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Theorem eulerpartlemn 31538
Description: Lemma for eulerpart 31539. (Contributed by Thierry Arnoux, 30-Aug-2018.)
Hypotheses
Ref Expression
eulerpart.p 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
eulerpart.o 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
eulerpart.j 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
eulerpart.r 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
eulerpart.t 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
eulerpart.g 𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
eulerpart.s 𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))
Assertion
Ref Expression
eulerpartlemn (𝐺𝑂):𝑂1-1-onto𝐷
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.
StepHypRef Expression
1 simpl 483 . . . . . . . . . . . . 13 ((𝑜 = 𝑞𝑘 ∈ ℕ) → 𝑜 = 𝑞)
21fveq1d 6665 . . . . . . . . . . . 12 ((𝑜 = 𝑞𝑘 ∈ ℕ) → (𝑜𝑘) = (𝑞𝑘))
32oveq1d 7160 . . . . . . . . . . 11 ((𝑜 = 𝑞𝑘 ∈ ℕ) → ((𝑜𝑘) · 𝑘) = ((𝑞𝑘) · 𝑘))
43sumeq2dv 15048 . . . . . . . . . 10 (𝑜 = 𝑞 → Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘))
54eqeq1d 2820 . . . . . . . . 9 (𝑜 = 𝑞 → (Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
65cbvrabv 3489 . . . . . . . 8 {𝑜 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁} = {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}
76a1i 11 . . . . . . 7 (𝑜 = 𝑞 → {𝑜 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁} = {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁})
87reseq2d 5846 . . . . . 6 (𝑜 = 𝑞 → (𝐺 ↾ {𝑜 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁}) = (𝐺 ↾ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}))
9 eqidd 2819 . . . . . 6 (𝑜 = 𝑞 → {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁} = {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁})
108, 7, 9f1oeq123d 6603 . . . . 5 (𝑜 = 𝑞 → ((𝐺 ↾ {𝑜 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁}):{𝑜 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁} ↔ (𝐺 ↾ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}):{𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁}))
1110imbi2d 342 . . . 4 (𝑜 = 𝑞 → ((⊤ → (𝐺 ↾ {𝑜 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁}):{𝑜 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁}) ↔ (⊤ → (𝐺 ↾ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}):{𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁})))
12 eulerpart.g . . . . 5 𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
13 eulerpart.p . . . . . . 7 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
14 eulerpart.o . . . . . . 7 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
15 eulerpart.d . . . . . . 7 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
16 eulerpart.j . . . . . . 7 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
17 eulerpart.f . . . . . . 7 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
18 eulerpart.h . . . . . . 7 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
19 eulerpart.m . . . . . . 7 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
20 eulerpart.r . . . . . . 7 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
21 eulerpart.t . . . . . . 7 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
