Theorem eulerpartlemn 34688

Description: Lemma for eulerpart 34689. (Contributed by Thierry Arnoux, 30-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))))))
eulerpart.s	⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))

Assertion

Ref	Expression
eulerpartlemn	⊢ (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷

Distinct variable groups:   𝑓,𝑔,𝑘,𝑛,𝑜,𝑟,𝑥,𝑦,𝑧   𝑘,𝐹,𝑛,𝑜,𝑥,𝑦   𝑓,𝐺,𝑘,𝑜   𝑜,𝐻,𝑟   𝑓,𝐽,𝑘,𝑛,𝑜,𝑟,𝑥,𝑦   𝑘,𝑀,𝑛,𝑜,𝑟,𝑥,𝑦   𝑓,𝑁,𝑔,𝑘,𝑛,𝑜,𝑥   𝑛,𝑂,𝑟,𝑥,𝑦   𝑃,𝑔,𝑘,𝑛   𝑅,𝑓,𝑘,𝑛,𝑜,𝑟,𝑥,𝑦   𝑇,𝑓,𝑘,𝑛,𝑜,𝑟,𝑥,𝑦

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

Proof of Theorem eulerpartlemn

Dummy variables 𝑑 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 487	. . . . . . . . . . . . 13 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → 𝑜 = 𝑞)
2	1	fveq1d 6873	. . . . . . . . . . . 12 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → (𝑜‘𝑘) = (𝑞‘𝑘))
3	2	oveq1d 7415	. . . . . . . . . . 11 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → ((𝑜‘𝑘) · 𝑘) = ((𝑞‘𝑘) · 𝑘))
4	3	sumeq2dv 15743	. . . . . . . . . 10 ⊢ (𝑜 = 𝑞 → Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘))
5	4	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑜 = 𝑞 → (Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
6	5	cbvrabv 3427	. . . . . . . 8 ⊢ {𝑜 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁} = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
7	6	a1i 11	. . . . . . 7 ⊢ (𝑜 = 𝑞 → {𝑜 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁} = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁})
8	7	reseq2d 5969	. . . . . 6 ⊢ (𝑜 = 𝑞 → (𝐺 ↾ {𝑜 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁}) = (𝐺 ↾ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}))
9		eqidd 2766	. . . . . 6 ⊢ (𝑜 = 𝑞 → {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁} = {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁})
10	8, 7, 9	f1oeq123d 6804	. . . . 5 ⊢ (𝑜 = 𝑞 → ((𝐺 ↾ {𝑜 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁}):{𝑜 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁} ↔ (𝐺 ↾ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}):{𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁}))
11	10	imbi2d 343	. . . 4 ⊢ (𝑜 = 𝑞 → ((⊤ → (𝐺 ↾ {𝑜 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁}):{𝑜 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁}) ↔ (⊤ → (𝐺 ↾ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}):{𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁})))
12		eulerpart.g	. . . . 5 ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
13		eulerpart.p	. . . . . . 7 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
14		eulerpart.o	. . . . . . 7 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
15		eulerpart.d	. . . . . . 7 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
16		eulerpart.j	. . . . . . 7 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
17		eulerpart.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
18		eulerpart.h	. . . . . . 7 ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
19		eulerpart.m	. . . . . . 7 ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
20		eulerpart.r	. . . . . . 7 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
21		eulerpart.t	. . . . . . 7 ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
22	13, 14, 15, 16, 17, 18, 19, 20, 21, 12	eulerpartgbij 34679	. . . . . 6 ⊢ 𝐺:(𝑇 ∩ 𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅)
23	22	a1i 11	. . . . 5 ⊢ (⊤ → 𝐺:(𝑇 ∩ 𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅))
24		fveq2 6871	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑜 → (𝐺‘𝑞) = (𝐺‘𝑜))
25		reseq1 5963	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = 𝑜 → (𝑞 ↾ 𝐽) = (𝑜 ↾ 𝐽))
26	25	coeq2d 5839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 𝑜 → (bits ∘ (𝑞 ↾ 𝐽)) = (bits ∘ (𝑜 ↾ 𝐽)))
27	26	fveq2d 6875	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 𝑜 → (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))) = (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))
28	27	imaeq2d 6053	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑜 → (𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))
