Theorem eulerpartlemn 32648

Description: Lemma for eulerpart 32649. (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 483	. . . . . . . . . . . . 13 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → 𝑜 = 𝑞)
2	1	fveq1d 6828	. . . . . . . . . . . 12 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → (𝑜‘𝑘) = (𝑞‘𝑘))
3	2	oveq1d 7353	. . . . . . . . . . 11 ⊢ ((𝑜 = 𝑞 ∧ 𝑘 ∈ ℕ) → ((𝑜‘𝑘) · 𝑘) = ((𝑞‘𝑘) · 𝑘))
4	3	sumeq2dv 15515	. . . . . . . . . 10 ⊢ (𝑜 = 𝑞 → Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘))
5	4	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑜 = 𝑞 → (Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
6	5	cbvrabv 3413	. . . . . . . 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 5924	. . . . . 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 2737	. . . . . 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 6762	. . . . 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 340	. . . 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 32639	. . . . . 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 6826	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑜 → (𝐺‘𝑞) = (𝐺‘𝑜))
25		reseq1 5918	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = 𝑜 → (𝑞 ↾ 𝐽) = (𝑜 ↾ 𝐽))
26	25	coeq2d 5805	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 𝑜 → (bits ∘ (𝑞 ↾ 𝐽)) = (bits ∘ (𝑜 ↾ 𝐽)))
27	26	fveq2d 6830	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 𝑜 → (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))) = (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))
28	27	imaeq2d 6000	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑜 → (𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))
29	28	fveq2d 6830	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑜 → ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))))) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
30	24, 29	eqeq12d 2752	. . . . . . . . . . . . 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 32640	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑞) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞 ↾ 𝐽))))))
32	30, 31	vtoclga 3522	. . . . . . . . . . . 12 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
33	32	3ad2ant2 1133	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (𝐺‘𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
34		simp3 1137	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
35	33, 34	eqtr4d 2779	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (𝐺‘𝑜) = 𝑑)
36	35	fveq1d 6828	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → ((𝐺‘𝑜)‘𝑘) = (𝑑‘𝑘))
37	36	oveq1d 7353	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑜 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) → (((𝐺‘𝑜)‘𝑘) · 𝑘) = ((𝑑‘𝑘) · 𝑘))
38	37	sumeq2sdv 15516	. . . . . . 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 6830	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑜 → (𝑆‘(𝐺‘𝑞)) = (𝑆‘(𝐺‘𝑜)))
40		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑜 → (𝑆‘𝑞) = (𝑆‘𝑜))
41	39, 40	eqeq12d 2752	. . . . . . . . . 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 32647	. . . . . . . . . 10 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝑞)) = (𝑆‘𝑞))
44	41, 43	vtoclga 3522	. . . . . . . . 9 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝑜)) = (𝑆‘𝑜))
45		nn0ex 12341	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
46		0nn0 12350	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
47		1nn0 12351	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ0
48		prssi 4769	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
49	46, 47, 48	mp2an 689	. . . . . . . . . . . . 13 ⊢ {0, 1} ⊆ ℕ0
50		mapss 8749	. . . . . . . . . . . . 13 ⊢ ((ℕ0 ∈ V ∧ {0, 1} ⊆ ℕ0) → ({0, 1} ↑m ℕ) ⊆ (ℕ0 ↑m ℕ))
51	45, 49, 50	mp2an 689	. . . . . . . . . . . 12 ⊢ ({0, 1} ↑m ℕ) ⊆ (ℕ0 ↑m ℕ)
52		ssrin 4181	. . . . . . . . . . . 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 6768	. . . . . . . . . . . . 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 7017	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅))
57	53, 56	sselid 3930	. . . . . . . . . 10 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝑜) ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅))
