Theorem eulerpartlemgh 34380

Description: Lemma for eulerpart 34384: The 𝐹 function is a bijection on the 𝑈 subsets. (Contributed by Thierry Arnoux, 15-Aug-2018.)

Hypotheses

Ref	Expression
eulerpart.p	⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
eulerpart.o	⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d	⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
eulerpart.j	⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f	⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h	⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m	⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
eulerpart.r	⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
eulerpart.t	⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
eulerpart.g	⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
eulerpartlemgh.1	⊢ 𝑈 = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))

Assertion

Ref	Expression
eulerpartlemgh	⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 ↾ 𝑈):𝑈–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})

Distinct variable groups:   𝑧,𝑡   𝑓,𝑔,𝑘,𝑛,𝑡,𝐴   𝑓,𝐽,𝑛,𝑡   𝑓,𝑁,𝑘,𝑛,𝑡   𝑛,𝑂,𝑡   𝑃,𝑔,𝑘   𝑅,𝑓,𝑘,𝑛,𝑡   𝑇,𝑛,𝑡   𝑥,𝑡,𝑦,𝑧   𝑓,𝑚,𝑥,𝑔,𝑘,𝑛,𝑡,𝐴   𝑛,𝐹,𝑡,𝑥   𝑦,𝑓,𝑛   𝑥,𝐽,𝑦   𝑡,𝑃

Allowed substitution hints:   𝐴(𝑦,𝑧,𝑜,𝑟)   𝐷(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑚,𝑛,𝑜,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑚,𝑛,𝑜,𝑟)   𝑅(𝑥,𝑦,𝑧,𝑔,𝑚,𝑜,𝑟)   𝑇(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑜,𝑟)   𝑈(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑚,𝑛,𝑜,𝑟)   𝐹(𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑜,𝑟)   𝐺(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑚,𝑛,𝑜,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑚,𝑛,𝑜,𝑟)   𝐽(𝑧,𝑔,𝑘,𝑚,𝑜,𝑟)   𝑀(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑚,𝑛,𝑜,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑔,𝑚,𝑜,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑜,𝑟)

