Theorem eulerpartlemn 29770

Description: Lemma for eulerpart 29771. (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 472	. . . . . . . . . . . . 13 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → 𝑜 = 𝑞)
2	1	fveq1d 6105	. . . . . . . . . . . 12 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → (𝑜‘𝑘) = (𝑞‘𝑘))
3	2	oveq1d 6564	. . . . . . . . . . 11 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → ((𝑜‘𝑘) · 𝑘) = ((𝑞‘𝑘) · 𝑘))
4	3	sumeq2dv 14281	. . . . . . . . . 10 ⊢ (𝑜 = 𝑞 → Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘))
5	4	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑜 = 𝑞 → (Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
6	5	cbvrabv 3172	. . . . . . . 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 5317	. . . . . 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 2611	. . . . . 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 6046	. . . . 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 329	. . . 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 29761	. . . . . 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 6103	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑜 → (𝐺‘𝑞) = (𝐺‘𝑜))
25		reseq1 5311	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = 𝑜 → (𝑞 ↾ 𝐽) = (𝑜 ↾ 𝐽))
26	25	coeq2d 5206	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 𝑜 → (bits ∘ (𝑞 ↾ 𝐽)) = (bits ∘ (𝑜 ↾ 𝐽)))
27	26	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 𝑜 → (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))) = (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))
28	27	imaeq2d 5385	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑜 → (𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))
29	28	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑜 → ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))))) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
30	24, 29	eqeq12d 2625	. . . . . . . . . . . . 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 29762	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑞) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))))))
32	30, 31	vtoclga 3245	. . . . . . . . . . . 12 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
33	32	3ad2ant2 1076	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (𝐺‘𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
34		simp3 1056	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
35	33, 34	eqtr4d 2647	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (𝐺‘𝑜) = 𝑑)
36	35	fveq1d 6105	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → ((𝐺‘𝑜)‘𝑘) = (𝑑‘𝑘))
37	36	oveq1d 6564	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (((𝐺‘𝑜)‘𝑘) · 𝑘) = ((𝑑‘𝑘) · 𝑘))
38	37	sumeq2sdv 14282	. . . . . . 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 6107	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑜 → (𝑆‘(𝐺‘𝑞)) = (𝑆‘(𝐺‘𝑜)))
40		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑜 → (𝑆‘𝑞) = (𝑆‘𝑜))
41	39, 40	eqeq12d 2625	. . . . . . . . . 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 29769	. . . . . . . . . 10 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝑞)) = (𝑆‘𝑞))
44	41, 43	vtoclga 3245	. . . . . . . . 9 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝑜)) = (𝑆‘𝑜))
45		nn0ex 11175	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
46		0nn0 11184	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
47		1nn0 11185	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ0
48		prssi 4293	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
49	46, 47, 48	mp2an 704	. . . . . . . . . . . . 13 ⊢ {0, 1} ⊆ ℕ0
50		mapss 7786	. . . . . . . . . . . . 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 3800	. . . . . . . . . . . 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 6050	. . . . . . . . . . . . 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	ffvelrni 6266	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅))
57	53, 56	sseldi 3566	. . . . . . . . . 10 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅))
58	20, 42	eulerpartlemsv1 29745	. . . . . . . . . 10 ⊢ ((𝐺‘𝑜) ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘(𝐺‘𝑜)) = Σ𝑘 ∈ ℕ (((𝐺‘𝑜)‘𝑘) · 𝑘))
59	57, 58	syl 17	. . . . . . . . 9 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝑜)) = Σ𝑘 ∈ ℕ (((𝐺‘𝑜)‘𝑘) · 𝑘))
60	13, 14, 15, 16, 17, 18, 19, 20, 21	eulerpartlemt0 29758	. . . . . . . . . . . 12 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) ↔ (𝑜 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝑜 “ ℕ) ∈ Fin ∧ (◡𝑜 “ ℕ) ⊆ 𝐽))
61	60	simp1bi 1069	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ (ℕ0 ↑𝑚 ℕ))
62		inss2 3796	. . . . . . . . . . . 12 ⊢ (𝑇 ∩ 𝑅) ⊆ 𝑅
63	62	sseli 3564	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ 𝑅)
64	61, 63	elind 3760	. . . . . . . . . 10 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅))
