Theorem pcprmpw2 15586

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 792	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. NN )
2	1	nnnn0d 11351	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. NN0 )
3		prmnn 15388	. . . . . . 7 |- ( P e. Prime -> P e. NN )
4	3	ad2antrr 762	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> P e. NN )
5		pccl 15554	. . . . . . 7 |- ( ( P e. Prime /\ A e. NN ) -> ( P pCnt A ) e. NN0 )
6	5	adantr 481	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. NN0 )
7	4, 6	nnexpcld 13030	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. NN )
8	7	nnnn0d 11351	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. NN0 )
9	6	nn0red 11352	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. RR )
10	9	leidd 10594	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) <_ ( P pCnt A ) )
11		simpll 790	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> P e. Prime )
12	6	nn0zd 11480	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. ZZ )
13		pcid 15577	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( P pCnt A ) e. ZZ ) -> ( P pCnt ( P ^ ( P pCnt A ) ) ) = ( P pCnt A ) )
14	11, 12, 13	syl2anc 693	. . . . . . . . . 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 4681	. . . . . . . . 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 762	. . . . . . . 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 477	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> p = P )
18	17	oveq1d 6665	. . . . . . . 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 6665	. . . . . . . 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 4685	. . . . . . 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 801	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A || ( P ^ n ) )
22		prmz 15389	. . . . . . . . . . . . . . 15 |- ( p e. Prime -> p e. ZZ )
23	22	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> p e. ZZ )
24	1	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A e. NN )
25	24	nnzd 11481	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A e. ZZ )
26		simprl 794	. . . . . . . . . . . . . . . . 17 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> n e. NN0 )
27	4, 26	nnexpcld 13030	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ n ) e. NN )
28	27	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( P ^ n ) e. NN )
29	28	nnzd 11481	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( P ^ n ) e. ZZ )
30		dvdstr 15018	. . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . 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 710	. . . . . . . . . . . 12 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || A -> p || ( P ^ n ) ) )
33		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> p e. Prime )
34	11	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> P e. Prime )
35		simplrl 800	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> n e. NN0 )
36		prmdvdsexpr 15429	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ P e. Prime /\ n e. NN0 ) -> ( p || ( P ^ n ) -> p = P ) )
37	33, 34, 35, 36	syl3anc 1326	. . . . . . . . . . . 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 47	. . . . . . . . . . 11 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || A -> p = P ) )
39	38	necon3ad 2807	. . . . . . . . . 10 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p =/= P -> -. p || A ) )
40	39	imp 445	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> -. p || A )
41		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> p e. Prime )
42	1	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> A e. NN )
43		pceq0 15575	. . . . . . . . . 10 |- ( ( p e. Prime /\ A e. NN ) -> ( ( p pCnt A ) = 0 <-> -. p || A ) )
44	41, 42, 43	syl2anc 693	. . . . . . . . 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 247	. . . . . . . 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 762	. . . . . . . . . 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 15555	. . . . . . . . 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 11354	. . . . . . . 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 4675	. . . . . . 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	20, 49	pm2.61dane 2881	. . . . . 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 ) ) ) )
51	50	ralrimiva 2966	. . . . 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 ) ) ) )
52	1	nnzd 11481	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. ZZ )
53	7	nnzd 11481	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
54		pc2dvds 15583	. . . . . 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 ) ) ) ) )
55	52, 53, 54	syl2anc 693	. . . . 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 ) ) ) ) )
56	51, 55	mpbird 247	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A || ( P ^ ( P pCnt A ) ) )
57		pcdvds 15568	. . . . 5 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || A )
58	57	adantr 481	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) || A )
59		dvdseq 15036	. . . 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 ) ) )
60	2, 8, 56, 58, 59	syl22anc 1327	. . 3 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A = ( P ^ ( P pCnt A ) ) )
61	60	rexlimdvaa 3032	. 2 |- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) -> A = ( P ^ ( P pCnt A ) ) ) )
62	3	adantr 481	. . . . . . 7 |- ( ( P e. Prime /\ A e. NN ) -> P e. NN )
63	62, 5	nnexpcld 13030	. . . . . 6 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) e. NN )
64	63	nnzd 11481	. . . . 5 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
65		iddvds 14995	. . . . 5 |- ( ( P ^ ( P pCnt A ) ) e. ZZ -> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) )
66	64, 65	syl 17	. . . 4 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) )
67		oveq2 6658	. . . . . 6 |- ( n = ( P pCnt A ) -> ( P ^ n ) = ( P ^ ( P pCnt A ) ) )
68	67	breq2d 4665	. . . . 5 |- ( n = ( P pCnt A ) -> ( ( P ^ ( P pCnt A ) ) || ( P ^ n ) <-> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) ) )
69	68	rspcev 3309	. . . 4 |- ( ( ( P pCnt A ) e. NN0 /\ ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) ) -> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) )
70	5, 66, 69	syl2anc 693	. . 3 |- ( ( P e. Prime /\ A e. NN ) -> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) )
71		breq1 4656	. . . 4 |- ( A = ( P ^ ( P pCnt A ) ) -> ( A || ( P ^ n ) <-> ( P ^ ( P pCnt A ) ) || ( P ^ n ) ) )
72	71	rexbidv 3052	. . 3 |- ( A = ( P ^ ( P pCnt A ) ) -> ( E. n e. NN0 A || ( P ^ n ) <-> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) ) )
73	70, 72	syl5ibrcom 237	. 2 |- ( ( P e. Prime /\ A e. NN ) -> ( A = ( P ^ ( P pCnt A ) ) -> E. n e. NN0 A || ( P ^ n ) ) )
74	61, 73	impbid 202	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   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   0cc0 9936   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZcz 11377   ^cexp 12860   || cdvds 14983   Primecprime 15385   pCnt cpc 15541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542

This theorem is referenced by:  pcprmpw  15587  dvdsprmpweq  15588  pgpfi1  18010  pgpfi  18020  sylow2alem2  18033  lt6abl  18296  pgpfac1lem3a  18475  dvdsppwf1o  24912