Proof of Theorem eulerpartlemgh

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eulerpart.j	. . . . 5 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
2		eulerpart.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
3	1, 2	oddpwdc 34356	. . . 4 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1of1 6847	. . . 4 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:(𝐽 × ℕ0)–1-1→ℕ)
5	3, 4	ax-mp 5	. . 3 ⊢ 𝐹:(𝐽 × ℕ0)–1-1→ℕ
6		eulerpartlemgh.1	. . . 4 ⊢ 𝑈 = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))
7		iunss 5045	. . . . 5 ⊢ (∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
8		inss2 4238	. . . . . . . 8 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
9	8	sseli 3979	. . . . . . 7 ⊢ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) → 𝑡 ∈ 𝐽)
10	9	snssd 4809	. . . . . 6 ⊢ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) → {𝑡} ⊆ 𝐽)
11		bitsss 16463	. . . . . 6 ⊢ (bits‘(𝐴‘𝑡)) ⊆ ℕ0
12		xpss12 5700	. . . . . 6 ⊢ (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴‘𝑡)) ⊆ ℕ0) → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
13	10, 11, 12	sylancl 586	. . . . 5 ⊢ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
14	7, 13	mprgbir 3068	. . . 4 ⊢ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0)
15	6, 14	eqsstri 4030	. . 3 ⊢ 𝑈 ⊆ (𝐽 × ℕ0)
16		f1ores 6862	. . 3 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1→ℕ ∧ 𝑈 ⊆ (𝐽 × ℕ0)) → (𝐹 ↾ 𝑈):𝑈–1-1-onto→(𝐹 “ 𝑈))
17	5, 15, 16	mp2an 692	. 2 ⊢ (𝐹 ↾ 𝑈):𝑈–1-1-onto→(𝐹 “ 𝑈)
18		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) = 𝑝)
19		2nn 12339	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 2 ∈ ℕ)
21	11	sseli 3979	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (bits‘(𝐴‘𝑡)) → 𝑛 ∈ ℕ0)
22	21	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 𝑛 ∈ ℕ0)
23	20, 22	nnexpcld 14284	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑛) ∈ ℕ)
24		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 𝑡 ∈ ℕ)
25	23, 24	nnmulcld 12319	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → ((2↑𝑛) · 𝑡) ∈ ℕ)
26	25	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) ∈ ℕ)
27	18, 26	eqeltrrd 2842	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → 𝑝 ∈ ℕ)
28	27	rexlimdva2 3157	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑝 ∈ ℕ))
29	28	rexlimdva 3155	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑝 ∈ ℕ))
30	29	pm4.71rd 562	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ (𝑝 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)))
31		rex0 4360	. . . . . . . . . . . . . . 15 ⊢ ¬ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝
32		simplr 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → 𝑡 ∈ ℕ)
33		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ 𝑡 ∈ (◡𝐴 “ ℕ))
34		eulerpart.p	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
35		eulerpart.o	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
36		eulerpart.d	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
37		eulerpart.h	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
38		eulerpart.m	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
39		eulerpart.r	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
40		eulerpart.t	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
41	34, 35, 36, 1, 2, 37, 38, 39, 40	eulerpartlemt0 34371	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
42	41	simp1bi 1146	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ (ℕ0 ↑m ℕ))
43		elmapi 8889	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴:ℕ⟶ℕ0)
44	42, 43	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
45	44	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → 𝐴:ℕ⟶ℕ0)
46		ffn 6736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴:ℕ⟶ℕ0 → 𝐴 Fn ℕ)
47		elpreima 7078	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 Fn ℕ → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ)))
48	45, 46, 47	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ)))
49	33, 48	mtbid 324	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ))
50		imnan 399	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑡 ∈ ℕ → ¬ (𝐴‘𝑡) ∈ ℕ) ↔ ¬ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ))
51	49, 50	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (𝑡 ∈ ℕ → ¬ (𝐴‘𝑡) ∈ ℕ))
52	32, 51	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ (𝐴‘𝑡) ∈ ℕ)
53	45, 32	ffvelcdmd 7105	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (𝐴‘𝑡) ∈ ℕ0)
54		elnn0 12528	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴‘𝑡) ∈ ℕ0 ↔ ((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0))
55	53, 54	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0))
56		orel1 889	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝐴‘𝑡) ∈ ℕ → (((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0) → (𝐴‘𝑡) = 0))
57	52, 55, 56	sylc 65	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (𝐴‘𝑡) = 0)
58	57	fveq2d 6910	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (bits‘(𝐴‘𝑡)) = (bits‘0))
59		0bits 16476	. . . . . . . . . . . . . . . . 17 ⊢ (bits‘0) = ∅
60	58, 59	eqtrdi 2793	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (bits‘(𝐴‘𝑡)) = ∅)
61	60	rexeqdv 3327	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝))
62	31, 61	mtbiri 327	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)
63	62	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (¬ 𝑡 ∈ (◡𝐴 “ ℕ) → ¬ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
64	63	con4d 115	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑡 ∈ (◡𝐴 “ ℕ)))
65	64	impr 454	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ (◡𝐴 “ ℕ))
66		eldif 3961	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ 𝐽))
67	34, 35, 36, 1, 2, 37, 38, 39, 40	eulerpartlemf 34372	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴‘𝑡) = 0)
68	66, 67	sylan2br 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ 𝐽)) → (𝐴‘𝑡) = 0)
69	68	anassrs 467	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (𝐴‘𝑡) = 0)
70	69	fveq2d 6910	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (bits‘(𝐴‘𝑡)) = (bits‘0))
71	70, 59	eqtrdi 2793	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (bits‘(𝐴‘𝑡)) = ∅)
72	71	rexeqdv 3327	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝))
73	31, 72	mtbiri 327	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → ¬ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)
74	73	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (¬ 𝑡 ∈ 𝐽 → ¬ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
75	74	con4d 115	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑡 ∈ 𝐽))
76	75	impr 454	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ 𝐽)
77	65, 76	elind 4200	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽))
78		simprr 773	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)
79	77, 78	jca 511	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