29	28	fveq2d 6875	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑜 → ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))))) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
30	24, 29	eqeq12d 2781	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑜 → ((𝐺‘𝑞) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))))) ↔ (𝐺‘𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))))
31	13, 14, 15, 16, 17, 18, 19, 20, 21, 12	eulerpartlemgv 34680	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑞) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))))))
32	30, 31	vtoclga 3544	. . . . . . . . . . . 12 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
33	32	3ad2ant2 1150	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (𝐺‘𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
34		simp3 1154	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
35	33, 34	eqtr4d 2803	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (𝐺‘𝑜) = 𝑑)
36	35	fveq1d 6873	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → ((𝐺‘𝑜)‘𝑘) = (𝑑‘𝑘))
37	36	oveq1d 7415	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (((𝐺‘𝑜)‘𝑘) · 𝑘) = ((𝑑‘𝑘) · 𝑘))
38	37	sumeq2sdv 15744	. . . . . . 7 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → Σ𝑘 ∈ ℕ (((𝐺‘𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘))
39	24	fveq2d 6875	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑜 → (𝑆‘(𝐺‘𝑞)) = (𝑆‘(𝐺‘𝑜)))
40		fveq2 6871	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑜 → (𝑆‘𝑞) = (𝑆‘𝑜))
41	39, 40	eqeq12d 2781	. . . . . . . . . 10 ⊢ (𝑞 = 𝑜 → ((𝑆‘(𝐺‘𝑞)) = (𝑆‘𝑞) ↔ (𝑆‘(𝐺‘𝑜)) = (𝑆‘𝑜)))
42		eulerpart.s	. . . . . . . . . . 11 ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))
43	13, 14, 15, 16, 17, 18, 19, 20, 21, 12, 42	eulerpartlemgs2 34687	. . . . . . . . . 10 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝑞)) = (𝑆‘𝑞))
44	41, 43	vtoclga 3544	. . . . . . . . 9 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝑜)) = (𝑆‘𝑜))
45		nn0ex 12501	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
46		0nn0 12510	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
47		1nn0 12511	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ0
48		prssi 4782	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
49	46, 47, 48	mp2an 704	. . . . . . . . . . . . 13 ⊢ {0, 1} ⊆ ℕ0
50		mapss 8875	. . . . . . . . . . . . 13 ⊢ ((ℕ0 ∈ V ∧ {0, 1} ⊆ ℕ0) → ({0, 1} ↑m ℕ) ⊆ (ℕ0 ↑m ℕ))
51	45, 49, 50	mp2an 704	. . . . . . . . . . . 12 ⊢ ({0, 1} ↑m ℕ) ⊆ (ℕ0 ↑m ℕ)
52		ssrin 4196	. . . . . . . . . . . 12 ⊢ (({0, 1} ↑m ℕ) ⊆ (ℕ0 ↑m ℕ) → (({0, 1} ↑m ℕ) ∩ 𝑅) ⊆ ((ℕ0 ↑m ℕ) ∩ 𝑅))
53	51, 52	ax-mp 5	. . . . . . . . . . 11 ⊢ (({0, 1} ↑m ℕ) ∩ 𝑅) ⊆ ((ℕ0 ↑m ℕ) ∩ 𝑅)
54		f1of 6810	. . . . . . . . . . . . 13 ⊢ (𝐺:(𝑇 ∩ 𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅) → 𝐺:(𝑇 ∩ 𝑅)⟶(({0, 1} ↑m ℕ) ∩ 𝑅))
55	22, 54	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝐺:(𝑇 ∩ 𝑅)⟶(({0, 1} ↑m ℕ) ∩ 𝑅)
56	55	ffvelcdmi 7068	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅))
57	53, 56	sselid 3937	. . . . . . . . . 10 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅))
58	20, 42	eulerpartlemsv1 34663	. . . . . . . . . 10 ⊢ ((𝐺‘𝑜) ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘(𝐺‘𝑜)) = Σ𝑘 ∈ ℕ (((𝐺‘𝑜)‘𝑘) · 𝑘))
59	57, 58	syl 18	. . . . . . . . 9 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝑜)) = Σ𝑘 ∈ ℕ (((𝐺‘𝑜)‘𝑘) · 𝑘))
60	13, 14, 15, 16, 17, 18, 19, 20, 21	eulerpartlemt0 34676	. . . . . . . . . . . 12 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) ↔ (𝑜 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑜 “ ℕ) ∈ Fin ∧ (◡𝑜 “ ℕ) ⊆ 𝐽))
61	60	simp1bi 1161	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ (ℕ0 ↑m ℕ))
62		inss2 4192	. . . . . . . . . . . 12 ⊢ (𝑇 ∩ 𝑅) ⊆ 𝑅
63	62	sseli 3935	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ 𝑅)
64	61, 63	elind 4155	. . . . . . . . . 10 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅))
65	20, 42	eulerpartlemsv1 34663	. . . . . . . . . 10 ⊢ (𝑜 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝑜) = Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘))
