Theorem eulerpartlemgh 34944

Description: Lemma for eulerpart 34948: 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 34920	. . . 4 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1of1 6811	. . . 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 5002	. . . . 5 ⊢ (∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
8		inss2 4182	. . . . . . . 8 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
9	8	sseli 3926	. . . . . . 7 ⊢ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) → 𝑡 ∈ 𝐽)
10	9	snssd 4746	. . . . . 6 ⊢ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) → {𝑡} ⊆ 𝐽)
11		bitsss 16563	. . . . . 6 ⊢ (bits‘(𝐴‘𝑡)) ⊆ ℕ0
12		xpss12 5662	. . . . . 6 ⊢ (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴‘𝑡)) ⊆ ℕ0) → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
13	10, 11, 12	sylancl 598	. . . . 5 ⊢ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
14	7, 13	mprgbir 3083	. . . 4 ⊢ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0)
15	6, 14	eqsstri 3976	. . 3 ⊢ 𝑈 ⊆ (𝐽 × ℕ0)
16		f1ores 6827	. . 3 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1→ℕ ∧ 𝑈 ⊆ (𝐽 × ℕ0)) → (𝐹 ↾ 𝑈):𝑈–1-1-onto→(𝐹 “ 𝑈))
17	5, 15, 16	mp2an 705	. 2 ⊢ (𝐹 ↾ 𝑈):𝑈–1-1-onto→(𝐹 “ 𝑈)
18		simpr 490	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) = 𝑝)
19		2nn 12385	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 2 ∈ ℕ)
21	11	sseli 3926	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (bits‘(𝐴‘𝑡)) → 𝑛 ∈ ℕ0)
22	21	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 𝑛 ∈ ℕ0)
23	20, 22	nnexpcld 14356	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑛) ∈ ℕ)
24		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 𝑡 ∈ ℕ)
25	23, 24	nnmulcld 12360	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → ((2↑𝑛) · 𝑡) ∈ ℕ)
26	25	adantr 486	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) ∈ ℕ)
27	18, 26	eqeltrrd 2861	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → 𝑝 ∈ ℕ)
28	27	rexlimdva2 3165	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑝 ∈ ℕ))
29	28	rexlimdva 3163	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑝 ∈ ℕ))
30	29	pm4.71rd 572	. . . . . 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 4307	. . . . . . . . . . . . . . 15 ⊢ ¬ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝
32		simplr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → 𝑡 ∈ ℕ)
33		simpr 490	. . . . . . . . . . . . . . . . . . . . . 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 34935	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
42	41	simp1bi 1163	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ (ℕ0 ↑m ℕ))
43		elmapi 8847	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴:ℕ⟶ℕ0)
44	42, 43	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
45	44	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → 𝐴:ℕ⟶ℕ0)
46		ffn 6697	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴:ℕ⟶ℕ0 → 𝐴 Fn ℕ)
47		elpreima 7045	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 Fn ℕ → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ)))
48	45, 46, 47	3syl 19	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (𝑡 ∈ (◡𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ)))
49	33, 48	mtbid 327	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ))
50		imnan 405	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑡 ∈ ℕ → ¬ (𝐴‘𝑡) ∈ ℕ) ↔ ¬ (𝑡 ∈ ℕ ∧ (𝐴‘𝑡) ∈ ℕ))
51	49, 50	sylibr 237	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (𝑡 ∈ ℕ → ¬ (𝐴‘𝑡) ∈ ℕ))
52	32, 51	mpd 16	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ (𝐴‘𝑡) ∈ ℕ)
53	45, 32	ffvelcdmd 7073	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (𝐴‘𝑡) ∈ ℕ0)
54		elnn0 12577	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴‘𝑡) ∈ ℕ0 ↔ ((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0))
55	53, 54	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0))
56		orel1 902	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝐴‘𝑡) ∈ ℕ → (((𝐴‘𝑡) ∈ ℕ ∨ (𝐴‘𝑡) = 0) → (𝐴‘𝑡) = 0))
57	52, 55, 56	sylc 66	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (𝐴‘𝑡) = 0)
58	57	fveq2d 6877	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (bits‘(𝐴‘𝑡)) = (bits‘0))
59		0bits 16576	. . . . . . . . . . . . . . . . 17 ⊢ (bits‘0) = ∅
60	58, 59	eqtrdi 2811	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → (bits‘(𝐴‘𝑡)) = ∅)
61	60	rexeqdv 3320	. . . . . . . . . . . . . . 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 330	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (◡𝐴 “ ℕ)) → ¬ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)
63	62	ex 418	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (¬ 𝑡 ∈ (◡𝐴 “ ℕ) → ¬ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
64	63	con4d 116	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑡 ∈ (◡𝐴 “ ℕ)))
65	64	impr 460	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ (◡𝐴 “ ℕ))
66		eldif 3908	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ 𝐽))
67	34, 35, 36, 1, 2, 37, 38, 39, 40	eulerpartlemf 34936	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴‘𝑡) = 0)
68	66, 67	sylan2br 607	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ 𝐽)) → (𝐴‘𝑡) = 0)
69	68	anassrs 473	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (𝐴‘𝑡) = 0)
70	69	fveq2d 6877	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (bits‘(𝐴‘𝑡)) = (bits‘0))
71	70, 59	eqtrdi 2811	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (bits‘(𝐴‘𝑡)) = ∅)
72	71	rexeqdv 3320	. . . . . . . . . . . . . . 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 330	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → ¬ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)
74	73	ex 418	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (¬ 𝑡 ∈ 𝐽 → ¬ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
75	74	con4d 116	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑡 ∈ 𝐽))
76	75	impr 460	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ 𝐽)
77	65, 76	elind 4145	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽))