65	20, 42	eulerpartlemsv1 29745	. . . . . . . . . 10 ⊢ (𝑜 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘𝑜) = Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘))
66	64, 65	syl 17	. . . . . . . . 9 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝑆‘𝑜) = Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘))
67	44, 59, 66	3eqtr3d 2652	. . . . . . . 8 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ ℕ (((𝐺‘𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘))
68	67	3ad2ant2 1076	. . . . . . 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 2646	. . . . . 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 2612	. . . . 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 6302	. . . 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	chvarv 2251	. . 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 5218	. . . . . . . . . 10 ⊢ (𝑔 = 𝑞 → ◡𝑔 = ◡𝑞)
74	73	imaeq1d 5384	. . . . . . . . 9 ⊢ (𝑔 = 𝑞 → (◡𝑔 “ ℕ) = (◡𝑞 “ ℕ))
75	74	raleqdv 3121	. . . . . . . 8 ⊢ (𝑔 = 𝑞 → (∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
76	75	cbvrabv 3172	. . . . . . 7 ⊢ {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
77		nfrab1 3099	. . . . . . . 8 ⊢ Ⅎ𝑞{𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
78		nfrab1 3099	. . . . . . . 8 ⊢ Ⅎ𝑞{𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
79		df-3an 1033	. . . . . . . . . . . 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 727	. . . . . . . . . . 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 29752	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑃 ↔ (𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
82	81	anbi1i 727	. . . . . . . . . . 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 835	. . . . . . . . . . 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 291	. . . . . . . . . 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 29758	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) ↔ (𝑞 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
86		nnex 10903	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
87	45, 86	elmap 7772	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ (ℕ0 ↑𝑚 ℕ) ↔ 𝑞:ℕ⟶ℕ0)
88	87	3anbi1i 1246	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽) ↔ (𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
89	85, 88	bitri 263	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) ↔ (𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
90		df-3an 1033	. . . . . . . . . . . 12 ⊢ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
91		cnvimass 5404	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑞 “ ℕ) ⊆ dom 𝑞
92		fdm 5964	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞:ℕ⟶ℕ0 → dom 𝑞 = ℕ)
93	91, 92	syl5sseq 3616	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞:ℕ⟶ℕ0 → (◡𝑞 “ ℕ) ⊆ ℕ)
94		dfss3 3558	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑞 “ ℕ) ⊆ ℕ ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ)
95	93, 94	sylib 207	. . . . . . . . . . . . . . . 16 ⊢ (𝑞:ℕ⟶ℕ0 → ∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ)
96	95	biantrurd 528	. . . . . . . . . . . . . . 15 ⊢ (𝑞:ℕ⟶ℕ0 → (∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛)))
97		dfss3 3558	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ 𝐽)
98		breq2 4587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
99	98	notbid 307	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
100	99, 16	elrab2 3333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
101	100	ralbii 2963	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
102		r19.26 3046	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (◡𝑞 “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
103	97, 101, 102	3bitri 285	. . . . . . . . . . . . . . 15 ⊢ ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
104	96, 103	syl6rbbr 278	. . . . . . . . . . . . . 14 ⊢ (𝑞:ℕ⟶ℕ0 → ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
105	104	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) → ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
106	105	pm5.32i 667	. . . . . . . . . . . 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 285	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
108	107	anbi1i 727	. . . . . . . . . 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 266	. . . . . . . . 9 ⊢ ((𝑞 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ (𝑞 ∈ (𝑇 ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
110		rabid 3095	. . . . . . . . 9 ⊢ (𝑞 ∈ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ (𝑞 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
111		rabid 3095	. . . . . . . . 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 291	. . . . . . . 8 ⊢ (𝑞 ∈ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ 𝑞 ∈ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁})
113	77, 78, 112	eqri 28735	. . . . . . 7 ⊢ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
