Theorem pcprmpw2 13056

Description: Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)

Assertion

Ref	Expression
pcprmpw2	|- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )

Distinct variable groups:   A, n   P, n

Proof of Theorem pcprmpw2

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 529	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. NN )
2	1	nnnn0d 9570	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. NN0 )
3		prmnn 12832	. . . . . . 7 |- ( P e. Prime -> P e. NN )
4	3	ad2antrr 488	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> P e. NN )
5		pccl 13022	. . . . . . 7 |- ( ( P e. Prime /\ A e. NN ) -> ( P pCnt A ) e. NN0 )
6	5	adantr 276	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. NN0 )
7	4, 6	nnexpcld 11082	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. NN )
8	7	nnnn0d 9570	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. NN0 )
9	6	nn0red 9571	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. RR )
10	9	leidd 8805	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) <_ ( P pCnt A ) )
11		simpll 527	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> P e. Prime )
12	6	nn0zd 9716	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. ZZ )
13		pcid 13047	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( P pCnt A ) e. ZZ ) -> ( P pCnt ( P ^ ( P pCnt A ) ) ) = ( P pCnt A ) )
14	11, 12, 13	syl2anc 411	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt ( P ^ ( P pCnt A ) ) ) = ( P pCnt A ) )
15	10, 14	breqtrrd 4142	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) <_ ( P pCnt ( P ^ ( P pCnt A ) ) ) )
16	15	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> ( P pCnt A ) <_ ( P pCnt ( P ^ ( P pCnt A ) ) ) )
17		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> p = P )
18	17	oveq1d 6073	. . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> ( p pCnt A ) = ( P pCnt A ) )
19	17	oveq1d 6073	. . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> ( p pCnt ( P ^ ( P pCnt A ) ) ) = ( P pCnt ( P ^ ( P pCnt A ) ) ) )
20	16, 18, 19	3brtr4d 4146	. . . . . . 7 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) )
21		simplrr 538	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A || ( P ^ n ) )
22		prmz 12833	. . . . . . . . . . . . . . 15 |- ( p e. Prime -> p e. ZZ )
23	22	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> p e. ZZ )
24	1	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A e. NN )
25	24	nnzd 9717	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A e. ZZ )
26		simprl 531	. . . . . . . . . . . . . . . . 17 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> n e. NN0 )
27	4, 26	nnexpcld 11082	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ n ) e. NN )
28	27	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( P ^ n ) e. NN )
29	28	nnzd 9717	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( P ^ n ) e. ZZ )
30		dvdstr 12539	. . . . . . . . . . . . . 14 |- ( ( p e. ZZ /\ A e. ZZ /\ ( P ^ n ) e. ZZ ) -> ( ( p || A /\ A || ( P ^ n ) ) -> p || ( P ^ n ) ) )
31	23, 25, 29, 30	syl3anc 1274	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( ( p || A /\ A || ( P ^ n ) ) -> p || ( P ^ n ) ) )
32	21, 31	mpan2d 428	. . . . . . . . . . . 12 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || A -> p || ( P ^ n ) ) )
33		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> p e. Prime )
34	11	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> P e. Prime )
35		simplrl 537	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> n e. NN0 )
36		prmdvdsexpr 12872	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ P e. Prime /\ n e. NN0 ) -> ( p || ( P ^ n ) -> p = P ) )
37	33, 34, 35, 36	syl3anc 1274	. . . . . . . . . . . 12 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || ( P ^ n ) -> p = P ) )
38	32, 37	syld 45	. . . . . . . . . . 11 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || A -> p = P ) )
39	38	necon3ad 2456	. . . . . . . . . 10 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p =/= P -> -. p || A ) )
40	39	imp 124	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> -. p || A )
41		simplr 529	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> p e. Prime )
42	1	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> A e. NN )
43		pceq0 13045	. . . . . . . . . 10 |- ( ( p e. Prime /\ A e. NN ) -> ( ( p pCnt A ) = 0 <-> -. p || A ) )
44	41, 42, 43	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> ( ( p pCnt A ) = 0 <-> -. p || A ) )