78		simprr 785	. . . . . . . . . 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 521	. . . . . . . . 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 418	. . . . . . . 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 3172	. . . . . . 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 4027	. . . . . . . . . 10 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
83	1, 82	eqsstri 3976	. . . . . . . . 9 ⊢ 𝐽 ⊆ ℕ
84	8, 83	sstri 3939	. . . . . . . 8 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ
85		ssrexv 4000	. . . . . . . 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 14	. . . . . . 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 215	. . . . . 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 284	. . . . 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 2772	. . . . . . . 8 ⊢ (𝑚 = 𝑝 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
90	89	2rexbidv 3227	. . . . . . 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 3644	. . . . . 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 6048	. . . . . . . . 9 ⊢ (𝐹 “ 𝑈) = (𝐹 “ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))))
94		imaiun 7237	. . . . . . . . 9 ⊢ (𝐹 “ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡))))
95	93, 94	eqtri 2783	. . . . . . . 8 ⊢ (𝐹 “ 𝑈) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡))))
96	95	eleq2i 2852	. . . . . . 7 ⊢ (𝑝 ∈ (𝐹 “ 𝑈) ↔ 𝑝 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))))
97		eliun 4954	. . . . . . 7 ⊢ (𝑝 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))))
98		f1ofn 6813	. . . . . . . . . . . . 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 4745	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝐽 → {𝑡} ⊆ 𝐽)
101	100, 11, 12	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝐽 → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
102		ovelimab 7587	. . . . . . . . . . . 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 599	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛)))
104		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
105		oveq1 7415	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (𝑥𝐹𝑛) = (𝑡𝐹𝑛))
106	105	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑝 = (𝑥𝐹𝑛) ↔ 𝑝 = (𝑡𝐹𝑛)))
107	106	rexbidv 3186	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛)))
108	104, 107	rexsn 4642	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛))
109	103, 108	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛)))
110		df-ov 7411	. . . . . . . . . . . . . . 15 ⊢ (𝑡𝐹𝑛) = (𝐹‘⟨𝑡, 𝑛⟩)
111	110	eqeq1i 2765	. . . . . . . . . . . . . 14 ⊢ ((𝑡𝐹𝑛) = 𝑝 ↔ (𝐹‘⟨𝑡, 𝑛⟩) = 𝑝)
112		eqcom 2767	. . . . . . . . . . . . . 14 ⊢ ((𝑡𝐹𝑛) = 𝑝 ↔ 𝑝 = (𝑡𝐹𝑛))
113	111, 112	bitr3i 280	. . . . . . . . . . . . 13 ⊢ ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝 ↔ 𝑝 = (𝑡𝐹𝑛))
114		opelxpi 5684	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ0) → ⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0))
115	1, 2	oddpwdcv 34921	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
116		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑛 ∈ V
117	104, 116	op2nd 7993	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
118	117	oveq2i 7419	. . . . . . . . . . . . . . . . 17 ⊢ (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
119	104, 116	op1st 7992	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
120	118, 119	oveq12i 7420	. . . . . . . . . . . . . . . 16 ⊢ ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
121	115, 120	eqtrdi 2811	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
122	114, 121	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
123	122	eqeq1d 2762	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ0) → ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
124	113, 123	bitr3id 288	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ0) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
125	21, 124	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
126	125	rexbidva 3184	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝐽 → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
127	109, 126	bitrd 282	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
128	9, 127	syl 18	. . . . . . . 8 ⊢ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝))
129	128	rexbiia 3107	. . . . . . 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 300	. . . . . 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 315	. . . 4 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑝 ∈ (𝐹 “ 𝑈) ↔ 𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
133	132	eqrdv 2758	. . 3 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 “ 𝑈) = {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
134		f1oeq3 6802	. . 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 18	. 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 236	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 ↾ 𝑈):𝑈–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  {crab 3412   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  𝒫 cpw 4556  {csn 4583  ⟨cop 4589  ∪ ciun 4950   class class class wbr 5102  {copab 5166   ↦ cmpt 5185   × cxp 5645  ^◡ccnv 5646   ↾ cres 5649   “ cima 5650   ∘ ccom 5651   Fn wfn 6522  ⟶wf 6523  –1-1→wf1 6524  –1-1-onto→wf1o 6526  ‘cfv 6527  (class class class)co 7408   ∈ cmpo 7410  1^st c1st 7982  2^nd c2nd 7983   supp csupp 8155   ↑_m cmap 8825  Fincfn 8951  0cc0 11171  1c1 11172   · cmul 11176   ≤ cle 11315  𝟭cind 12289  ℕcn 12304  2c2 12366  ℕ₀cn0 12575  ↑cexp 14172  Σcsu 15820   ∥ cdvds 16389  bitscbits 16556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248  ax-pre-sup 11249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-map 8827  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-sup 9412  df-inf 9413  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-div 11943  df-nn 12305  df-2 12374  df-n0 12576  df-z 12663  df-uz 12935  df-rp 13090  df-fz 13609  df-fzo 13757  df-fl 13900  df-seq 14113  df-exp 14173  df-dvds 16390  df-bits 16559

This theorem is used by:  eulerpartlemgs2  34946