Theorem dvdsppwf1o 24912

Description: A bijection from the divisors of a prime power to the integers less than the prime count. (Contributed by Mario Carneiro, 5-May-2016.)

Hypothesis

Ref	Expression
dvdsppwf1o.f	|- F = ( n e. ( 0 ... A ) |-> ( P ^ n ) )

Assertion

Ref	Expression
dvdsppwf1o	|- ( ( P e. Prime /\ A e. NN0 ) -> F : ( 0 ... A ) -1-1-onto-> { x e. NN | x || ( P ^ A ) } )

Distinct variable groups:   x, n, A   P, n, x

Allowed substitution hints:   F( x, n)

Proof of Theorem dvdsppwf1o

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsppwf1o.f	. 2 |- F = ( n e. ( 0 ... A ) |-> ( P ^ n ) )
2		prmnn 15388	. . . . 5 |- ( P e. Prime -> P e. NN )
3	2	adantr 481	. . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. NN )
4		elfznn0 12433	. . . 4 |- ( n e. ( 0 ... A ) -> n e. NN0 )
5		nnexpcl 12873	. . . 4 |- ( ( P e. NN /\ n e. NN0 ) -> ( P ^ n ) e. NN )
6	3, 4, 5	syl2an 494	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P ^ n ) e. NN )
7		prmz 15389	. . . . 5 |- ( P e. Prime -> P e. ZZ )
8	7	ad2antrr 762	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> P e. ZZ )
9	4	adantl 482	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> n e. NN0 )
10		elfzuz3 12339	. . . . 5 |- ( n e. ( 0 ... A ) -> A e. ( ZZ>= ` n ) )
11	10	adantl 482	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> A e. ( ZZ>= ` n ) )
12		dvdsexp 15049	. . . 4 |- ( ( P e. ZZ /\ n e. NN0 /\ A e. ( ZZ>= ` n ) ) -> ( P ^ n ) || ( P ^ A ) )
13	8, 9, 11, 12	syl3anc 1326	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P ^ n ) || ( P ^ A ) )
14		breq1 4656	. . . 4 |- ( x = ( P ^ n ) -> ( x || ( P ^ A ) <-> ( P ^ n ) || ( P ^ A ) ) )
15	14	elrab 3363	. . 3 |- ( ( P ^ n ) e. { x e. NN | x || ( P ^ A ) } <-> ( ( P ^ n ) e. NN /\ ( P ^ n ) || ( P ^ A ) ) )
16	6, 13, 15	sylanbrc 698	. 2 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P ^ n ) e. { x e. NN | x || ( P ^ A ) } )
17		simpl 473	. . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. Prime )
18		elrabi 3359	. . . 4 |- ( m e. { x e. NN | x || ( P ^ A ) } -> m e. NN )
19		pccl 15554	. . . 4 |- ( ( P e. Prime /\ m e. NN ) -> ( P pCnt m ) e. NN0 )
20	17, 18, 19	syl2an 494	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) e. NN0 )
21	17	adantr 481	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> P e. Prime )
22	18	adantl 482	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> m e. NN )
23	22	nnzd 11481	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> m e. ZZ )
24	7	ad2antrr 762	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> P e. ZZ )
25		simplr 792	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> A e. NN0 )
26		zexpcl 12875	. . . . . 6 |- ( ( P e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. ZZ )
27	24, 25, 26	syl2anc 693	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P ^ A ) e. ZZ )
28		breq1 4656	. . . . . . . 8 |- ( x = m -> ( x || ( P ^ A ) <-> m || ( P ^ A ) ) )
29	28	elrab 3363	. . . . . . 7 |- ( m e. { x e. NN | x || ( P ^ A ) } <-> ( m e. NN /\ m || ( P ^ A ) ) )
30	29	simprbi 480	. . . . . 6 |- ( m e. { x e. NN | x || ( P ^ A ) } -> m || ( P ^ A ) )
31	30	adantl 482	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> m || ( P ^ A ) )
32		pcdvdstr 15580	. . . . 5 |- ( ( P e. Prime /\ ( m e. ZZ /\ ( P ^ A ) e. ZZ /\ m || ( P ^ A ) ) ) -> ( P pCnt m ) <_ ( P pCnt ( P ^ A ) ) )
33	21, 23, 27, 31, 32	syl13anc 1328	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) <_ ( P pCnt ( P ^ A ) ) )