45	40, 44	mpbird 167	. . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> ( p pCnt A ) = 0 )
46	7	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> ( P ^ ( P pCnt A ) ) e. NN )
47	41, 46	pccld 13023	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> ( p pCnt ( P ^ ( P pCnt A ) ) ) e. NN0 )
48	47	nn0ge0d 9573	. . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> 0 <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) )
49	45, 48	eqbrtrd 4136	. . . . . . 7 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) )
50		prmz 12833	. . . . . . . . . . 11 |- ( P e. Prime -> P e. ZZ )
51	50	adantr 276	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. NN ) -> P e. ZZ )
52	51	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> P e. ZZ )
53		zdceq 9670	. . . . . . . . 9 |- ( ( p e. ZZ /\ P e. ZZ ) -> DECID p = P )
54	23, 52, 53	syl2anc 411	. . . . . . . 8 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> DECID p = P )
55		dcne 2425	. . . . . . . 8 |- (DECID p = P <-> ( p = P \/ p =/= P ) )
56	54, 55	sylib 122	. . . . . . 7 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p = P \/ p =/= P ) )
57	20, 49, 56	mpjaodan 806	. . . . . 6 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) )
58	57	ralrimiva 2617	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A. p e. Prime ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) )
59	1	nnzd 9717	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. ZZ )
60	7	nnzd 9717	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
61		pc2dvds 13053	. . . . . 6 |- ( ( A e. ZZ /\ ( P ^ ( P pCnt A ) ) e. ZZ ) -> ( A || ( P ^ ( P pCnt A ) ) <-> A. p e. Prime ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) ) )
62	59, 60, 61	syl2anc 411	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( A || ( P ^ ( P pCnt A ) ) <-> A. p e. Prime ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) ) )
63	58, 62	mpbird 167	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A || ( P ^ ( P pCnt A ) ) )
64		pcdvds 13038	. . . . 5 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || A )
65	64	adantr 276	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) || A )
66		dvdseq 12559	. . . 4 |- ( ( ( A e. NN0 /\ ( P ^ ( P pCnt A ) ) e. NN0 ) /\ ( A || ( P ^ ( P pCnt A ) ) /\ ( P ^ ( P pCnt A ) ) || A ) ) -> A = ( P ^ ( P pCnt A ) ) )
67	2, 8, 63, 65, 66	syl22anc 1275	. . 3 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A = ( P ^ ( P pCnt A ) ) )
68	67	rexlimdvaa 2663	. 2 |- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) -> A = ( P ^ ( P pCnt A ) ) ) )
69	3	adantr 276	. . . . . . 7 |- ( ( P e. Prime /\ A e. NN ) -> P e. NN )
70	69, 5	nnexpcld 11082	. . . . . 6 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) e. NN )
71	70	nnzd 9717	. . . . 5 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
72		iddvds 12515	. . . . 5 |- ( ( P ^ ( P pCnt A ) ) e. ZZ -> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) )
73	71, 72	syl 14	. . . 4 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) )
74		oveq2 6066	. . . . . 6 |- ( n = ( P pCnt A ) -> ( P ^ n ) = ( P ^ ( P pCnt A ) ) )
75	74	breq2d 4126	. . . . 5 |- ( n = ( P pCnt A ) -> ( ( P ^ ( P pCnt A ) ) || ( P ^ n ) <-> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) ) )
76	75	rspcev 2923	. . . 4 |- ( ( ( P pCnt A ) e. NN0 /\ ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) ) -> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) )
77	5, 73, 76	syl2anc 411	. . 3 |- ( ( P e. Prime /\ A e. NN ) -> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) )
78		breq1 4117	. . . 4 |- ( A = ( P ^ ( P pCnt A ) ) -> ( A || ( P ^ n ) <-> ( P ^ ( P pCnt A ) ) || ( P ^ n ) ) )
79	78	rexbidv 2545	. . 3 |- ( A = ( P ^ ( P pCnt A ) ) -> ( E. n e. NN0 A || ( P ^ n ) <-> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) ) )
80	77, 79	syl5ibrcom 157	. 2 |- ( ( P e. Prime /\ A e. NN ) -> ( A = ( P ^ ( P pCnt A ) ) -> E. n e. NN0 A || ( P ^ n ) ) )
81	68, 80	impbid 129	1 |- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   E.wrex 2523   class class class wbr 4114  (class class class)co 6058   0cc0 8143   <_ cle 8325   NNcn 9254   NN0cn0 9513   ZZcz 9594   ^cexp 10924   || cdvds 12498   Primecprime 12829   pCnt cpc 13007

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-xnn0 9581  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675  df-prm 12830  df-pc 13008

This theorem is referenced by:  pcprmpw  13057  dvdsprmpweq  13058  dvdsppwf1o  15983