66	64, 65	syl 18	. . . . . . . . 9 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝑆‘𝑜) = Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘))
67	44, 59, 66	3eqtr3d 2808	. . . . . . . 8 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ ℕ (((𝐺‘𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘))
68	67	3ad2ant2 1150	. . . . . . 7 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → Σ𝑘 ∈ ℕ (((𝐺‘𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘))
69	38, 68	eqtr3d 2802	. . . . . 6 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘))
70	69	eqeq1d 2767	. . . . 5 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁))
71	12, 23, 70	f1oresrab 7113	. . . 4 ⊢ (⊤ → (𝐺 ↾ {𝑜 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁}):{𝑜 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁})
72	11, 71	chvarvv 2012	. . 3 ⊢ (⊤ → (𝐺 ↾ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}):{𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁})
73		cnveq 5850	. . . . . . . . . 10 ⊢ (𝑔 = 𝑞 → ◡𝑔 = ◡𝑞)
74	73	imaeq1d 6052	. . . . . . . . 9 ⊢ (𝑔 = 𝑞 → (◡𝑔 “ ℕ) = (◡𝑞 “ ℕ))
75	74	raleqdv 3323	. . . . . . . 8 ⊢ (𝑔 = 𝑞 → (∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
76	75	cbvrabv 3427	. . . . . . 7 ⊢ {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
77		nfrab1 3437	. . . . . . . 8 ⊢ Ⅎ𝑞{𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
78		nfrab1 3437	. . . . . . . 8 ⊢ Ⅎ𝑞{𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
79		df-3an 1103	. . . . . . . . . . . 12 ⊢ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
80	79	anbi1i 635	. . . . . . . . . . 11 ⊢ (((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ (((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
81	13	eulerpartleme 34670	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑃 ↔ (𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
82	81	anbi1i 635	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
83		an32 658	. . . . . . . . . . 11 ⊢ ((((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁) ↔ (((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
84	80, 82, 83	3bitr4i 306	. . . . . . . . . 10 ⊢ ((𝑞 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ (((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
85	13, 14, 15, 16, 17, 18, 19, 20, 21	eulerpartlemt0 34676	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) ↔ (𝑞 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
86		nnex 12230	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
87	45, 86	elmap 8857	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ (ℕ0 ↑m ℕ) ↔ 𝑞:ℕ⟶ℕ0)
88	87	3anbi1i 1173	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽) ↔ (𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
89	85, 88	bitri 278	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) ↔ (𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
90		df-3an 1103	. . . . . . . . . . . 12 ⊢ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
91		dfss3 3928	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ 𝐽)
92		breq2 5109	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
93	92	notbid 321	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
94	93, 16	elrab2 3657	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
95	94	ralbii 3111	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
96		r19.26 3125	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (◡𝑞 “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
97	91, 95, 96	3bitri 300	. . . . . . . . . . . . . . 15 ⊢ ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
98		cnvimass 6075	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑞 “ ℕ) ⊆ dom 𝑞
99		fdm 6705	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞:ℕ⟶ℕ0 → dom 𝑞 = ℕ)
100	98, 99	sseqtrid 3981	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞:ℕ⟶ℕ0 → (◡𝑞 “ ℕ) ⊆ ℕ)
101		dfss3 3928	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑞 “ ℕ) ⊆ ℕ ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ)
102	100, 101	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (𝑞:ℕ⟶ℕ0 → ∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ)