34		pcidlem 15576	. . . . 5 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) = A )
35	34	adantr 481	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt ( P ^ A ) ) = A )
36	33, 35	breqtrd 4679	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) <_ A )
37		fznn0 12432	. . . 4 |- ( A e. NN0 -> ( ( P pCnt m ) e. ( 0 ... A ) <-> ( ( P pCnt m ) e. NN0 /\ ( P pCnt m ) <_ A ) ) )
38	25, 37	syl 17	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( ( P pCnt m ) e. ( 0 ... A ) <-> ( ( P pCnt m ) e. NN0 /\ ( P pCnt m ) <_ A ) ) )
39	20, 36, 38	mpbir2and 957	. 2 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) e. ( 0 ... A ) )
40		oveq2 6658	. . . . . . . . 9 |- ( n = A -> ( P ^ n ) = ( P ^ A ) )
41	40	breq2d 4665	. . . . . . . 8 |- ( n = A -> ( m || ( P ^ n ) <-> m || ( P ^ A ) ) )
42	41	rspcev 3309	. . . . . . 7 |- ( ( A e. NN0 /\ m || ( P ^ A ) ) -> E. n e. NN0 m || ( P ^ n ) )
43	25, 31, 42	syl2anc 693	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> E. n e. NN0 m || ( P ^ n ) )
44		pcprmpw2 15586	. . . . . . 7 |- ( ( P e. Prime /\ m e. NN ) -> ( E. n e. NN0 m || ( P ^ n ) <-> m = ( P ^ ( P pCnt m ) ) ) )
45	17, 18, 44	syl2an 494	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( E. n e. NN0 m || ( P ^ n ) <-> m = ( P ^ ( P pCnt m ) ) ) )
46	43, 45	mpbid 222	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> m = ( P ^ ( P pCnt m ) ) )
47	46	adantrl 752	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ ( n e. ( 0 ... A ) /\ m e. { x e. NN | x || ( P ^ A ) } ) ) -> m = ( P ^ ( P pCnt m ) ) )
48		oveq2 6658	. . . . 5 |- ( n = ( P pCnt m ) -> ( P ^ n ) = ( P ^ ( P pCnt m ) ) )
49	48	eqeq2d 2632	. . . 4 |- ( n = ( P pCnt m ) -> ( m = ( P ^ n ) <-> m = ( P ^ ( P pCnt m ) ) ) )
50	47, 49	syl5ibrcom 237	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ ( n e. ( 0 ... A ) /\ m e. { x e. NN | x || ( P ^ A ) } ) ) -> ( n = ( P pCnt m ) -> m = ( P ^ n ) ) )
51		elfzelz 12342	. . . . . . 7 |- ( n e. ( 0 ... A ) -> n e. ZZ )
52		pcid 15577	. . . . . . 7 |- ( ( P e. Prime /\ n e. ZZ ) -> ( P pCnt ( P ^ n ) ) = n )
53	17, 51, 52	syl2an 494	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P pCnt ( P ^ n ) ) = n )
54	53	eqcomd 2628	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> n = ( P pCnt ( P ^ n ) ) )
55	54	adantrr 753	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ ( n e. ( 0 ... A ) /\ m e. { x e. NN | x || ( P ^ A ) } ) ) -> n = ( P pCnt ( P ^ n ) ) )
56		oveq2 6658	. . . . 5 |- ( m = ( P ^ n ) -> ( P pCnt m ) = ( P pCnt ( P ^ n ) ) )
57	56	eqeq2d 2632	. . . 4 |- ( m = ( P ^ n ) -> ( n = ( P pCnt m ) <-> n = ( P pCnt ( P ^ n ) ) ) )
58	55, 57	syl5ibrcom 237	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ ( n e. ( 0 ... A ) /\ m e. { x e. NN | x || ( P ^ A ) } ) ) -> ( m = ( P ^ n ) -> n = ( P pCnt m ) ) )
59	50, 58	impbid 202	. 2 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ ( n e. ( 0 ... A ) /\ m e. { x e. NN | x || ( P ^ A ) } ) ) -> ( n = ( P pCnt m ) <-> m = ( P ^ n ) ) )
60	1, 16, 39, 59	f1o2d 6887	1 |- ( ( P e. Prime /\ A e. NN0 ) -> F : ( 0 ... A ) -1-1-onto-> { x e. NN | x || ( P ^ A ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   class class class wbr 4653   |-> cmpt 4729   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   ^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:  sgmppw  24922  0sgmppw  24923  dchrisum0flblem1  25197