2213, 14, 15, 16, 17, 18, 19, 20, 21, 12eulerpartgbij 31529 . . . . . 6 𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅)
2322a1i 11 . . . . 5 (⊤ → 𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅))
24 fveq2 6663 . . . . . . . . . . . . . 14 (𝑞 = 𝑜 → (𝐺𝑞) = (𝐺𝑜))
25 reseq1 5840 . . . . . . . . . . . . . . . . . 18 (𝑞 = 𝑜 → (𝑞𝐽) = (𝑜𝐽))
2625coeq2d 5726 . . . . . . . . . . . . . . . . 17 (𝑞 = 𝑜 → (bits ∘ (𝑞𝐽)) = (bits ∘ (𝑜𝐽)))
2726fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑞 = 𝑜 → (𝑀‘(bits ∘ (𝑞𝐽))) = (𝑀‘(bits ∘ (𝑜𝐽))))
2827imaeq2d 5922 . . . . . . . . . . . . . . 15 (𝑞 = 𝑜 → (𝐹 “ (𝑀‘(bits ∘ (𝑞𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))
2928fveq2d 6667 . . . . . . . . . . . . . 14 (𝑞 = 𝑜 → ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞𝐽))))) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
3024, 29eqeq12d 2834 . . . . . . . . . . . . 13 (𝑞 = 𝑜 → ((𝐺𝑞) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞𝐽))))) ↔ (𝐺𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))))
3113, 14, 15, 16, 17, 18, 19, 20, 21, 12eulerpartlemgv 31530 . . . . . . . . . . . . 13 (𝑞 ∈ (𝑇𝑅) → (𝐺𝑞) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞𝐽))))))
3230, 31vtoclga 3571 . . . . . . . . . . . 12 (𝑜 ∈ (𝑇𝑅) → (𝐺𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
33323ad2ant2 1126 . . . . . . . . . . 11 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → (𝐺𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
34 simp3 1130 . . . . . . . . . . 11 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
3533, 34eqtr4d 2856 . . . . . . . . . 10 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → (𝐺𝑜) = 𝑑)
3635fveq1d 6665 . . . . . . . . 9 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → ((𝐺𝑜)‘𝑘) = (𝑑𝑘))
3736oveq1d 7160 . . . . . . . 8 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → (((𝐺𝑜)‘𝑘) · 𝑘) = ((𝑑𝑘) · 𝑘))
3837sumeq2sdv 15049 . . . . . . 7 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → Σ𝑘 ∈ ℕ (((𝐺𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘))
3924fveq2d 6667 . . . . . . . . . . 11 (𝑞 = 𝑜 → (𝑆‘(𝐺𝑞)) = (𝑆‘(𝐺𝑜)))
40 fveq2 6663 . . . . . . . . . . 11 (𝑞 = 𝑜 → (𝑆𝑞) = (𝑆𝑜))
4139, 40eqeq12d 2834 . . . . . . . . . 10 (𝑞 = 𝑜 → ((𝑆‘(𝐺𝑞)) = (𝑆𝑞) ↔ (𝑆‘(𝐺𝑜)) = (𝑆𝑜)))
42 eulerpart.s . . . . . . . . . . 11 𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))
4313, 14, 15, 16, 17, 18, 19, 20, 21, 12, 42eulerpartlemgs2 31537 . . . . . . . . . 10 (𝑞 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝑞)) = (𝑆𝑞))
4441, 43vtoclga 3571 . . . . . . . . 9 (𝑜 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝑜)) = (𝑆𝑜))
45 nn0ex 11891 . . . . . . . . . . . . 13 0 ∈ V
46 0nn0 11900 . . . . . . . . . . . . . 14 0 ∈ ℕ0
47 1nn0 11901 . . . . . . . . . . . . . 14 1 ∈ ℕ0
48 prssi 4746 . . . . . . . . . . . . . 14 ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
4946, 47, 48mp2an 688 . . . . . . . . . . . . 13 {0, 1} ⊆ ℕ0
50 mapss 8441 . . . . . . . . . . . . 13 ((ℕ0 ∈ V ∧ {0, 1} ⊆ ℕ0) → ({0, 1} ↑m ℕ) ⊆ (ℕ0m ℕ))
5145, 49, 50mp2an 688 . . . . . . . . . . . 12 ({0, 1} ↑m ℕ) ⊆ (ℕ0m ℕ)
52 ssrin 4207 . . . . . . . . . . . 12 (({0, 1} ↑m ℕ) ⊆ (ℕ0m ℕ) → (({0, 1} ↑m ℕ) ∩ 𝑅) ⊆ ((ℕ0m ℕ) ∩ 𝑅))