103	102	biantrurd 541	. . . . . . . . . . . . . . 15 ⊢ (𝑞:ℕ⟶ℕ0 → (∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛)))
104	97, 103	bitr4id 293	. . . . . . . . . . . . . 14 ⊢ (𝑞:ℕ⟶ℕ0 → ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
105	104	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) → ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
106	105	pm5.32i 584	. . . . . . . . . . . 12 ⊢ (((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ (◡𝑞 “ ℕ) ⊆ 𝐽) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
107	89, 90, 106	3bitri 300	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
108	107	anbi1i 635	. . . . . . . . . 10 ⊢ ((𝑞 ∈ (𝑇 ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁) ↔ (((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
109	84, 108	bitr4i 281	. . . . . . . . 9 ⊢ ((𝑞 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ (𝑞 ∈ (𝑇 ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
110		rabid 3438	. . . . . . . . 9 ⊢ (𝑞 ∈ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ (𝑞 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
111		rabid 3438	. . . . . . . . 9 ⊢ (𝑞 ∈ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁} ↔ (𝑞 ∈ (𝑇 ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
112	109, 110, 111	3bitr4i 306	. . . . . . . 8 ⊢ (𝑞 ∈ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ 𝑞 ∈ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁})
113	77, 78, 112	eqri 3959	. . . . . . 7 ⊢ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
114	14, 76, 113	3eqtri 2792	. . . . . 6 ⊢ 𝑂 = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
115	114	reseq2i 5966	. . . . 5 ⊢ (𝐺 ↾ 𝑂) = (𝐺 ↾ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁})
116	115	a1i 11	. . . 4 ⊢ (⊤ → (𝐺 ↾ 𝑂) = (𝐺 ↾ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}))
117	114	a1i 11	. . . 4 ⊢ (⊤ → 𝑂 = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁})
118		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑑𝐷
119		nfrab1 3437	. . . . . 6 ⊢ Ⅎ𝑑{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁}
120		fnima 6655	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 Fn ℕ → (𝑑 “ ℕ) = ran 𝑑)
121	120	sseq1d 3970	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 Fn ℕ → ((𝑑 “ ℕ) ⊆ {0, 1} ↔ ran 𝑑 ⊆ {0, 1}))
122	121	anbi2d 641	. . . . . . . . . . . . . . 15 ⊢ (𝑑 Fn ℕ → ((ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (ran 𝑑 ⊆ ℕ0 ∧ ran 𝑑 ⊆ {0, 1})))
123		sstr 3947	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝑑 ⊆ {0, 1} ∧ {0, 1} ⊆ ℕ0) → ran 𝑑 ⊆ ℕ0)
124	49, 123	mpan2 703	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝑑 ⊆ {0, 1} → ran 𝑑 ⊆ ℕ0)
125	124	pm4.71ri 569	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑑 ⊆ {0, 1} ↔ (ran 𝑑 ⊆ ℕ0 ∧ ran 𝑑 ⊆ {0, 1}))
126	122, 125	bitr4di 292	. . . . . . . . . . . . . 14 ⊢ (𝑑 Fn ℕ → ((ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ran 𝑑 ⊆ {0, 1}))
127	126	pm5.32i 584	. . . . . . . . . . . . 13 ⊢ ((𝑑 Fn ℕ ∧ (ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1})) ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ {0, 1}))
128		anass 473	. . . . . . . . . . . . 13 ⊢ (((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (𝑑 Fn ℕ ∧ (ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1})))
129		df-f 6529	. . . . . . . . . . . . 13 ⊢ (𝑑:ℕ⟶{0, 1} ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ {0, 1}))
130	127, 128, 129	3bitr4ri 307	. . . . . . . . . . . 12 ⊢ (𝑑:ℕ⟶{0, 1} ↔ ((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
131		prex 5400	. . . . . . . . . . . . 13 ⊢ {0, 1} ∈ V
132	131, 86	elmap 8857	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ({0, 1} ↑m ℕ) ↔ 𝑑:ℕ⟶{0, 1})
133		df-f 6529	. . . . . . . . . . . . 13 ⊢ (𝑑:ℕ⟶ℕ0 ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0))
134	133	anbi1i 635	. . . . . . . . . . . 12 ⊢ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
135	130, 132, 134	3bitr4i 306	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ({0, 1} ↑m ℕ) ↔ (𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
136		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑑 ∈ V
137		cnveq 5850	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑑 → ◡𝑓 = ◡𝑑)
