Theorem dvdsppwf1o 16086

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 4131	. . 3 |- ( x = ( P ^ n ) -> ( x || ( P ^ A ) <-> ( P ^ n ) || ( P ^ A ) ) )
3		prmnn 12871	. . . . 5 |- ( P e. Prime -> P e. NN )
4	3	adantr 276	. . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. NN )
5		elfznn0 10504	. . . 4 |- ( n e. ( 0 ... A ) -> n e. NN0 )
6		nnexpcl 10972	. . . 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 12872	. . . . 5 |- ( P e. Prime -> P e. ZZ )
9	8	ad2antrr 492	. . . 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 10408	. . . . 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 12611	. . . 4 |- ( ( P e. ZZ /\ n e. NN0 /\ A e. ( ZZ>= ` n ) ) -> ( P ^ n ) || ( P ^ A ) )
14	9, 10, 12, 13	syl3anc 1278	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P ^ n ) || ( P ^ A ) )
15	2, 7, 14	elrabd 2984	. 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 2979	. . . 4 |- ( m e. { x e. NN | x || ( P ^ A ) } -> m e. NN )
18		pccl 13061	. . . 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 9750	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> m e. ZZ )
23	8	ad2antrr 492	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> P e. ZZ )
24		simplr 533	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> A e. NN0 )
25		zexpcl 10974	. . . . . 6 |- ( ( P e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. ZZ )
26	23, 24, 25	syl2anc 415	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P ^ A ) e. ZZ )
27		breq1 4131	. . . . . . . 8 |- ( x = m -> ( x || ( P ^ A ) <-> m || ( P ^ A ) ) )
28	27	elrab 2982	. . . . . . 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 13089	. . . . 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 1280	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) <_ ( P pCnt ( P ^ A ) ) )
33		pcidlem 13085	. . . . 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 4154	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) <_ A )
36		fznn0 10503	. . . 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 957	. 2 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) e. ( 0 ... A ) )
39		oveq2 6087	. . . . . . . . 9 |- ( n = A -> ( P ^ n ) = ( P ^ A ) )
40	39	breq2d 4140	. . . . . . . 8 |- ( n = A -> ( m || ( P ^ n ) <-> m || ( P ^ A ) ) )
41	40	rspcev 2929	. . . . . . 7 |- ( ( A e. NN0 /\ m || ( P ^ A ) ) -> E. n e. NN0 m || ( P ^ n ) )
42	24, 30, 41	syl2anc 415	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> E. n e. NN0 m || ( P ^ n ) )
43		pcprmpw2 13095	. . . . . . 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 482	. . . 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 6087	. . . . 5 |- ( n = ( P pCnt m ) -> ( P ^ n ) = ( P ^ ( P pCnt m ) ) )
48	47	eqeq2d 2250	. . . 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 10411	. . . . . . 7 |- ( n e. ( 0 ... A ) -> n e. ZZ )
51		pcid 13086	. . . . . . 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 2244	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> n = ( P pCnt ( P ^ n ) ) )
54	53	adantrr 483	. . . 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 6087	. . . . 5 |- ( m = ( P ^ n ) -> ( P pCnt m ) = ( P pCnt ( P ^ n ) ) )
56	55	eqeq2d 2250	. . . 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 6289	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 1402   e. wcel 2209   E.wrex 2529   {crab 2532   class class class wbr 4128   |-> cmpt 4190   -1-1-onto->wf1o 5374   ` cfv 5375  (class class class)co 6079   0cc0 8173   <_ cle 8355   NNcn 9287   NN0cn0 9546   ZZcz 9627   ZZ>=cuz 9904   ...cfz 10394   ^cexp 10958   || cdvds 12537   Primecprime 12868   pCnt cpc 13046

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-2o 6682  df-er 6801  df-en 7017  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-xnn0 9614  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-dvds 12538  df-gcd 12714  df-prm 12869  df-pc 13047

This theorem is referenced by:  sgmppw  16089  0sgmppw  16090