5351, 52ax-mp 5 . . . . . . . . . . 11 (({0, 1} ↑m ℕ) ∩ 𝑅) ⊆ ((ℕ0m ℕ) ∩ 𝑅)
54 f1of 6608 . . . . . . . . . . . . 13 (𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅) → 𝐺:(𝑇𝑅)⟶(({0, 1} ↑m ℕ) ∩ 𝑅))
5522, 54ax-mp 5 . . . . . . . . . . . 12 𝐺:(𝑇𝑅)⟶(({0, 1} ↑m ℕ) ∩ 𝑅)
5655ffvelrni 6842 . . . . . . . . . . 11 (𝑜 ∈ (𝑇𝑅) → (𝐺𝑜) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅))
5753, 56sseldi 3962 . . . . . . . . . 10 (𝑜 ∈ (𝑇𝑅) → (𝐺𝑜) ∈ ((ℕ0m ℕ) ∩ 𝑅))
5820, 42eulerpartlemsv1 31513 . . . . . . . . . 10 ((𝐺𝑜) ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆‘(𝐺𝑜)) = Σ𝑘 ∈ ℕ (((𝐺𝑜)‘𝑘) · 𝑘))
5957, 58syl 17 . . . . . . . . 9 (𝑜 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝑜)) = Σ𝑘 ∈ ℕ (((𝐺𝑜)‘𝑘) · 𝑘))
6013, 14, 15, 16, 17, 18, 19, 20, 21eulerpartlemt0 31526 . . . . . . . . . . . 12 (𝑜 ∈ (𝑇𝑅) ↔ (𝑜 ∈ (ℕ0m ℕ) ∧ (𝑜 “ ℕ) ∈ Fin ∧ (𝑜 “ ℕ) ⊆ 𝐽))
6160simp1bi 1137 . . . . . . . . . . 11 (𝑜 ∈ (𝑇𝑅) → 𝑜 ∈ (ℕ0m ℕ))
62 inss2 4203 . . . . . . . . . . . 12 (𝑇𝑅) ⊆ 𝑅
6362sseli 3960 . . . . . . . . . . 11 (𝑜 ∈ (𝑇𝑅) → 𝑜𝑅)
6461, 63elind 4168 . . . . . . . . . 10 (𝑜 ∈ (𝑇𝑅) → 𝑜 ∈ ((ℕ0m ℕ) ∩ 𝑅))
6520, 42eulerpartlemsv1 31513 . . . . . . . . . 10 (𝑜 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝑜) = Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘))
6664, 65syl 17 . . . . . . . . 9 (𝑜 ∈ (𝑇𝑅) → (𝑆𝑜) = Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘))
6744, 59, 663eqtr3d 2861 . . . . . . . 8 (𝑜 ∈ (𝑇𝑅) → Σ𝑘 ∈ ℕ (((𝐺𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘))
68673ad2ant2 1126 . . . . . . 7 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → Σ𝑘 ∈ ℕ (((𝐺𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘))
6938, 68eqtr3d 2855 . . . . . 6 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘))
7069eqeq1d 2820 . . . . 5 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → (Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁))
7112, 23, 70f1oresrab 6881 . . . 4 (⊤ → (𝐺 ↾ {𝑜 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁}):{𝑜 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁})
7211, 71chvarvv 1996 . . 3 (⊤ → (𝐺 ↾ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}):{𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁})
73 cnveq 5737 . . . . . . . . . 10 (𝑔 = 𝑞𝑔 = 𝑞)
7473imaeq1d 5921 . . . . . . . . 9 (𝑔 = 𝑞 → (𝑔 “ ℕ) = (𝑞 “ ℕ))
7574raleqdv 3413 . . . . . . . 8 (𝑔 = 𝑞 → (∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
7675cbvrabv 3489 . . . . . . 7 {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞𝑃 ∣ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
77 nfrab1 3382 . . . . . . . 8 𝑞{𝑞𝑃 ∣ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
78 nfrab1 3382 . . . . . . . 8 𝑞{𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}
79 df-3an 1081 . . . . . . . . . . . 12 ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
8079anbi1i 623 . . . . . . . . . . 11 (((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁) ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ (((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁) ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
8113eulerpartleme 31520 . . . . . . . . . . . 12 (𝑞𝑃 ↔ (𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