114	14, 76, 113	3eqtri 2636	. . . . . 6 ⊢ 𝑂 = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
115	114	reseq2i 5314	. . . . 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 2751	. . . . . 6 ⊢ Ⅎ𝑑𝐷
119		nfrab1 3099	. . . . . 6 ⊢ Ⅎ𝑑{𝑑 ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁}
120		fnima 5923	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 Fn ℕ → (𝑑 “ ℕ) = ran 𝑑)
121	120	sseq1d 3595	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 Fn ℕ → ((𝑑 “ ℕ) ⊆ {0, 1} ↔ ran 𝑑 ⊆ {0, 1}))
122	121	anbi2d 736	. . . . . . . . . . . . . . 15 ⊢ (𝑑 Fn ℕ → ((ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (ran 𝑑 ⊆ ℕ0 ∧ ran 𝑑 ⊆ {0, 1})))
123		sstr 3576	. . . . . . . . . . . . . . . . 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 663	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑑 ⊆ {0, 1} ↔ (ran 𝑑 ⊆ ℕ0 ∧ ran 𝑑 ⊆ {0, 1}))
126	122, 125	syl6bbr 277	. . . . . . . . . . . . . 14 ⊢ (𝑑 Fn ℕ → ((ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ran 𝑑 ⊆ {0, 1}))
127	126	pm5.32i 667	. . . . . . . . . . . . 13 ⊢ ((𝑑 Fn ℕ ∧ (ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1})) ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ {0, 1}))
128		anass 679	. . . . . . . . . . . . 13 ⊢ (((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (𝑑 Fn ℕ ∧ (ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1})))
129		df-f 5808	. . . . . . . . . . . . 13 ⊢ (𝑑:ℕ⟶{0, 1} ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ {0, 1}))
130	127, 128, 129	3bitr4ri 292	. . . . . . . . . . . 12 ⊢ (𝑑:ℕ⟶{0, 1} ↔ ((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
131		prex 4836	. . . . . . . . . . . . 13 ⊢ {0, 1} ∈ V
132	131, 86	elmap 7772	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ({0, 1} ↑𝑚 ℕ) ↔ 𝑑:ℕ⟶{0, 1})
133		df-f 5808	. . . . . . . . . . . . 13 ⊢ (𝑑:ℕ⟶ℕ0 ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0))
134	133	anbi1i 727	. . . . . . . . . . . 12 ⊢ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
135	130, 132, 134	3bitr4i 291	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ({0, 1} ↑𝑚 ℕ) ↔ (𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
136		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑑 ∈ V
137		cnveq 5218	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑑 → ◡𝑓 = ◡𝑑)
138	137	imaeq1d 5384	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑑 → (◡𝑓 “ ℕ) = (◡𝑑 “ ℕ))
139	138	eleq1d 2672	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑑 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝑑 “ ℕ) ∈ Fin))
140	136, 139, 20	elab2 3323	. . . . . . . . . . 11 ⊢ (𝑑 ∈ 𝑅 ↔ (◡𝑑 “ ℕ) ∈ Fin)
141	135, 140	anbi12i 729	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ({0, 1} ↑𝑚 ℕ) ∧ 𝑑 ∈ 𝑅) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ (◡𝑑 “ ℕ) ∈ Fin))
142		elin 3758	. . . . . . . . . 10 ⊢ (𝑑 ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) ↔ (𝑑 ∈ ({0, 1} ↑𝑚 ℕ) ∧ 𝑑 ∈ 𝑅))
143		an32 835	. . . . . . . . . 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 291	. . . . . . . . 9 ⊢ (𝑑 ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
145	144	anbi1i 727	. . . . . . . 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 29752	. . . . . . . . . 10 ⊢ (𝑑 ∈ 𝑃 ↔ (𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁))
147	146	anbi1i 727	. . . . . . . . 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 1033	. . . . . . . . . 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 727	. . . . . . . . 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 835	. . . . . . . . 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 285	. . . . . . . 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 266	. . . . . . 7 ⊢ ((𝑑 ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁) ↔ (𝑑 ∈ 𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
153		rabid 3095	. . . . . . 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 29755	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 ↔ (𝑑 ∈ 𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
155	152, 153, 154	3bitr4ri 292	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 ↔ 𝑑 ∈ {𝑑 ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁})
156	118, 119, 155	eqri 28735	. . . . 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 6046	. . 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 246	. 2 ⊢ (⊤ → (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷)
160	159	trud 1484	1 ⊢ (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {cpr 4127   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   supp csupp 7182   ↑_𝑚 cmap 7744  Fincfn 7841  0cc0 9815  1c1 9816   · cmul 9820   ≤ cle 9954  ℕcn 10897  2c2 10947  ℕ₀cn0 11169  ↑cexp 12722  Σcsu 14264   ∥ cdvds 14821  bitscbits 14979  𝟭cind 29400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-bits 14982  df-ind 29401

This theorem is referenced by:  eulerpart  29771