80	79	ex 412	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝) → (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)))
81	80	reximdv2 3164	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → ∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
82		ssrab2 4080	. . . . . . . . . 10 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
83	1, 82	eqsstri 4030	. . . . . . . . 9 ⊢ 𝐽 ⊆ ℕ
84	8, 83	sstri 3993	. . . . . . . 8 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ
85		ssrexv 4053	. . . . . . . 8 ⊢ (((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ → (∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
86	84, 85	mp1i 13	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
87	81, 86	impbid 212	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
88	30, 87	bitr3d 281	. . . . 5 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((𝑝 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝) ↔ ∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
89		eqeq2 2749	. . . . . . . 8 ⊢ (𝑚 = 𝑝 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
90	89	2rexbidv 3222	. . . . . . 7 ⊢ (𝑚 = 𝑝 → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
91	90	elrab 3692	. . . . . 6 ⊢ (𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ↔ (𝑝 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
92	91	a1i 11	. . . . 5 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ↔ (𝑝 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)))
93	6	imaeq2i 6076	. . . . . . . . 9 ⊢ (𝐹 “ 𝑈) = (𝐹 “ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))))
94		imaiun 7265	. . . . . . . . 9 ⊢ (𝐹 “ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡))))
95	93, 94	eqtri 2765	. . . . . . . 8 ⊢ (𝐹 “ 𝑈) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡))))
96	95	eleq2i 2833	. . . . . . 7 ⊢ (𝑝 ∈ (𝐹 “ 𝑈) ↔ 𝑝 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))))
97		eliun 4995	. . . . . . 7 ⊢ (𝑝 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))))
98		f1ofn 6849	. . . . . . . . . . . . 13 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹 Fn (𝐽 × ℕ0))
99	3, 98	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝐹 Fn (𝐽 × ℕ0)
100		snssi 4808	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝐽 → {𝑡} ⊆ 𝐽)
101	100, 11, 12	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝐽 → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
102		ovelimab 7611	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn (𝐽 × ℕ0) ∧ ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0)) → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛)))
103	99, 101, 102	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛)))
104		vex 3484	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
105		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (𝑥𝐹𝑛) = (𝑡𝐹𝑛))
106	105	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑝 = (𝑥𝐹𝑛) ↔ 𝑝 = (𝑡𝐹𝑛)))
107	106	rexbidv 3179	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛)))
108	104, 107	rexsn 4682	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛))
109	103, 108	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛)))
110		df-ov 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑡𝐹𝑛) = (𝐹‘⟨𝑡, 𝑛⟩)
111	110	eqeq1i 2742	. . . . . . . . . . . . . 14 ⊢ ((𝑡𝐹𝑛) = 𝑝 ↔ (𝐹‘⟨𝑡, 𝑛⟩) = 𝑝)
112		eqcom 2744	. . . . . . . . . . . . . 14 ⊢ ((𝑡𝐹𝑛) = 𝑝 ↔ 𝑝 = (𝑡𝐹𝑛))
113	111, 112	bitr3i 277	. . . . . . . . . . . . 13 ⊢ ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝 ↔ 𝑝 = (𝑡𝐹𝑛))
114		opelxpi 5722	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ0) → ⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0))
115	1, 2	oddpwdcv 34357	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
116		vex 3484	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑛 ∈ V
117	104, 116	op2nd 8023	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
118	117	oveq2i 7442	. . . . . . . . . . . . . . . . 17 ⊢ (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
119	104, 116	op1st 8022	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
120	118, 119	oveq12i 7443	. . . . . . . . . . . . . . . 16 ⊢ ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
121	115, 120	eqtrdi 2793	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
122	114, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
123	122	eqeq1d 2739	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ0) → ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
124	113, 123	bitr3id 285	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ0) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
125	21, 124	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
126	125	rexbidva 3177	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝐽 → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
127	109, 126	bitrd 279	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
128	9, 127	syl 17	. . . . . . . 8 ⊢ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
129	128	rexbiia 3092	. . . . . . 7 ⊢ (∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)
130	96, 97, 129	3bitri 297	. . . . . 6 ⊢ (𝑝 ∈ (𝐹 “ 𝑈) ↔ ∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)
131	130	a1i 11	. . . . 5 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑝 ∈ (𝐹 “ 𝑈) ↔ ∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
132	88, 92, 131	3bitr4rd 312	. . . 4 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑝 ∈ (𝐹 “ 𝑈) ↔ 𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
133	132	eqrdv 2735	. . 3 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 “ 𝑈) = {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
134		f1oeq3 6838	. . 3 ⊢ ((𝐹 “ 𝑈) = {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → ((𝐹 ↾ 𝑈):𝑈–1-1-onto→(𝐹 “ 𝑈) ↔ (𝐹 ↾ 𝑈):𝑈–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
135	133, 134	syl 17	. 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((𝐹 ↾ 𝑈):𝑈–1-1-onto→(𝐹 “ 𝑈) ↔ (𝐹 ↾ 𝑈):𝑈–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
136	17, 135	mpbii 233	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 ↾ 𝑈):𝑈–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  {crab 3436   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {csn 4626  ⟨cop 4632  ∪ ciun 4991   class class class wbr 5143  {copab 5205   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684   ↾ cres 5687   “ cima 5688   ∘ ccom 5689   Fn wfn 6556  ⟶wf 6557  –1-1→wf1 6558  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  1^st c1st 8012  2^nd c2nd 8013   supp csupp 8185   ↑_m cmap 8866  Fincfn 8985  0cc0 11155  1c1 11156   · cmul 11160   ≤ cle 11296  ℕcn 12266  2c2 12321  ℕ₀cn0 12526  ↑cexp 14102  Σcsu 15722   ∥ cdvds 16290  bitscbits 16456  𝟭cind 32835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-dvds 16291  df-bits 16459

This theorem is referenced by:  eulerpartlemgs2  34382