eulerpartlemgh - Mathbox for Thierry Arnoux

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Theorem eulerpartlemgh 33866

Description: Lemma for eulerpart 33870: 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 33842	. . . 4 ⊢ 𝐹:(𝐽 × ℕ₀)–1-1-onto→ℕ
4		f1of1 6822	. . . 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 5038	. . . . 5 ⊢ (∪ 𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ₀) ↔ ∀𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ₀))
8		inss2 4221	. . . . . . . 8 ⊢ ((^◡𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
9	8	sseli 3970	. . . . . . 7 ⊢ (𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽) → 𝑡 ∈ 𝐽)
10	9	snssd 4804	. . . . . 6 ⊢ (𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽) → {𝑡} ⊆ 𝐽)
11		bitsss 16364	. . . . . 6 ⊢ (bits‘(𝐴‘𝑡)) ⊆ ℕ₀
12		xpss12 5681	. . . . . 6 ⊢ (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴‘𝑡)) ⊆ ℕ₀) → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ₀))
13	10, 11, 12	sylancl 585	. . . . 5 ⊢ (𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽) → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ₀))
14	7, 13	mprgbir 3060	. . . 4 ⊢ ∪ 𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ₀)
15	6, 14	eqsstri 4008	. . 3 ⊢ 𝑈 ⊆ (𝐽 × ℕ₀)
16		f1ores 6837	. . 3 ⊢ ((𝐹:(𝐽 × ℕ₀)–1-1→ℕ ∧ 𝑈 ⊆ (𝐽 × ℕ₀)) → (𝐹 ↾ 𝑈):𝑈–1-1-onto→(𝐹 “ 𝑈))
17	5, 15, 16	mp2an 689	. 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 12282	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 2 ∈ ℕ)
21	11	sseli 3970	. . . . . . . . . . . . . 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 14205	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑛) ∈ ℕ)
24		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → 𝑡 ∈ ℕ)
25	23, 24	nnmulcld 12262	. . . . . . . . . . 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 2826	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → 𝑝 ∈ ℕ)
28	27	rexlimdva2 3149	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝 → 𝑝 ∈ ℕ))
29	28	rexlimdva 3147	. . . . . . 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 4349	. . . . . . . . . . . . . . 15 ⊢ ¬ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝
32		simplr 766	. . . . . . . . . . . . . . . . . . . 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 33857	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ₀ ↑_m ℕ) ∧ (^◡𝐴 “ ℕ) ∈ Fin ∧ (^◡𝐴 “ ℕ) ⊆ 𝐽))
42	41	simp1bi 1142	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ (ℕ₀ ↑_m ℕ))
43		elmapi 8839	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ (ℕ₀ ↑_m ℕ) → 𝐴:ℕ⟶ℕ₀)
44	42, 43	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴:ℕ⟶ℕ₀)
45	44	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (^◡𝐴 “ ℕ)) → 𝐴:ℕ⟶ℕ₀)
46		ffn 6707	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴:ℕ⟶ℕ₀ → 𝐴 Fn ℕ)
47		elpreima 7049	. . . . . . . . . . . . . . . . . . . . . . 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 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 7077	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (^◡𝐴 “ ℕ)) → (𝐴‘𝑡) ∈ ℕ₀)
54		elnn0 12471	. . . . . . . . . . . . . . . . . . . 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 885	. . . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (^◡𝐴 “ ℕ)) → (bits‘(𝐴‘𝑡)) = (bits‘0))
59		0bits 16377	. . . . . . . . . . . . . . . . 17 ⊢ (bits‘0) = ∅
60	58, 59	eqtrdi 2780	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (^◡𝐴 “ ℕ)) → (bits‘(𝐴‘𝑡)) = ∅)
61	60	rexeqdv 3318	. . . . . . . . . . . . . . 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 3950	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ 𝐽))
67	34, 35, 36, 1, 2, 37, 38, 39, 40	eulerpartlemf 33858	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴‘𝑡) = 0)
68	66, 67	sylan2br 594	. . . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (bits‘(𝐴‘𝑡)) = (bits‘0))
71	70, 59	eqtrdi 2780	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ 𝐽) → (bits‘(𝐴‘𝑡)) = ∅)
72	71	rexeqdv 3318	. . . . . . . . . . . . . . 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 4186	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽))
78		simprr 770	. . . . . . . . . 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 3156	. . . . . . 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 4069	. . . . . . . . . 10 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
83	1, 82	eqsstri 4008	. . . . . . . . 9 ⊢ 𝐽 ⊆ ℕ
84	8, 83	sstri 3983	. . . . . . . 8 ⊢ ((^◡𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ
85		ssrexv 4043	. . . . . . . 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 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 2736	. . . . . . . 8 ⊢ (𝑚 = 𝑝 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
90	89	2rexbidv 3211	. . . . . . 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 3675	. . . . . 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 6047	. . . . . . . . 9 ⊢ (𝐹 “ 𝑈) = (𝐹 “ ∪ 𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))))
94		imaiun 7236	. . . . . . . . 9 ⊢ (𝐹 “ ∪ 𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))) = ∪ 𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡))))
95	93, 94	eqtri 2752	. . . . . . . 8 ⊢ (𝐹 “ 𝑈) = ∪ 𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡))))
96	95	eleq2i 2817	. . . . . . 7 ⊢ (𝑝 ∈ (𝐹 “ 𝑈) ↔ 𝑝 ∈ ∪ 𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))))
97		eliun 4991	. . . . . . 7 ⊢ (𝑝 ∈ ∪ 𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑡 ∈ ((^◡𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))))
98		f1ofn 6824	. . . . . . . . . . . . 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 4803	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝐽 → {𝑡} ⊆ 𝐽)
101	100, 11, 12	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝐽 → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ₀))
102		ovelimab 7578	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴‘𝑡)))) ↔ ∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛)))
104		vex 3470	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
105		oveq1 7408	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (𝑥𝐹𝑛) = (𝑡𝐹𝑛))
106	105	eqeq2d 2735	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑝 = (𝑥𝐹𝑛) ↔ 𝑝 = (𝑡𝐹𝑛)))
107	106	rexbidv 3170	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))𝑝 = (𝑡𝐹𝑛)))
108	104, 107	rexsn 4678	. . . . . . . . . . 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 7404	. . . . . . . . . . . . . . 15 ⊢ (𝑡𝐹𝑛) = (𝐹‘⟨𝑡, 𝑛⟩)
111	110	eqeq1i 2729	. . . . . . . . . . . . . 14 ⊢ ((𝑡𝐹𝑛) = 𝑝 ↔ (𝐹‘⟨𝑡, 𝑛⟩) = 𝑝)
112		eqcom 2731	. . . . . . . . . . . . . 14 ⊢ ((𝑡𝐹𝑛) = 𝑝 ↔ 𝑝 = (𝑡𝐹𝑛))
113	111, 112	bitr3i 277	. . . . . . . . . . . . 13 ⊢ ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝 ↔ 𝑝 = (𝑡𝐹𝑛))
114		opelxpi 5703	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ ℕ₀) → ⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ₀))
115	1, 2	oddpwdcv 33843	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ₀) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2^nd ‘⟨𝑡, 𝑛⟩)) · (1^st ‘⟨𝑡, 𝑛⟩)))
116		vex 3470	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑛 ∈ V
117	104, 116	op2nd 7977	. . . . . . . . . . . . . . . . . 18 ⊢ (2^nd ‘⟨𝑡, 𝑛⟩) = 𝑛
118	117	oveq2i 7412	. . . . . . . . . . . . . . . . 17 ⊢ (2↑(2^nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
119	104, 116	op1st 7976	. . . . . . . . . . . . . . . . 17 ⊢ (1^st ‘⟨𝑡, 𝑛⟩) = 𝑡
120	118, 119	oveq12i 7413	. . . . . . . . . . . . . . . 16 ⊢ ((2↑(2^nd ‘⟨𝑡, 𝑛⟩)) · (1^st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
121	115, 120	eqtrdi 2780	. . . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . 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 592	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝐽 ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
126	125	rexbidva 3168	. . . . . . . . . 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 3084	. . . . . . 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 2722	. . 3 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 “ 𝑈) = {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
134		f1oeq3 6813	. . 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})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 {cab 2701 ∀wral 3053 ∃wrex 3062 {crab 3424 ∖ cdif 3937 ∩ cin 3939 ⊆ wss 3940 ∅c0 4314 𝒫 cpw 4594 {csn 4620 ⟨cop 4626 ∪ ciun 4987 class class class wbr 5138 {copab 5200 ↦ cmpt 5221 × cxp 5664 ^◡ccnv 5665 ↾ cres 5668 “ cima 5669 ∘ ccom 5670 Fn wfn 6528 ⟶wf 6529 –1-1→wf1 6530 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7401 ∈ cmpo 7403 1^st c1st 7966 2^nd c2nd 7967 supp csupp 8140 ↑_m cmap 8816 Fincfn 8935 0cc0 11106 1c1 11107 · cmul 11111 ≤ cle 11246 ℕcn 12209 2c2 12264 ℕ₀cn0 12469 ↑cexp 14024 Σcsu 15629 ∥ cdvds 16194 bitscbits 16357 𝟭cind 33497

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-dvds 16195 df-bits 16360

This theorem is referenced by: eulerpartlemgs2 33868

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