Theorem dvdsppwf1o 15969

Description: A bijection between the divisors of a prime power and the integers less than or equal to the exponent. (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		breq1 4117	. . 3 |- ( x = ( P ^ n ) -> ( x || ( P ^ A ) <-> ( P ^ n ) || ( P ^ A ) ) )
3		prmnn 12832	. . . . 5 |- ( P e. Prime -> P e. NN )
4	3	adantr 276	. . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. NN )
5		elfznn0 10470	. . . 4 |- ( n e. ( 0 ... A ) -> n e. NN0 )
6		nnexpcl 10938	. . . 4 |- ( ( P e. NN /\ n e. NN0 ) -> ( P ^ n ) e. NN )
7	4, 5, 6	syl2an 289	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P ^ n ) e. NN )
8		prmz 12833	. . . . 5 |- ( P e. Prime -> P e. ZZ )
9	8	ad2antrr 488	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> P e. ZZ )
10	5	adantl 277	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> n e. NN0 )
11		elfzuz3 10375	. . . . 5 |- ( n e. ( 0 ... A ) -> A e. ( ZZ>= ` n ) )
12	11	adantl 277	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> A e. ( ZZ>= ` n ) )
13		dvdsexp 12572	. . . 4 |- ( ( P e. ZZ /\ n e. NN0 /\ A e. ( ZZ>= ` n ) ) -> ( P ^ n ) || ( P ^ A ) )
14	9, 10, 12, 13	syl3anc 1274	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P ^ n ) || ( P ^ A ) )
15	2, 7, 14	elrabd 2978	. 2 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P ^ n ) e. { x e. NN | x || ( P ^ A ) } )
16		simpl 109	. . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. Prime )
17		elrabi 2973	. . . 4 |- ( m e. { x e. NN | x || ( P ^ A ) } -> m e. NN )
18		pccl 13022	. . . 4 |- ( ( P e. Prime /\ m e. NN ) -> ( P pCnt m ) e. NN0 )
19	16, 17, 18	syl2an 289	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) e. NN0 )
20	16	adantr 276	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> P e. Prime )
21	17	adantl 277	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> m e. NN )
22	21	nnzd 9717	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> m e. ZZ )
23	8	ad2antrr 488	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> P e. ZZ )
24		simplr 529	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> A e. NN0 )
25		zexpcl 10940	. . . . . 6 |- ( ( P e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. ZZ )
26	23, 24, 25	syl2anc 411	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P ^ A ) e. ZZ )
27		breq1 4117	. . . . . . . 8 |- ( x = m -> ( x || ( P ^ A ) <-> m || ( P ^ A ) ) )
28	27	elrab 2976	. . . . . . 7 |- ( m e. { x e. NN | x || ( P ^ A ) } <-> ( m e. NN /\ m || ( P ^ A ) ) )
29	28	simprbi 275	. . . . . 6 |- ( m e. { x e. NN | x || ( P ^ A ) } -> m || ( P ^ A ) )
30	29	adantl 277	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> m || ( P ^ A ) )
31		pcdvdstr 13050	. . . . 5 |- ( ( P e. Prime /\ ( m e. ZZ /\ ( P ^ A ) e. ZZ /\ m || ( P ^ A ) ) ) -> ( P pCnt m ) <_ ( P pCnt ( P ^ A ) ) )
32	20, 22, 26, 30, 31	syl13anc 1276	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) <_ ( P pCnt ( P ^ A ) ) )
33		pcidlem 13046	. . . . 5 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) = A )
34	33	adantr 276	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt ( P ^ A ) ) = A )
35	32, 34	breqtrd 4140	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) <_ A )
36		fznn0 10469	. . . 4 |- ( A e. NN0 -> ( ( P pCnt m ) e. ( 0 ... A ) <-> ( ( P pCnt m ) e. NN0 /\ ( P pCnt m ) <_ A ) ) )
37	24, 36	syl 14	. . 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 ) ) )
38	19, 35, 37	mpbir2and 953	. 2 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) e. ( 0 ... A ) )
39		oveq2 6066	. . . . . . . . 9 |- ( n = A -> ( P ^ n ) = ( P ^ A ) )
40	39	breq2d 4126	. . . . . . . 8 |- ( n = A -> ( m || ( P ^ n ) <-> m || ( P ^ A ) ) )
41	40	rspcev 2923	. . . . . . 7 |- ( ( A e. NN0 /\ m || ( P ^ A ) ) -> E. n e. NN0 m || ( P ^ n ) )
42	24, 30, 41	syl2anc 411	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> E. n e. NN0 m || ( P ^ n ) )
43		pcprmpw2 13056	. . . . . . 7 |- ( ( P e. Prime /\ m e. NN ) -> ( E. n e. NN0 m || ( P ^ n ) <-> m = ( P ^ ( P pCnt m ) ) ) )
44	16, 17, 43	syl2an 289	. . . . . 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 ) ) ) )
45	42, 44	mpbid 147	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> m = ( P ^ ( P pCnt m ) ) )
46	45	adantrl 478	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ ( n e. ( 0 ... A ) /\ m e. { x e. NN | x || ( P ^ A ) } ) ) -> m = ( P ^ ( P pCnt m ) ) )
47		oveq2 6066	. . . . 5 |- ( n = ( P pCnt m ) -> ( P ^ n ) = ( P ^ ( P pCnt m ) ) )
48	47	eqeq2d 2246	. . . 4 |- ( n = ( P pCnt m ) -> ( m = ( P ^ n ) <-> m = ( P ^ ( P pCnt m ) ) ) )
49	46, 48	syl5ibrcom 157	. . 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 ) ) )
50		elfzelz 10378	. . . . . . 7 |- ( n e. ( 0 ... A ) -> n e. ZZ )
51		pcid 13047	. . . . . . 7 |- ( ( P e. Prime /\ n e. ZZ ) -> ( P pCnt ( P ^ n ) ) = n )
52	16, 50, 51	syl2an 289	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P pCnt ( P ^ n ) ) = n )
53	52	eqcomd 2240	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> n = ( P pCnt ( P ^ n ) ) )
54	53	adantrr 479	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ ( n e. ( 0 ... A ) /\ m e. { x e. NN | x || ( P ^ A ) } ) ) -> n = ( P pCnt ( P ^ n ) ) )
55		oveq2 6066	. . . . 5 |- ( m = ( P ^ n ) -> ( P pCnt m ) = ( P pCnt ( P ^ n ) ) )
56	55	eqeq2d 2246	. . . 4 |- ( m = ( P ^ n ) -> ( n = ( P pCnt m ) <-> n = ( P pCnt ( P ^ n ) ) ) )
57	54, 56	syl5ibrcom 157	. . 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 ) ) )
58	49, 57	impbid 129	. 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 ) ) )
59	1, 15, 38, 58	f1o2d 6268	1 |- ( ( P e. Prime /\ A e. NN0 ) -> F : ( 0 ... A ) -1-1-onto-> { x e. NN | x || ( P ^ A ) } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   E.wrex 2523   {crab 2526   class class class wbr 4114   |-> cmpt 4176   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   0cc0 8143   <_ cle 8325   NNcn 9254   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   ^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:  sgmppw  15972  0sgmppw  15973