Theorem nnmaxpw 12969

Description: The function F that decomposes a number into its "odd" and "even" parts, which is to say the largest power of a base and largest divisor of the number not divisible by that base, is a bijection from pairs of a nonnegative integer and a number not divisible by that base to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.) (Revised by Jim Kingdon, 19-Aug-2026.)

Hypotheses

Ref	Expression
nnmaxpw.j	|- J = { z e. NN | -. B || z }
nnmaxpw.f	|- F = ( x e. J , y e. NN0 |-> ( ( B ^ y ) x. x ) )

Assertion

Ref	Expression
nnmaxpw	|- ( B e. ( ZZ>= ` 2 ) -> F : ( J X. NN0 ) -1-1-onto-> NN )

Distinct variable groups:   x, y, z   x, J, y   x, B, y, z

Allowed substitution hints:   F( x, y, z)   J( z)

Proof of Theorem nnmaxpw

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnmaxpw.f	. 2 |- F = ( x e. J , y e. NN0 |-> ( ( B ^ y ) x. x ) )
2		eluz2nn 9975	. . . . . 6 |- ( B e. ( ZZ>= ` 2 ) -> B e. NN )
3	2	adantr 276	. . . . 5 |- ( ( B e. ( ZZ>= ` 2 ) /\ ( x e. J /\ y e. NN0 ) ) -> B e. NN )
4	3	nncnd 9320	. . . 4 |- ( ( B e. ( ZZ>= ` 2 ) /\ ( x e. J /\ y e. NN0 ) ) -> B e. CC )
5		simprr 537	. . . 4 |- ( ( B e. ( ZZ>= ` 2 ) /\ ( x e. J /\ y e. NN0 ) ) -> y e. NN0 )
6	4, 5	expcld 11124	. . 3 |- ( ( B e. ( ZZ>= ` 2 ) /\ ( x e. J /\ y e. NN0 ) ) -> ( B ^ y ) e. CC )
7		breq2 4134	. . . . . . . 8 |- ( z = x -> ( B || z <-> B || x ) )
8	7	notbid 677	. . . . . . 7 |- ( z = x -> ( -. B || z <-> -. B || x ) )
9		nnmaxpw.j	. . . . . . 7 |- J = { z e. NN | -. B || z }
10	8, 9	elrab2 2985	. . . . . 6 |- ( x e. J <-> ( x e. NN /\ -. B || x ) )
11	10	simplbi 274	. . . . 5 |- ( x e. J -> x e. NN )
12	11	ad2antrl 494	. . . 4 |- ( ( B e. ( ZZ>= ` 2 ) /\ ( x e. J /\ y e. NN0 ) ) -> x e. NN )
13	12	nncnd 9320	. . 3 |- ( ( B e. ( ZZ>= ` 2 ) /\ ( x e. J /\ y e. NN0 ) ) -> x e. CC )
14	6, 13	mulcld 8346	. 2 |- ( ( B e. ( ZZ>= ` 2 ) /\ ( x e. J /\ y e. NN0 ) ) -> ( ( B ^ y ) x. x ) e. CC )
15		simpl 109	. . . . . 6 |- ( ( a e. NN /\ B e. ( ZZ>= ` 2 ) ) -> a e. NN )
16	15	nnnn0d 9624	. . . . 5 |- ( ( a e. NN /\ B e. ( ZZ>= ` 2 ) ) -> a e. NN0 )
17	2	adantl 277	. . . . . 6 |- ( ( a e. NN /\ B e. ( ZZ>= ` 2 ) ) -> B e. NN )
18		pwbdvdseu 12963	. . . . . . 7 |- ( ( a e. NN /\ B e. ( ZZ>= ` 2 ) ) -> E! z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) )
19		riotacl 6054	. . . . . . 7 |- ( E! z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) -> ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) e. NN0 )
20	18, 19	syl 14	. . . . . 6 |- ( ( a e. NN /\ B e. ( ZZ>= ` 2 ) ) -> ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) e. NN0 )
21	17, 20	nnexpcld 11146	. . . . 5 |- ( ( a e. NN /\ B e. ( ZZ>= ` 2 ) ) -> ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) ) e. NN )
22		nn0nndivcl 9633	. . . . 5 |- ( ( a e. NN0 /\ ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) ) e. NN ) -> ( a / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) ) ) e. RR )
23	16, 21, 22	syl2anc 415	. . . 4 |- ( ( a e. NN /\ B e. ( ZZ>= ` 2 ) ) -> ( a / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) ) ) e. RR )
24	23, 20	jca 306	. . 3 |- ( ( a e. NN /\ B e. ( ZZ>= ` 2 ) ) -> ( ( a / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) ) ) e. RR /\ ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) e. NN0 ) )
25	24	ancoms 268	. 2 |- ( ( B e. ( ZZ>= ` 2 ) /\ a e. NN ) -> ( ( a / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) ) ) e. RR /\ ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) e. NN0 ) )
26	10	anbi1i 462	. . . 4 |- ( ( x e. J /\ y e. NN0 ) <-> ( ( x e. NN /\ -. B || x ) /\ y e. NN0 ) )
27	26	anbi1i 462	. . 3 |- ( ( ( x e. J /\ y e. NN0 ) /\ a = ( ( B ^ y ) x. x ) ) <-> ( ( ( x e. NN /\ -. B || x ) /\ y e. NN0 ) /\ a = ( ( B ^ y ) x. x ) ) )
28		nnmaxpwlemparts 12968	. . 3 |- ( B e. ( ZZ>= ` 2 ) -> ( ( ( ( x e. NN /\ -. B || x ) /\ y e. NN0 ) /\ a = ( ( B ^ y ) x. x ) ) <-> ( a e. NN /\ ( x = ( a / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) ) ) /\ y = ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) ) ) ) )
29	27, 28	bitrid 192	. 2 |- ( B e. ( ZZ>= ` 2 ) -> ( ( ( x e. J /\ y e. NN0 ) /\ a = ( ( B ^ y ) x. x ) ) <-> ( a e. NN /\ ( x = ( a / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) ) ) /\ y = ( iota_ z e. NN0 ( ( B ^ z ) || a /\ -. ( B ^ ( z + 1 ) ) || a ) ) ) ) ) )
30	1, 14, 25, 29	f1od2 6471	1 |- ( B e. ( ZZ>= ` 2 ) -> F : ( J X. NN0 ) -1-1-onto-> NN )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E!wreu 2530   {crab 2532   class class class wbr 4130   X. cxp 4772   -1-1-onto->wf1o 5376   ` cfv 5377   iota_crio 6037  (class class class)co 6085   e. cmpo 6087   CCcc 8177   RRcr 8178   1c1 8180   + caddc 8182   x. cmul 8184   / cdiv 9004   NNcn 9306   2c2 9357   NN0cn0 9567   ZZ>=cuz 9930   ^cexp 10988   || cdvds 12570

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-fz 10422  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-dvds 12571

This theorem is used by:  oddpwdc  12970  zprmlogbaplem3  16136