58	20, 42	eulerpartlemsv1 32623	. . . . . . . . . 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 32636	. . . . . . . . . . . 12 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) ↔ (𝑜 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑜 “ ℕ) ∈ Fin ∧ (◡𝑜 “ ℕ) ⊆ 𝐽))
61	60	simp1bi 1144	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ (ℕ0 ↑m ℕ))
62		inss2 4177	. . . . . . . . . . . 12 ⊢ (𝑇 ∩ 𝑅) ⊆ 𝑅
63	62	sseli 3928	. . . . . . . . . . 11 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ 𝑅)
64	61, 63	elind 4142	. . . . . . . . . 10 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → 𝑜 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅))
65	20, 42	eulerpartlemsv1 32623	. . . . . . . . . 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 2784	. . . . . . . 8 ⊢ (𝑜 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ ℕ (((𝐺‘𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜‘𝑘) · 𝑘))
68	67	3ad2ant2 1133	. . . . . . 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 2778	. . . . . 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 2738	. . . . 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 7056	. . . 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 2001	. . 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 5816	. . . . . . . . . 10 ⊢ (𝑔 = 𝑞 → ◡𝑔 = ◡𝑞)
74	73	imaeq1d 5999	. . . . . . . . 9 ⊢ (𝑔 = 𝑞 → (◡𝑔 “ ℕ) = (◡𝑞 “ ℕ))
75	74	raleqdv 3309	. . . . . . . 8 ⊢ (𝑔 = 𝑞 → (∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
76	75	cbvrabv 3413	. . . . . . 7 ⊢ {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
77		nfrab1 3422	. . . . . . . 8 ⊢ Ⅎ𝑞{𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
78		nfrab1 3422	. . . . . . . 8 ⊢ Ⅎ𝑞{𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
79		df-3an 1088	. . . . . . . . . . . 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 624	. . . . . . . . . . 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 32630	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑃 ↔ (𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
82	81	anbi1i 624	. . . . . . . . . . 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 643	. . . . . . . . . . 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 302	. . . . . . . . . 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 32636	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) ↔ (𝑞 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
86		nnex 12081	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
87	45, 86	elmap 8731	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ (ℕ0 ↑m ℕ) ↔ 𝑞:ℕ⟶ℕ0)
88	87	3anbi1i 1156	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽) ↔ (𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
89	85, 88	bitri 274	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) ↔ (𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
90		df-3an 1088	. . . . . . . . . . . 12 ⊢ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin ∧ (◡𝑞 “ ℕ) ⊆ 𝐽) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ (◡𝑞 “ ℕ) ⊆ 𝐽))
91		dfss3 3920	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ 𝐽)
92		breq2 5097	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
93	92	notbid 317	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
94	93, 16	elrab2 3637	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
95	94	ralbii 3092	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
96		r19.26 3110	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (◡𝑞 “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
97	91, 95, 96	3bitri 296	. . . . . . . . . . . . . . 15 ⊢ ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
98		cnvimass 6020	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑞 “ ℕ) ⊆ dom 𝑞
99		fdm 6661	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞:ℕ⟶ℕ0 → dom 𝑞 = ℕ)
100	98, 99	sseqtrid 3984	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞:ℕ⟶ℕ0 → (◡𝑞 “ ℕ) ⊆ ℕ)
101		dfss3 3920	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑞 “ ℕ) ⊆ ℕ ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ)
102	100, 101	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑞:ℕ⟶ℕ0 → ∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ)
