Theorem dvdsppwf1o 15671

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 4086	. . 3 |- ( x = ( P ^ n ) -> ( x || ( P ^ A ) <-> ( P ^ n ) || ( P ^ A ) ) )
3		prmnn 12640	. . . . 5 |- ( P e. Prime -> P e. NN )
4	3	adantr 276	. . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. NN )
5		elfznn0 10318	. . . 4 |- ( n e. ( 0 ... A ) -> n e. NN0 )
6		nnexpcl 10782	. . . 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 12641	. . . . 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 10226	. . . . 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 12380	. . . 4 |- ( ( P e. ZZ /\ n e. NN0 /\ A e. ( ZZ>= ` n ) ) -> ( P ^ n ) || ( P ^ A ) )
14	9, 10, 12, 13	syl3anc 1271	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ n e. ( 0 ... A ) ) -> ( P ^ n ) || ( P ^ A ) )
15	2, 7, 14	elrabd 2961	. 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 2956	. . . 4 |- ( m e. { x e. NN | x || ( P ^ A ) } -> m e. NN )
18		pccl 12830	. . . 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 9576	. . . . 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 10784	. . . . . 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 4086	. . . . . . . 8 |- ( x = m -> ( x || ( P ^ A ) <-> m || ( P ^ A ) ) )
28	27	elrab 2959	. . . . . . 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 12858	. . . . 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 1273	. . . 4 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) <_ ( P pCnt ( P ^ A ) ) )
33		pcidlem 12854	. . . . 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 4109	. . 3 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) <_ A )
36		fznn0 10317	. . . 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 950	. 2 |- ( ( ( P e. Prime /\ A e. NN0 ) /\ m e. { x e. NN | x || ( P ^ A ) } ) -> ( P pCnt m ) e. ( 0 ... A ) )
39		oveq2 6015	. . . . . . . . 9 |- ( n = A -> ( P ^ n ) = ( P ^ A ) )
40	39	breq2d 4095	. . . . . . . 8 |- ( n = A -> ( m || ( P ^ n ) <-> m || ( P ^ A ) ) )
41	40	rspcev 2907	. . . . . . 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 12864	. . . . . . 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 6015	. . . . 5 |- ( n = ( P pCnt m ) -> ( P ^ n ) = ( P ^ ( P pCnt m ) ) )
48	47	eqeq2d 2241	. . . 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 10229	. . . . . . 7 |- ( n e. ( 0 ... A ) -> n e. ZZ )
51		pcid 12855	. . . . . . 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 2235	. . . . 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 6015	. . . . 5 |- ( m = ( P ^ n ) -> ( P pCnt m ) = ( P pCnt ( P ^ n ) ) )
56	55	eqeq2d 2241	. . . 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 6217	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 1395   e. wcel 2200   E.wrex 2509   {crab 2512   class class class wbr 4083   |-> cmpt 4145   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6007   0cc0 8007   <_ cle 8190   NNcn 9118   NN0cn0 9377   ZZcz 9454   ZZ>=cuz 9730   ...cfz 10212   ^cexp 10768   || cdvds 12306   Primecprime 12637   pCnt cpc 12815

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sup 7159  df-inf 7160  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-xnn0 9441  df-z 9455  df-uz 9731  df-q 9823  df-rp 9858  df-fz 10213  df-fzo 10347  df-fl 10498  df-mod 10553  df-seqfrec 10678  df-exp 10769  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518  df-dvds 12307  df-gcd 12483  df-prm 12638  df-pc 12816

This theorem is referenced by:  sgmppw  15674  0sgmppw  15675