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 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑧}
nnmaxpw.f	⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((𝐵↑𝑦) · 𝑥))

Assertion

Ref	Expression
nnmaxpw	⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ)

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 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((𝐵↑𝑦) · 𝑥))
2		eluz2nn 9975	. . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℕ)
3	2	adantr 276	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0)) → 𝐵 ∈ ℕ)
4	3	nncnd 9320	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0)) → 𝐵 ∈ ℂ)
5		simprr 537	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0)) → 𝑦 ∈ ℕ0)
6	4, 5	expcld 11124	. . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0)) → (𝐵↑𝑦) ∈ ℂ)
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 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑧}
10	8, 9	elrab2 2985	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 ↔ (𝑥 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑥))
11	10	simplbi 274	. . . . 5 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ∈ ℕ)
12	11	ad2antrl 494	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0)) → 𝑥 ∈ ℕ)
13	12	nncnd 9320	. . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0)) → 𝑥 ∈ ℂ)
14	6, 13	mulcld 8346	. 2 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0)) → ((𝐵↑𝑦) · 𝑥) ∈ ℂ)
15		simpl 109	. . . . . 6 ⊢ ((𝑎 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝑎 ∈ ℕ)
16	15	nnnn0d 9624	. . . . 5 ⊢ ((𝑎 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝑎 ∈ ℕ0)
17	2	adantl 277	. . . . . 6 ⊢ ((𝑎 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐵 ∈ ℕ)
18		pwbdvdseu 12963	. . . . . . 7 ⊢ ((𝑎 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → ∃!𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎))
19		riotacl 6054	. . . . . . 7 ⊢ (∃!𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎) → (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎)) ∈ ℕ0)
20	18, 19	syl 14	. . . . . 6 ⊢ ((𝑎 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎)) ∈ ℕ0)
21	17, 20	nnexpcld 11146	. . . . 5 ⊢ ((𝑎 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎))) ∈ ℕ)
22		nn0nndivcl 9633	. . . . 5 ⊢ ((𝑎 ∈ ℕ0 ∧ (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎))) ∈ ℕ) → (𝑎 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎)))) ∈ ℝ)
23	16, 21, 22	syl2anc 415	. . . 4 ⊢ ((𝑎 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → (𝑎 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎)))) ∈ ℝ)
24	23, 20	jca 306	. . 3 ⊢ ((𝑎 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → ((𝑎 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎)))) ∈ ℝ ∧ (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎)) ∈ ℕ0))
25	24	ancoms 268	. 2 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑎 ∈ ℕ) → ((𝑎 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎)))) ∈ ℝ ∧ (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎)) ∈ ℕ0))
26	10	anbi1i 462	. . . 4 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) ↔ ((𝑥 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑥) ∧ 𝑦 ∈ ℕ0))
27	26	anbi1i 462	. . 3 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) ∧ 𝑎 = ((𝐵↑𝑦) · 𝑥)) ↔ (((𝑥 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑥) ∧ 𝑦 ∈ ℕ0) ∧ 𝑎 = ((𝐵↑𝑦) · 𝑥)))
28		nnmaxpwlemparts 12968	. . 3 ⊢ (𝐵 ∈ (ℤ≥‘2) → ((((𝑥 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑥) ∧ 𝑦 ∈ ℕ0) ∧ 𝑎 = ((𝐵↑𝑦) · 𝑥)) ↔ (𝑎 ∈ ℕ ∧ (𝑥 = (𝑎 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎)))) ∧ 𝑦 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎))))))
29	27, 28	bitrid 192	. 2 ⊢ (𝐵 ∈ (ℤ≥‘2) → (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) ∧ 𝑎 = ((𝐵↑𝑦) · 𝑥)) ↔ (𝑎 ∈ ℕ ∧ (𝑥 = (𝑎 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎)))) ∧ 𝑦 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝑎 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝑎))))))
30	1, 14, 25, 29	f1od2 6471	1 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∃!wreu 2530  {crab 2532   class class class wbr 4130   × cxp 4772  –1-1-onto→wf1o 5376  ‘cfv 5377  ℩crio 6037  (class class class)co 6085   ∈ cmpo 6087  ℂcc 8177  ℝcr 8178  1c1 8180   + caddc 8182   · cmul 8184   / cdiv 9004  ℕcn 9306  2c2 9357  ℕ₀cn0 9567  ℤ_≥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