8281anbi1i 623 . . . . . . . . . . 11 ((𝑞𝑃 ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁) ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
83 an32 642 . . . . . . . . . . 11 ((((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁) ↔ (((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁) ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
8480, 82, 833bitr4i 304 . . . . . . . . . 10 ((𝑞𝑃 ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ (((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
8513, 14, 15, 16, 17, 18, 19, 20, 21eulerpartlemt0 31526 . . . . . . . . . . . . 13 (𝑞 ∈ (𝑇𝑅) ↔ (𝑞 ∈ (ℕ0m ℕ) ∧ (𝑞 “ ℕ) ∈ Fin ∧ (𝑞 “ ℕ) ⊆ 𝐽))
86 nnex 11632 . . . . . . . . . . . . . . 15 ℕ ∈ V
8745, 86elmap 8424 . . . . . . . . . . . . . 14 (𝑞 ∈ (ℕ0m ℕ) ↔ 𝑞:ℕ⟶ℕ0)
88873anbi1i 1149 . . . . . . . . . . . . 13 ((𝑞 ∈ (ℕ0m ℕ) ∧ (𝑞 “ ℕ) ∈ Fin ∧ (𝑞 “ ℕ) ⊆ 𝐽) ↔ (𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin ∧ (𝑞 “ ℕ) ⊆ 𝐽))
8985, 88bitri 276 . . . . . . . . . . . 12 (𝑞 ∈ (𝑇𝑅) ↔ (𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin ∧ (𝑞 “ ℕ) ⊆ 𝐽))
90 df-3an 1081 . . . . . . . . . . . 12 ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin ∧ (𝑞 “ ℕ) ⊆ 𝐽) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ (𝑞 “ ℕ) ⊆ 𝐽))
91 cnvimass 5942 . . . . . . . . . . . . . . . . . 18 (𝑞 “ ℕ) ⊆ dom 𝑞
92 fdm 6515 . . . . . . . . . . . . . . . . . 18 (𝑞:ℕ⟶ℕ0 → dom 𝑞 = ℕ)
9391, 92sseqtrid 4016 . . . . . . . . . . . . . . . . 17 (𝑞:ℕ⟶ℕ0 → (𝑞 “ ℕ) ⊆ ℕ)
94 dfss3 3953 . . . . . . . . . . . . . . . . 17 ((𝑞 “ ℕ) ⊆ ℕ ↔ ∀𝑛 ∈ (𝑞 “ ℕ)𝑛 ∈ ℕ)
9593, 94sylib 219 . . . . . . . . . . . . . . . 16 (𝑞:ℕ⟶ℕ0 → ∀𝑛 ∈ (𝑞 “ ℕ)𝑛 ∈ ℕ)
9695biantrurd 533 . . . . . . . . . . . . . . 15 (𝑞:ℕ⟶ℕ0 → (∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ (𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛)))
97 dfss3 3953 . . . . . . . . . . . . . . . 16 ((𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (𝑞 “ ℕ)𝑛𝐽)
98 breq2 5061 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
9998notbid 319 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
10099, 16elrab2 3680 . . . . . . . . . . . . . . . . 17 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
101100ralbii 3162 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (𝑞 “ ℕ)𝑛𝐽 ↔ ∀𝑛 ∈ (𝑞 “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
102 r19.26 3167 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (𝑞 “ ℕ)(𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛) ↔ (∀𝑛 ∈ (𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
10397, 101, 1023bitri 298 . . . . . . . . . . . . . . 15 ((𝑞 “ ℕ) ⊆ 𝐽 ↔ (∀𝑛 ∈ (𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
10496, 103syl6rbbr 291 . . . . . . . . . . . . . 14 (𝑞:ℕ⟶ℕ0 → ((𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
105104adantr 481 . . . . . . . . . . . . 13 ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) → ((𝑞 “ ℕ) ⊆ 𝐽 ↔ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
106105pm5.32i 575 . . . . . . . . . . . 12 (((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ (𝑞 “ ℕ) ⊆ 𝐽) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
10789, 90, 1063bitri 298 . . . . . . . . . . 11 (𝑞 ∈ (𝑇𝑅) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
