . . . 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 9766	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) + 1 ) e. ZZ )
61		eluz 9935	. . . . . . . 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 415	. . . . . . 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 12628	. . . . . . . . 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 1236	. . . . . . . 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 415	. . . . . . 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 13095	. . . . . . . . 9 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
68	26, 27, 28, 67	syl12anc 1276	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
69	34, 59	nnexpcld 11133	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) e. NN )
70	69	nnzd 9767	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) e. ZZ )
71		dvdstr 12595	. . . . . . . . 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 1278	. . . . . . . 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 673	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> -. ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) /\ ( P ^ A ) || N ) )
74		imnan 701	. . . . . . 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 865	. . . 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 1029	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> N e. ZZ )
82		0zd 9656	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> 0 e. ZZ )
83		zdceq 9720	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
84	81, 82, 83	syl2anc 415	. . 3 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> DECID N = 0 )
85		dcne 2431	. . 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 810	1 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   +oocpnf 8357   RR*cxr 8359   < clt 8360   <_ cle 8361   NNcn 9304   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   ^cexp 10975   || cdvds 12554   Primecprime 12885   pCnt cpc 13063

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 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

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-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731  df-prm 12886  df-pc 13064

This theorem is used by:  pcelnn  13100  pcidlem  13102  pcdvdstr  13106  pcgcd1  13107  pcfac  13129  pockthlem  13135  pockthg  13136