Theorem oddpwdc 12896

Description: The function 𝐹 that decomposes a number into its "odd" and "even" parts, which is to say the largest power of two and largest odd divisor of a number, is a bijection from pairs of a nonnegative integer and an odd number to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.)

Hypotheses

Ref	Expression
oddpwdc.j	⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
oddpwdc.f	⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))

Assertion

Ref	Expression
oddpwdc	⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ

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

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

Proof of Theorem oddpwdc

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oddpwdc.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
2		2cnd 9327	. . . . . 6 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) → 2 ∈ ℂ)
3		simpr 110	. . . . . 6 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈ ℕ0)
4	2, 3	expcld 11060	. . . . 5 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
5		breq2 4118	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑥))
6	5	notbid 673	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑥))
7		oddpwdc.j	. . . . . . . . 9 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
8	6, 7	elrab2 2979	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐽 ↔ (𝑥 ∈ ℕ ∧ ¬ 2 ∥ 𝑥))
9	8	simplbi 274	. . . . . . 7 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ∈ ℕ)
10	9	adantr 276	. . . . . 6 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) → 𝑥 ∈ ℕ)
11	10	nncnd 9268	. . . . 5 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) → 𝑥 ∈ ℂ)
12	4, 11	mulcld 8310	. . . 4 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) → ((2↑𝑦) · 𝑥) ∈ ℂ)
13	12	adantl 277	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0)) → ((2↑𝑦) · 𝑥) ∈ ℂ)
14		nnnn0 9520	. . . . . 6 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ0)
15		2nn 9416	. . . . . . 7 ⊢ 2 ∈ ℕ
16		pw2dvdseu 12890	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ → ∃!𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎))
17		riotacl 6027	. . . . . . . 8 ⊢ (∃!𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎) → (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)) ∈ ℕ0)
18	16, 17	syl 14	. . . . . . 7 ⊢ (𝑎 ∈ ℕ → (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)) ∈ ℕ0)
19		nnexpcl 10938	. . . . . . 7 ⊢ ((2 ∈ ℕ ∧ (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)) ∈ ℕ0) → (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎))) ∈ ℕ)
20	15, 18, 19	sylancr 414	. . . . . 6 ⊢ (𝑎 ∈ ℕ → (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎))) ∈ ℕ)
21		nn0nndivcl 9579	. . . . . 6 ⊢ ((𝑎 ∈ ℕ0 ∧ (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎))) ∈ ℕ) → (𝑎 / (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)))) ∈ ℝ)
22	14, 20, 21	syl2anc 411	. . . . 5 ⊢ (𝑎 ∈ ℕ → (𝑎 / (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)))) ∈ ℝ)
23	22, 18	jca 306	. . . 4 ⊢ (𝑎 ∈ ℕ → ((𝑎 / (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)))) ∈ ℝ ∧ (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)) ∈ ℕ0))
24	23	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑎 ∈ ℕ) → ((𝑎 / (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)))) ∈ ℝ ∧ (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)) ∈ ℕ0))
25	8	anbi1i 458	. . . . . 6 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) ↔ ((𝑥 ∈ ℕ ∧ ¬ 2 ∥ 𝑥) ∧ 𝑦 ∈ ℕ0))
26	25	anbi1i 458	. . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) ∧ 𝑎 = ((2↑𝑦) · 𝑥)) ↔ (((𝑥 ∈ ℕ ∧ ¬ 2 ∥ 𝑥) ∧ 𝑦 ∈ ℕ0) ∧ 𝑎 = ((2↑𝑦) · 𝑥)))
27		oddpwdclemdc 12895	. . . . 5 ⊢ ((((𝑥 ∈ ℕ ∧ ¬ 2 ∥ 𝑥) ∧ 𝑦 ∈ ℕ0) ∧ 𝑎 = ((2↑𝑦) · 𝑥)) ↔ (𝑎 ∈ ℕ ∧ (𝑥 = (𝑎 / (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)))) ∧ 𝑦 = (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)))))
28	26, 27	bitri 184	. . . 4 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) ∧ 𝑎 = ((2↑𝑦) · 𝑥)) ↔ (𝑎 ∈ ℕ ∧ (𝑥 = (𝑎 / (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)))) ∧ 𝑦 = (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)))))
29	28	a1i 9	. . 3 ⊢ (⊤ → (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ ℕ0) ∧ 𝑎 = ((2↑𝑦) · 𝑥)) ↔ (𝑎 ∈ ℕ ∧ (𝑥 = (𝑎 / (2↑(℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎)))) ∧ 𝑦 = (℩𝑧 ∈ ℕ0 ((2↑𝑧) ∥ 𝑎 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝑎))))))
30	1, 13, 24, 29	f1od2 6444	. 2 ⊢ (⊤ → 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ)
31	30	mptru 1407	1 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   = wceq 1398  ⊤wtru 1399   ∈ wcel 2205  ∃!wreu 2524  {crab 2526   class class class wbr 4114   × cxp 4752  –1-1-onto→wf1o 5356  ℩crio 6010  (class class class)co 6058   ∈ cmpo 6060  ℂcc 8141  ℝcr 8142  1c1 8144   + caddc 8146   · cmul 8148   / cdiv 8963  ℕcn 9254  2c2 9305  ℕ₀cn0 9513  ↑cexp 10924   ∥ cdvds 12498

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-dvds 12499

This theorem is referenced by:  sqpweven  12897  2sqpwodd  12898  xpnnen  13229