Theorem eulerpartlemgh 32978

Description: Lemma for eulerpart 32982: 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 32954	. . . 4 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1of1 6783	. . . 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 5005	. . . . 5 ⊢ (∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
8		inss2 4189	. . . . . . . 8 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
9	8	sseli 3940	. . . . . . 7 ⊢ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) → 𝑡 ∈ 𝐽)
10	9	snssd 4769	. . . . . 6 ⊢ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) → {𝑡} ⊆ 𝐽)
11		bitsss 16306	. . . . . 6 ⊢ (bits‘(𝐴‘𝑡)) ⊆ ℕ0
12		xpss12 5648	. . . . . 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 3071	. . . 4 ⊢ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0)
15	6, 14	eqsstri 3978	. . 3 ⊢ 𝑈 ⊆ (𝐽 × ℕ0)
16		f1ores 6798	. . 3 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1→ℕ ∧ 𝑈 ⊆ (𝐽 × ℕ0)) → (𝐹 ↾ 𝑈):𝑈–1-1-onto→(𝐹 “ 𝑈))
17	5, 15, 16	mp2an 690	. 2 ⊢ (𝐹 ↾ 𝑈):𝑈–1-1-onto→(𝐹 “ 𝑈)
18		simpr 485	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) = 𝑝)
19		2nn 12226	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 2 ∈ ℕ)
21	11	sseli 3940	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (bits‘(𝐴‘𝑡)) → 𝑛 ∈ ℕ0)
22	21	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 𝑛 ∈ ℕ0)
23	20, 22	nnexpcld 14148	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑛) ∈ ℕ)
24		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 𝑡 ∈ ℕ)
25	23, 24	nnmulcld 12206	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → ((2↑𝑛) · 𝑡) ∈ ℕ)
26	25	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) ∈ ℕ)
27	18, 26	eqeltrrd 2839	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → 𝑝 ∈ ℕ)
28	27	rexlimdva2 3154	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑝 ∈ ℕ))
29	28	rexlimdva 3152	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑝 ∈ ℕ))
30	29	pm4.71rd 563	. . . . . 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 4317	. . . . . . . . . . . . . . 15 ⊢ ¬ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝
32		simplr 767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → 𝑡 ∈ ℕ)
33		simpr 485	. . . . . . . . . . . . . . . . . . . . . 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 32969	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
42	41	simp1bi 1145	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ (ℕ0 ↑m ℕ))
43		elmapi 8787	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴:ℕ⟶ℕ0)
44	42, 43	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
45	44	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → 𝐴:ℕ⟶ℕ0)
46		ffn 6668	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴:ℕ⟶ℕ0 → 𝐴 Fn ℕ)
47		elpreima 7008	. . . . . . . . . . . . . . . . . . . . . . 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 323	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ))
50		imnan 400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑡 ∈ ℕ → ¬ (𝐴‘𝑡) ∈ ℕ) ↔ ¬ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ))
51	49, 50	sylibr 233	. . . . . . . . . . . . . . . . . . . 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 7036	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (𝐴‘𝑡) ∈ ℕ0)
54		elnn0 12415	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴‘𝑡) ∈ ℕ0 ↔ ((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0))
55	53, 54	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0))
56		orel1 887	. . . . . . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (bits‘(𝐴‘𝑡)) = (bits‘0))
59		0bits 16319	. . . . . . . . . . . . . . . . 17 ⊢ (bits‘0) = ∅
60	58, 59	eqtrdi 2792	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (bits‘(𝐴‘𝑡)) = ∅)
61	60	rexeqdv 3314	. . . . . . . . . . . . . . 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 326	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)
63	62	ex 413	. . . . . . . . . . . . 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 455	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ (◡𝐴 “ ℕ))
66		eldif 3920	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ 𝐽))
67	34, 35, 36, 1, 2, 37, 38, 39, 40	eulerpartlemf 32970	. . . . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (𝐴‘𝑡) = 0)
70	69	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (bits‘(𝐴‘𝑡)) = (bits‘0))
71	70, 59	eqtrdi 2792	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (bits‘(𝐴‘𝑡)) = ∅)
72	71	rexeqdv 3314	. . . . . . . . . . . . . . 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 326	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → ¬ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)
74	73	ex 413	. . . . . . . . . . . . 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 455	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ 𝐽)
77	65, 76	elind 4154	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽))