138	137	imaeq1d 6052	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑑 → (◡𝑓 “ ℕ) = (◡𝑑 “ ℕ))
139	138	eleq1d 2850	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑑 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝑑 “ ℕ) ∈ Fin))
140	136, 139, 20	elab2 3644	. . . . . . . . . . 11 ⊢ (𝑑 ∈ 𝑅 ↔ (◡𝑑 “ ℕ) ∈ Fin)
141	135, 140	anbi12i 639	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ({0, 1} ↑m ℕ) ∧ 𝑑 ∈ 𝑅) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ (◡𝑑 “ ℕ) ∈ Fin))
142		elin 3923	. . . . . . . . . 10 ⊢ (𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ (𝑑 ∈ ({0, 1} ↑m ℕ) ∧ 𝑑 ∈ 𝑅))
143		an32 658	. . . . . . . . . 10 ⊢ (((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ (◡𝑑 “ ℕ) ∈ Fin))
144	141, 142, 143	3bitr4i 306	. . . . . . . . 9 ⊢ (𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
145	144	anbi1i 635	. . . . . . . 8 ⊢ ((𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁) ↔ (((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁))
146	13	eulerpartleme 34670	. . . . . . . . . 10 ⊢ (𝑑 ∈ 𝑃 ↔ (𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁))
147	146	anbi1i 635	. . . . . . . . 9 ⊢ ((𝑑 ∈ 𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
148		df-3an 1103	. . . . . . . . . 10 ⊢ ((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁))
149	148	anbi1i 635	. . . . . . . . 9 ⊢ (((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
150		an32 658	. . . . . . . . 9 ⊢ ((((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁))
151	147, 149, 150	3bitri 300	. . . . . . . 8 ⊢ ((𝑑 ∈ 𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁))
152	145, 151	bitr4i 281	. . . . . . 7 ⊢ ((𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁) ↔ (𝑑 ∈ 𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
153		rabid 3438	. . . . . . 7 ⊢ (𝑑 ∈ {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁} ↔ (𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁))
154	13, 14, 15	eulerpartlemd 34673	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 ↔ (𝑑 ∈ 𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
155	152, 153, 154	3bitr4ri 307	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 ↔ 𝑑 ∈ {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁})
156	118, 119, 155	eqri 3959	. . . . 5 ⊢ 𝐷 = {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁}
157	156	a1i 11	. . . 4 ⊢ (⊤ → 𝐷 = {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁})
158	116, 117, 157	f1oeq123d 6804	. . 3 ⊢ (⊤ → ((𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷 ↔ (𝐺 ↾ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}):{𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁}))
159	72, 158	mpbird 260	. 2 ⊢ (⊤ → (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷)
160	159	mptru 1570	1 ⊢ (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ⊤wtru 1564   ∈ wcel 2145  {cab 2743  ∀wral 3079  {crab 3417  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558  {cpr 4587   class class class wbr 5105  {copab 5167   ↦ cmpt 5186  ^◡ccnv 5651  dom cdm 5652  ran crn 5653   ↾ cres 5654   “ cima 5655   ∘ ccom 5656   Fn wfn 6520  ⟶wf 6521  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402   supp csupp 8144   ↑_m cmap 8812  Fincfn 8931  0cc0 11088  1c1 11089   · cmul 11093   ≤ cle 11232  𝟭cind 12209  ℕcn 12224  2c2 12286  ℕ₀cn0 12495  ↑cexp 14088  Σcsu 15727   ∥ cdvds 16300  bitscbits 16467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-ac2 10435  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-disj 5073  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-er 8682  df-map 8814  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-inf 9391  df-oi 9460  df-dju 9875  df-card 9913  df-acn 9916  df-ac 10088  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-ind 12210  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-xnn0 12569  df-z 12583  df-uz 12854  df-rp 13008  df-fz 13527  df-fzo 13674  df-fl 13816  df-mod 13894  df-seq 14029  df-exp 14089  df-hash 14358  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-clim 15529  df-sum 15728  df-dvds 16301  df-bits 16470

This theorem is referenced by:  eulerpart  34689