103	102	biantrurd 533	. . . . . . . . . . . . . . 15 ⊢ (𝑞:ℕ⟶ℕ0 → (∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ (◡𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛)))
104	97, 103	bitr4id 289	. . . . . . . . . . . . . 14 ⊢ (𝑞:ℕ⟶ℕ0 → ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
105	104	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) → ((◡𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
106	105	pm5.32i 575	. . . . . . . . . . . 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 296	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (𝑇 ∩ 𝑅) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (◡𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
108	107	anbi1i 624	. . . . . . . . . 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 277	. . . . . . . . 9 ⊢ ((𝑞 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ (𝑞 ∈ (𝑇 ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁))
110		rabid 3423	. . . . . . . . 9 ⊢ (𝑞 ∈ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ (𝑞 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
111		rabid 3423	. . . . . . . . 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 302	. . . . . . . 8 ⊢ (𝑞 ∈ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ 𝑞 ∈ {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁})
113	77, 78, 112	eqri 3952	. . . . . . 7 ⊢ {𝑞 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑞 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
114	14, 76, 113	3eqtri 2768	. . . . . 6 ⊢ 𝑂 = {𝑞 ∈ (𝑇 ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞‘𝑘) · 𝑘) = 𝑁}
115	114	reseq2i 5921	. . . . 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 2904	. . . . . 6 ⊢ Ⅎ𝑑𝐷
119		nfrab1 3422	. . . . . 6 ⊢ Ⅎ𝑑{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁}
120		fnima 6615	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 Fn ℕ → (𝑑 “ ℕ) = ran 𝑑)
121	120	sseq1d 3963	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 Fn ℕ → ((𝑑 “ ℕ) ⊆ {0, 1} ↔ ran 𝑑 ⊆ {0, 1}))
122	121	anbi2d 629	. . . . . . . . . . . . . . 15 ⊢ (𝑑 Fn ℕ → ((ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (ran 𝑑 ⊆ ℕ0 ∧ ran 𝑑 ⊆ {0, 1})))
123		sstr 3940	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝑑 ⊆ {0, 1} ∧ {0, 1} ⊆ ℕ0) → ran 𝑑 ⊆ ℕ0)
124	49, 123	mpan2 688	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝑑 ⊆ {0, 1} → ran 𝑑 ⊆ ℕ0)
125	124	pm4.71ri 561	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑑 ⊆ {0, 1} ↔ (ran 𝑑 ⊆ ℕ0 ∧ ran 𝑑 ⊆ {0, 1}))
126	122, 125	bitr4di 288	. . . . . . . . . . . . . 14 ⊢ (𝑑 Fn ℕ → ((ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ran 𝑑 ⊆ {0, 1}))
127	126	pm5.32i 575	. . . . . . . . . . . . 13 ⊢ ((𝑑 Fn ℕ ∧ (ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1})) ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ {0, 1}))
128		anass 469	. . . . . . . . . . . . 13 ⊢ (((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (𝑑 Fn ℕ ∧ (ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1})))
129		df-f 6484	. . . . . . . . . . . . 13 ⊢ (𝑑:ℕ⟶{0, 1} ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ {0, 1}))
130	127, 128, 129	3bitr4ri 303	. . . . . . . . . . . 12 ⊢ (𝑑:ℕ⟶{0, 1} ↔ ((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
131		prex 5378	. . . . . . . . . . . . 13 ⊢ {0, 1} ∈ V
132	131, 86	elmap 8731	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ({0, 1} ↑m ℕ) ↔ 𝑑:ℕ⟶{0, 1})
133		df-f 6484	. . . . . . . . . . . . 13 ⊢ (𝑑:ℕ⟶ℕ0 ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0))
134	133	anbi1i 624	. . . . . . . . . . . 12 ⊢ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
135	130, 132, 134	3bitr4i 302	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ({0, 1} ↑m ℕ) ↔ (𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
136		vex 3445	. . . . . . . . . . . 12 ⊢ 𝑑 ∈ V
137		cnveq 5816	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑑 → ◡𝑓 = ◡𝑑)
138	137	imaeq1d 5999	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑑 → (◡𝑓 “ ℕ) = (◡𝑑 “ ℕ))
139	138	eleq1d 2821	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑑 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝑑 “ ℕ) ∈ Fin))
140	136, 139, 20	elab2 3623	. . . . . . . . . . 11 ⊢ (𝑑 ∈ 𝑅 ↔ (◡𝑑 “ ℕ) ∈ Fin)
