Theorem eulerpartlemn 34803

Description: Lemma for eulerpart 34804. (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 488	. . . . . . . . . . . . 13 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → 𝑜 = 𝑞)
2	1	fveq1d 6890	. . . . . . . . . . . 12 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → (𝑜‘𝑘) = (𝑞‘𝑘))
3	2	oveq1d 7438	. . . . . . . . . . 11 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → ((𝑜‘𝑘) · 𝑘) = ((𝑞‘𝑘) · 𝑘))
4	3	sumeq2dv 15779	. . . . . . . . . 10 ⊢ (𝑜 = 𝑞 → Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘))
5	4	eqeq1d 2768	. . . . . . . . 9 ⊢ (𝑜 = 𝑞 → (Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
6	5	cbvrabv 3429	. . . . . . . 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 5983	. . . . . 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 2767	. . . . . 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 6821	. . . . 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 34794	. . . . . 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 6888	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑜 → (𝐺‘𝑞) = (𝐺‘𝑜))
25		reseq1 5977	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = 𝑜 → (𝑞 ↾ 𝐽) = (𝑜 ↾ 𝐽))
26	25	coeq2d 5853	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 𝑜 → (bits ∘ (𝑞 ↾ 𝐽)) = (bits ∘ (𝑜 ↾ 𝐽)))
27	26	fveq2d 6892	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 𝑜 → (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))) = (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))
28	27	imaeq2d 6067	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑜 → (𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))
29	28	fveq2d 6892	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑜 → ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))))) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
30	24, 29	eqeq12d 2782	. . . . . . . . . . . . 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 34795	. . . . . . . . . . . . 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 1152	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (𝐺‘𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
34		simp3 1156	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
35	33, 34	eqtr4d 2804	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (𝐺‘𝑜) = 𝑑)
36	35	fveq1d 6890	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → ((𝐺‘𝑜)‘𝑘) = (𝑑‘𝑘))
37	36	oveq1d 7438	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (((𝐺‘𝑜)‘𝑘) · 𝑘) = ((𝑑‘𝑘) · 𝑘))
38	37	sumeq2sdv 15780	. . . . . . 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 6892	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑜 → (𝑆‘(𝐺‘𝑞)) = (𝑆‘(𝐺‘𝑜)))
40		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑜 → (𝑆‘𝑞) = (𝑆‘𝑜))
41	39, 40	eqeq12d 2782	. . . . . . . . . 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 34802	. . . . . . . . . 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 12528	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
46		0nn0 12537	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
47		1nn0 12538	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ0
48		prssi 4792	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
49	46, 47, 48	mp2an 705	. . . . . . . . . . . . 13 ⊢ {0, 1} ⊆ ℕ0
50		mapss 8896	. . . . . . . . . . . . 13 ⊢ ((ℕ0 ∈ V ∧ {0, 1} ⊆ ℕ0) → ({0, 1} ↑m ℕ) ⊆ (ℕ0 ↑m ℕ))
51	45, 49, 50	mp2an 705	. . . . . . . . . . . 12 ⊢ ({0, 1} ↑m ℕ) ⊆ (ℕ0 ↑m ℕ)
52		ssrin 4197	. . . . . . . . . . . 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 6827	. . . . . . . . . . . . 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 7085	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅))
57	53, 56	sselid 3938	. . . . . . . . . 10 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅))
58	20, 42	eulerpartlemsv1 34778	. . . . . . . . . 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 34791	. . . . . . . . . . . 12 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) ↔ (𝑜 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑜 “ ℕ) ∈ Fin ∧ (◡𝑜 “ ℕ) ⊆ 𝐽))
61	60	simp1bi 1163	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ (ℕ0 ↑m ℕ))
62		inss2 4193	. . . . . . . . . . . 12 ⊢ (𝑇 ∩ 𝑅) ⊆ 𝑅
63	62	sseli 3936	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ 𝑅)
64	61, 63	elind 4156	. . . . . . . . . 10 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅))
65	20, 42	eulerpartlemsv1 34778	. . . . . . . . . 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 2809	. . . . . . . 8 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ ℕ (((𝐺‘𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘))
68	67	3ad2ant2 1152	. . . . . . 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 2803	. . . . . 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 2768	. . . . 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 7130	. . . 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 2022	. . 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 5864	. . . . . . . . . 10 ⊢ (𝑔 = 𝑞 → ◡𝑔 = ◡𝑞)
74	73	imaeq1d 6066	. . . . . . . . 9 ⊢ (𝑔 = 𝑞 → (◡𝑔 “ ℕ) = (◡𝑞 “ ℕ))
75	74	raleqdv 3326	. . . . . . . 8 ⊢ (𝑔 = 𝑞 → (∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
76	75	cbvrabv 3429	. . . . . . 7 ⊢ {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
77		nfrab1 3439	. . . . . . . 8 ⊢ Ⅎ𝑞{𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
78		nfrab1 3439	. . . . . . . 8 ⊢ Ⅎ𝑞{𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
79		df-3an 1105	. . . . . . . . . . . 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 636	. . . . . . . . . . 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 34785	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑃 ↔ (𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
82	81	anbi1i 636	. . . . . . . . . . 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 659	. . . . . . . . . . 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 34791	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) ↔ (𝑞 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
86		nnex 12257	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
87	45, 86	elmap 8878	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ (ℕ0 ↑m ℕ) ↔ 𝑞:ℕ⟶ℕ0)