108107anbi1i 623 . . . . . . . . . 10 ((𝑞 ∈ (𝑇𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁) ↔ (((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
10984, 108bitr4i 279 . . . . . . . . 9 ((𝑞𝑃 ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ (𝑞 ∈ (𝑇𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
110 rabid 3376 . . . . . . . . 9 (𝑞 ∈ {𝑞𝑃 ∣ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ (𝑞𝑃 ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
111 rabid 3376 . . . . . . . . 9 (𝑞 ∈ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁} ↔ (𝑞 ∈ (𝑇𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
112109, 110, 1113bitr4i 304 . . . . . . . 8 (𝑞 ∈ {𝑞𝑃 ∣ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ 𝑞 ∈ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁})
11377, 78, 112eqri 3984 . . . . . . 7 {𝑞𝑃 ∣ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}
11414, 76, 1133eqtri 2845 . . . . . 6 𝑂 = {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}
115114reseq2i 5843 . . . . 5 (𝐺𝑂) = (𝐺 ↾ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁})
116115a1i 11 . . . 4 (⊤ → (𝐺𝑂) = (𝐺 ↾ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}))
117114a1i 11 . . . 4 (⊤ → 𝑂 = {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁})
118 nfcv 2974 . . . . . 6 𝑑𝐷
119 nfrab1 3382 . . . . . 6 𝑑{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁}
120 fnima 6471 . . . . . . . . . . . . . . . . 17 (𝑑 Fn ℕ → (𝑑 “ ℕ) = ran 𝑑)
121120sseq1d 3995 . . . . . . . . . . . . . . . 16 (𝑑 Fn ℕ → ((𝑑 “ ℕ) ⊆ {0, 1} ↔ ran 𝑑 ⊆ {0, 1}))
122121anbi2d 628 . . . . . . . . . . . . . . 15 (𝑑 Fn ℕ → ((ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (ran 𝑑 ⊆ ℕ0 ∧ ran 𝑑 ⊆ {0, 1})))
123 sstr 3972 . . . . . . . . . . . . . . . . 17 ((ran 𝑑 ⊆ {0, 1} ∧ {0, 1} ⊆ ℕ0) → ran 𝑑 ⊆ ℕ0)
12449, 123mpan2 687 . . . . . . . . . . . . . . . 16 (ran 𝑑 ⊆ {0, 1} → ran 𝑑 ⊆ ℕ0)
125124pm4.71ri 561 . . . . . . . . . . . . . . 15 (ran 𝑑 ⊆ {0, 1} ↔ (ran 𝑑 ⊆ ℕ0 ∧ ran 𝑑 ⊆ {0, 1}))
126122, 125syl6bbr 290 . . . . . . . . . . . . . 14 (𝑑 Fn ℕ → ((ran 𝑑 ⊆ ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ran 𝑑 ⊆ {0, 1}))
127126pm5.32i 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 6352 . . . . . . . . . . . . 13 (𝑑:ℕ⟶{0, 1} ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ {0, 1}))
130127, 128, 1293bitr4ri 305 . . . . . . . . . . . 12 (𝑑:ℕ⟶{0, 1} ↔ ((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
131 prex 5323 . . . . . . . . . . . . 13 {0, 1} ∈ V
132131, 86elmap 8424 . . . . . . . . . . . 12 (𝑑 ∈ ({0, 1} ↑m ℕ) ↔ 𝑑:ℕ⟶{0, 1})
133 df-f 6352 . . . . . . . . . . . . 13 (𝑑:ℕ⟶ℕ0 ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0))
134133anbi1i 623 . . . . . . . . . . . 12 ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
135130, 132, 1343bitr4i 304 . . . . . . . . . . 11 (𝑑 ∈ ({0, 1} ↑m ℕ) ↔ (𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
136 vex 3495 . . . . . . . . . . . 12 𝑑 ∈ V
137 cnveq 5737 . . . . . . . . . . . . . 14 (𝑓 = 𝑑𝑓 = 𝑑)
138137imaeq1d 5921 . . . . . . . . . . . . 13 (𝑓 = 𝑑 → (𝑓 “ ℕ) = (𝑑 “ ℕ))
139138eleq1d 2894 . . . . . . . . . . . 12 (𝑓 = 𝑑 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝑑 “ ℕ) ∈ Fin))