78		simprr 771	. . . . . . . . . 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 512	. . . . . . . . 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 413	. . . . . . . 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 3161	. . . . . . 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 4037	. . . . . . . . . 10 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
83	1, 82	eqsstri 3978	. . . . . . . . 9 ⊢ 𝐽 ⊆ ℕ
84	8, 83	sstri 3953	. . . . . . . 8 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ
85		ssrexv 4011	. . . . . . . 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 211	. . . . . 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 280	. . . . 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 2748	. . . . . . . 8 ⊢ (𝑚 = 𝑝 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
90	89	2rexbidv 3213	. . . . . . 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 3645	. . . . . 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 6011	. . . . . . . . 9 ⊢ (𝐹 “ 𝑈) = (𝐹 “ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))))
94		imaiun 7192	. . . . . . . . 9 ⊢ (𝐹 “ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡))))
95	93, 94	eqtri 2764	. . . . . . . 8 ⊢ (𝐹 “ 𝑈) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡))))
96	95	eleq2i 2829	. . . . . . 7 ⊢ (𝑝 ∈ (𝐹 “ 𝑈) ↔ 𝑝 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))))
97		eliun 4958	. . . . . . 7 ⊢ (𝑝 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))))
98		f1ofn 6785	. . . . . . . . . . . . 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 4768	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝐽 → {𝑡} ⊆ 𝐽)
101	100, 11, 12	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝐽 → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
102		ovelimab 7532	. . . . . . . . . . . 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 3449	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
105		oveq1 7364	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (𝑥𝐹𝑛) = (𝑡𝐹𝑛))
106	105	eqeq2d 2747	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑝 = (𝑥𝐹𝑛) ↔ 𝑝 = (𝑡𝐹𝑛)))
107	106	rexbidv 3175	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛)))
108	104, 107	rexsn 4643	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛))
109	103, 108	bitrdi 286	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛)))
110		df-ov 7360	. . . . . . . . . . . . . . 15 ⊢ (𝑡𝐹𝑛) = (𝐹‘⟨𝑡, 𝑛⟩)
111	110	eqeq1i 2741	. . . . . . . . . . . . . 14 ⊢ ((𝑡𝐹𝑛) = 𝑝 ↔ (𝐹‘⟨𝑡, 𝑛⟩) = 𝑝)
112		eqcom 2743	. . . . . . . . . . . . . 14 ⊢ ((𝑡𝐹𝑛) = 𝑝 ↔ 𝑝 = (𝑡𝐹𝑛))
113	111, 112	bitr3i 276	. . . . . . . . . . . . 13 ⊢ ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝 ↔ 𝑝 = (𝑡𝐹𝑛))
114		opelxpi 5670	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ0) → ⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0))
115	1, 2	oddpwdcv 32955	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
116		vex 3449	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑛 ∈ V
117	104, 116	op2nd 7930	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
118	117	oveq2i 7368	. . . . . . . . . . . . . . . . 17 ⊢ (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
119	104, 116	op1st 7929	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
120	118, 119	oveq12i 7369	. . . . . . . . . . . . . . . 16 ⊢ ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
121	115, 120	eqtrdi 2792	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ0) → ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
124	113, 123	bitr3id 284	. . . . . . . . . . . 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 3173	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝐽 → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
127	109, 126	bitrd 278	. . . . . . . . 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 3095	. . . . . . 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 296	. . . . . 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 311	. . . 4 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑝 ∈ (𝐹 “ 𝑈) ↔ 𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
133	132	eqrdv 2734	. . 3 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 “ 𝑈) = {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
134		f1oeq3 6774	. . 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 232	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 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {cab 2713  ∀wral 3064  ∃wrex 3073  {crab 3407   ∖ cdif 3907   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  {csn 4586  ⟨cop 4592  ∪ ciun 4954   class class class wbr 5105  {copab 5167   ↦ cmpt 5188   × cxp 5631  ^◡ccnv 5632   ↾ cres 5635   “ cima 5636   ∘ ccom 5637   Fn wfn 6491  ⟶wf 6492  –1-1→wf1 6493  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  1^st c1st 7919  2^nd c2nd 7920   supp csupp 8092   ↑_m cmap 8765  Fincfn 8883  0cc0 11051  1c1 11052   · cmul 11056   ≤ cle 11190  ℕcn 12153  2c2 12208  ℕ₀cn0 12413  ↑cexp 13967  Σcsu 15570   ∥ cdvds 16136  bitscbits 16299  𝟭cind 32609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-dvds 16137  df-bits 16302

This theorem is referenced by:  eulerpartlemgs2  32980