88	87	3anbi1i 1175	. . . . . . . . . . . . 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 1105	. . . . . . . . . . . 12 ⊢ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
91		dfss3 3929	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ 𝐽)
92		breq2 5118	. . . . . . . . . . . . . . . . . . 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 3114	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
96		r19.26 3128	. . . . . . . . . . . . . . . 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 6089	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑞 “ ℕ) ⊆ dom 𝑞
99		fdm 6722	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞:ℕ⟶ℕ0 → dom 𝑞 = ℕ)
100	98, 99	sseqtrid 3982	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞:ℕ⟶ℕ0 → (◡𝑞 “ ℕ) ⊆ ℕ)
101		dfss3 3929	. . . . . . . . . . . . . . . . 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 542	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) → ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
106	105	pm5.32i 585	. . . . . . . . . . . 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 636	. . . . . . . . . 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 3440	. . . . . . . . 9 ⊢ (𝑞 ∈ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ (𝑞 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
111		rabid 3440	. . . . . . . . 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 3960	. . . . . . 7 ⊢ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
114	14, 76, 113	3eqtri 2793	. . . . . 6 ⊢ 𝑂 = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
115	114	reseq2i 5980	. . . . 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 2928	. . . . . 6 ⊢ Ⅎ𝑑𝐷
119		nfrab1 3439	. . . . . 6 ⊢ Ⅎ𝑑{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁}
120		fnima 6672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 Fn ℕ → (𝑑 “ ℕ) = ran 𝑑)
121	120	sseq1d 3971	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 Fn ℕ → ((𝑑 “ ℕ) ⊆ {0, 1} ↔ ran 𝑑 ⊆ {0, 1}))
122	121	anbi2d 642	. . . . . . . . . . . . . . 15 ⊢ (𝑑 Fn ℕ → ((ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (ran 𝑑 ⊆ ℕ0 ∧ ran 𝑑 ⊆ {0, 1})))
123		sstr 3948	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝑑 ⊆ {0, 1} ∧ {0, 1} ⊆ ℕ0) → ran 𝑑 ⊆ ℕ0)
124	49, 123	mpan2 704	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝑑 ⊆ {0, 1} → ran 𝑑 ⊆ ℕ0)
125	124	pm4.71ri 570	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ ((𝑑 Fn ℕ ∧ (ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1})) ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ {0, 1}))
128		anass 474	. . . . . . . . . . . . 13 ⊢ (((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (𝑑 Fn ℕ ∧ (ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1})))
129		df-f 6547	. . . . . . . . . . . . 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 5414	. . . . . . . . . . . . 13 ⊢ {0, 1} ∈ V
132	131, 86	elmap 8878	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ({0, 1} ↑m ℕ) ↔ 𝑑:ℕ⟶{0, 1})
133		df-f 6547	. . . . . . . . . . . . 13 ⊢ (𝑑:ℕ⟶ℕ0 ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0))
134	133	anbi1i 636	. . . . . . . . . . . 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 3462	. . . . . . . . . . . 12 ⊢ 𝑑 ∈ V
137		cnveq 5864	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑑 → ◡𝑓 = ◡𝑑)
138	137	imaeq1d 6066	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑑 → (◡𝑓 “ ℕ) = (◡𝑑 “ ℕ))
139	138	eleq1d 2851	. . . . . . . . . . . 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 640	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ({0, 1} ↑m ℕ) ∧ 𝑑 ∈ 𝑅) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ (◡𝑑 “ ℕ) ∈ Fin))
142		elin 3924	. . . . . . . . . 10 ⊢ (𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ (𝑑 ∈ ({0, 1} ↑m ℕ) ∧ 𝑑 ∈ 𝑅))
143		an32 659	. . . . . . . . . 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 636	. . . . . . . 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 34785	. . . . . . . . . 10 ⊢ (𝑑 ∈ 𝑃 ↔ (𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁))
147	146	anbi1i 636	. . . . . . . . 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 1105	. . . . . . . . . 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 636	. . . . . . . . 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 659	. . . . . . . . 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 3440	. . . . . . 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 34788	. . . . . . 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 3960	. . . . 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 6821	. . 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 1577	1 ⊢ (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  {cab 2744  ∀wral 3082  {crab 3419  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567  {cpr 4596   class class class wbr 5114  {copab 5178   ↦ cmpt 5197  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   ↾ cres 5668   “ cima 5669   ∘ ccom 5670   Fn wfn 6538  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425   supp csupp 8165   ↑_m cmap 8833  Fincfn 8952  0cc0 11118  1c1 11119   · cmul 11123   ≤ cle 11262  𝟭cind 12236  ℕcn 12251  2c2 12313  ℕ₀cn0 12522  ↑cexp 14117  Σcsu 15763   ∥ cdvds 16335  bitscbits 16502

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-inf2 9620  ax-ac2 10465  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-pre-sup 11196

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-disj 5082  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  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 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8703  df-map 8835  df-pm 8836  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-sup 9412  df-inf 9413  df-oi 9482  df-dju 9906  df-card 9944  df-acn 9947  df-ac 10119  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-div 11890  df-ind 12237  df-nn 12252  df-2 12321  df-3 12322  df-n0 12523  df-xnn0 12596  df-z 12610  df-uz 12881  df-rp 13035  df-fz 13554  df-fzo 13702  df-fl 13845  df-mod 13923  df-seq 14058  df-exp 14118  df-hash 14387  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-clim 15565  df-sum 15764  df-dvds 16336  df-bits 16505

This theorem is used by:  eulerpart  34804