140136, 139, 20elab2 3667 . . . . . . . . . . 11 (𝑑𝑅 ↔ (𝑑 “ ℕ) ∈ Fin)
141135, 140anbi12i 626 . . . . . . . . . 10 ((𝑑 ∈ ({0, 1} ↑m ℕ) ∧ 𝑑𝑅) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ (𝑑 “ ℕ) ∈ Fin))
142 elin 4166 . . . . . . . . . 10 (𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ (𝑑 ∈ ({0, 1} ↑m ℕ) ∧ 𝑑𝑅))
143 an32 642 . . . . . . . . . 10 (((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ (𝑑 “ ℕ) ∈ Fin))
144141, 142, 1433bitr4i 304 . . . . . . . . 9 (𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
145144anbi1i 623 . . . . . . . 8 ((𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁) ↔ (((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁))
14613eulerpartleme 31520 . . . . . . . . . 10 (𝑑𝑃 ↔ (𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁))
147146anbi1i 623 . . . . . . . . 9 ((𝑑𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
148 df-3an 1081 . . . . . . . . . 10 ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁))
149148anbi1i 623 . . . . . . . . 9 (((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
150 an32 642 . . . . . . . . 9 ((((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁))
151147, 149, 1503bitri 298 . . . . . . . 8 ((𝑑𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁))
152145, 151bitr4i 279 . . . . . . 7 ((𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁) ↔ (𝑑𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
153 rabid 3376 . . . . . . 7 (𝑑 ∈ {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁} ↔ (𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁))
15413, 14, 15eulerpartlemd 31523 . . . . . . 7 (𝑑𝐷 ↔ (𝑑𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
155152, 153, 1543bitr4ri 305 . . . . . 6 (𝑑𝐷𝑑 ∈ {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁})
156118, 119, 155eqri 3984 . . . . 5 𝐷 = {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁}
157156a1i 11 . . . 4 (⊤ → 𝐷 = {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁})
158116, 117, 157f1oeq123d 6603 . . 3 (⊤ → ((𝐺𝑂):𝑂1-1-onto𝐷 ↔ (𝐺 ↾ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}):{𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}–1-1-onto→{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁}))
15972, 158mpbird 258 . 2 (⊤ → (𝐺𝑂):𝑂1-1-onto𝐷)
160159mptru 1535 1 (𝐺𝑂):𝑂1-1-onto𝐷
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wtru 1529  wcel 2105  {cab 2796  wral 3135  {crab 3139  Vcvv 3492  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  {cpr 4559   class class class wbr 5057  {copab 5119  cmpt 5137  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  ccom 5552   Fn wfn 6343  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cmpo 7147   supp csupp 7819  m cmap 8395  Fincfn 8497  0cc0 10525  1c1 10526   · cmul 10530  cle 10664  cn 11626  2c2 11680  0cn0 11885  cexp 13417  Σcsu 15030  cdvds 15595  bitscbits 15756  𝟭cind 31168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-ac2 9873  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-ac 9530  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-dvds 15596  df-bits 15759  df-ind 31169
This theorem is referenced by:  eulerpart  31539
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