141	135, 140	anbi12i 627	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ({0, 1} ↑m ℕ) ∧ 𝑑 ∈ 𝑅) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ (◡𝑑 “ ℕ) ∈ Fin))
142		elin 3914	. . . . . . . . . 10 ⊢ (𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ (𝑑 ∈ ({0, 1} ↑m ℕ) ∧ 𝑑 ∈ 𝑅))
143		an32 643	. . . . . . . . . 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 302	. . . . . . . . 9 ⊢ (𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
145	144	anbi1i 624	. . . . . . . 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 32630	. . . . . . . . . 10 ⊢ (𝑑 ∈ 𝑃 ↔ (𝑑:ℕ⟶ℕ0 ∧ (◡𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁))
147	146	anbi1i 624	. . . . . . . . 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 1088	. . . . . . . . . 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 624	. . . . . . . . 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 643	. . . . . . . . 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 296	. . . . . . . 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 277	. . . . . . 7 ⊢ ((𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁) ↔ (𝑑 ∈ 𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
153		rabid 3423	. . . . . . 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 32633	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 ↔ (𝑑 ∈ 𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
155	152, 153, 154	3bitr4ri 303	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 ↔ 𝑑 ∈ {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑‘𝑘) · 𝑘) = 𝑁})
156	118, 119, 155	eqri 3952	. . . . 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 6762	. . 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 256	. 2 ⊢ (⊤ → (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷)
160	159	mptru 1547	1 ⊢ (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541   ∈ wcel 2105  {cab 2713  ∀wral 3061  {crab 3403  Vcvv 3441   ∩ cin 3897   ⊆ wss 3898  ∅c0 4270  𝒫 cpw 4548  {cpr 4576   class class class wbr 5093  {copab 5155   ↦ cmpt 5176  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   ↾ cres 5623   “ cima 5624   ∘ ccom 5625   Fn wfn 6475  ⟶wf 6476  –1-1-onto→wf1o 6479  ‘cfv 6480  (class class class)co 7338   ∈ cmpo 7340   supp csupp 8048   ↑_m cmap 8687  Fincfn 8805  0cc0 10973  1c1 10974   · cmul 10978   ≤ cle 11112  ℕcn 12075  2c2 12130  ℕ₀cn0 12335  ↑cexp 13884  Σcsu 15497   ∥ cdvds 16063  bitscbits 16226  𝟭cind 32276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651  ax-inf2 9499  ax-ac2 10321  ax-cnex 11029  ax-resscn 11030  ax-1cn 11031  ax-icn 11032  ax-addcl 11033  ax-addrcl 11034  ax-mulcl 11035  ax-mulrcl 11036  ax-mulcom 11037  ax-addass 11038  ax-mulass 11039  ax-distr 11040  ax-i2m1 11041  ax-1ne0 11042  ax-1rid 11043  ax-rnegex 11044  ax-rrecex 11045  ax-cnre 11046  ax-pre-lttri 11047  ax-pre-lttrn 11048  ax-pre-ltadd 11049  ax-pre-mulgt0 11050  ax-pre-sup 11051

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-int 4896  df-iun 4944  df-disj 5059  df-br 5094  df-opab 5156  df-mpt 5177  df-tr 5211  df-id 5519  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5576  df-se 5577  df-we 5578  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6239  df-ord 6306  df-on 6307  df-lim 6308  df-suc 6309  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-isom 6489  df-riota 7294  df-ov 7341  df-oprab 7342  df-mpo 7343  df-om 7782  df-1st 7900  df-2nd 7901  df-supp 8049  df-frecs 8168  df-wrecs 8199  df-recs 8273  df-rdg 8312  df-1o 8368  df-2o 8369  df-oadd 8372  df-er 8570  df-map 8689  df-pm 8690  df-en 8806  df-dom 8807  df-sdom 8808  df-fin 8809  df-fsupp 9228  df-sup 9300  df-inf 9301  df-oi 9368  df-dju 9759  df-card 9797  df-acn 9800  df-ac 9974  df-pnf 11113  df-mnf 11114  df-xr 11115  df-ltxr 11116  df-le 11117  df-sub 11309  df-neg 11310  df-div 11735  df-nn 12076  df-2 12138  df-3 12139  df-n0 12336  df-xnn0 12408  df-z 12422  df-uz 12685  df-rp 12833  df-fz 13342  df-fzo 13485  df-fl 13614  df-mod 13692  df-seq 13824  df-exp 13885  df-hash 14147  df-cj 14910  df-re 14911  df-im 14912  df-sqrt 15046  df-abs 15047  df-clim 15297  df-sum 15498  df-dvds 16064  df-bits 16229  df-ind 32277

This theorem is referenced by:  eulerpart  32649