Theorem dvdsppwf1o 15511

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 4051	. . 3 |- ( x = ( P ^ n ) -> ( x || ( P ^ A ) <-> ( P ^ n ) || ( P ^ A ) ) )
3		prmnn 12482	. . . . 5 |- ( P e. Prime -> P e. NN )
4	3	adantr 276	. . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. NN )
5		elfznn0 10249	. . . 4 |- ( n e. ( 0 ... A ) -> n e. NN0 )
6		nnexpcl 10710	. . . 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 12483	. . . . 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 10157	. . . . 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 12222	. . . 4 |- ( ( P e. ZZ /\ n e. NN0 /\ A e. ( ZZ>= ` n ) ) -> ( P ^ n ) || ( P ^ A ) )
14	9, 10, 12, 13	syl3anc 1250	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P ^ n ) || ( P ^ A ) )
15	2, 7, 14	elrabd 2933	. 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 2928	. . . 4 |- ( m e. { x e. NN | x || ( P ^ A ) } -> m e. NN )
18		pccl 12672	. . . 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 9507	. . . . 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 528	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> A e. NN0 )
25		zexpcl 10712	. . . . . 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 4051	. . . . . . . 8 |- ( x = m -> ( x || ( P ^ A ) <-> m || ( P ^ A ) ) )
28	27	elrab 2931	. . . . . . 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 12700	. . . . 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 1252	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) <_ ( P pCnt ( P ^ A ) ) )
33		pcidlem 12696	. . . . 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 4074	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) <_ A )
36		fznn0 10248	. . . 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 947	. 2 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) e. ( 0 ... A ) )
39		oveq2 5962	. . . . . . . . 9 |- ( n = A -> ( P ^ n ) = ( P ^ A ) )
40	39	breq2d 4060	. . . . . . . 8 |- ( n = A -> ( m || ( P ^ n ) <-> m || ( P ^ A ) ) )
41	40	rspcev 2879	. . . . . . 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 12706	. . . . . . 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 5962	. . . . 5 |- ( n = ( P pCnt m ) -> ( P ^ n ) = ( P ^ ( P pCnt m ) ) )
48	47	eqeq2d 2218	. . . 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 10160	. . . . . . 7 |- ( n e. ( 0 ... A ) -> n e. ZZ )
51		pcid 12697	. . . . . . 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 2212	. . . . 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 5962	. . . . 5 |- ( m = ( P ^ n ) -> ( P pCnt m ) = ( P pCnt ( P ^ n ) ) )
56	55	eqeq2d 2218	. . . 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 6161	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 1373   e. wcel 2177   E.wrex 2486   {crab 2489   class class class wbr 4048   |-> cmpt 4110   -1-1-onto->wf1o 5276   ` cfv 5277  (class class class)co 5954   0cc0 7938   <_ cle 8121   NNcn 9049   NN0cn0 9308   ZZcz 9385   ZZ>=cuz 9661   ...cfz 10143   ^cexp 10696   || cdvds 12148   Primecprime 12479   pCnt cpc 12657

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-mulrcl 8037  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-0lt1 8044  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-precex 8048  ax-cnre 8049  ax-pre-ltirr 8050  ax-pre-ltwlin 8051  ax-pre-lttrn 8052  ax-pre-apti 8053  ax-pre-ltadd 8054  ax-pre-mulgt0 8055  ax-pre-mulext 8056  ax-arch 8057  ax-caucvg 8058

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-id 4345  df-po 4348  df-iso 4349  df-iord 4418  df-on 4420  df-ilim 4421  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-isom 5286  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-recs 6401  df-frec 6487  df-1o 6512  df-2o 6513  df-er 6630  df-en 6838  df-sup 7098  df-inf 7099  df-pnf 8122  df-mnf 8123  df-xr 8124  df-ltxr 8125  df-le 8126  df-sub 8258  df-neg 8259  df-reap 8661  df-ap 8668  df-div 8759  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-n0 9309  df-xnn0 9372  df-z 9386  df-uz 9662  df-q 9754  df-rp 9789  df-fz 10144  df-fzo 10278  df-fl 10426  df-mod 10481  df-seqfrec 10606  df-exp 10697  df-cj 11203  df-re 11204  df-im 11205  df-rsqrt 11359  df-abs 11360  df-dvds 12149  df-gcd 12325  df-prm 12480  df-pc 12658

This theorem is referenced by:  sgmppw  15514  0sgmppw  15515