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Theorem eulerpartlemn 32648
Description: Lemma for eulerpart 32649. (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 6828 . . . . . . . . . . . 12 ((𝑜 = 𝑞𝑘 ∈ ℕ) → (𝑜𝑘) = (𝑞𝑘))
32oveq1d 7353 . . . . . . . . . . 11 ((𝑜 = 𝑞𝑘 ∈ ℕ) → ((𝑜𝑘) · 𝑘) = ((𝑞𝑘) · 𝑘))
43sumeq2dv 15515 . . . . . . . . . 10 (𝑜 = 𝑞 → Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘))
54eqeq1d 2738 . . . . . . . . 9 (𝑜 = 𝑞 → (Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
65cbvrabv 3413 . . . . . . . 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 5924 . . . . . 6 (𝑜 = 𝑞 → (𝐺 ↾ {𝑜 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁}) = (𝐺 ↾ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}))
9 eqidd 2737 . . . . . 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 6762 . . . . 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 340 . . . 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 32639 . . . . . 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 6826 . . . . . . . . . . . . . 14 (𝑞 = 𝑜 → (𝐺𝑞) = (𝐺𝑜))
25 reseq1 5918 . . . . . . . . . . . . . . . . . 18 (𝑞 = 𝑜 → (𝑞𝐽) = (𝑜𝐽))
2625coeq2d 5805 . . . . . . . . . . . . . . . . 17 (𝑞 = 𝑜 → (bits ∘ (𝑞𝐽)) = (bits ∘ (𝑜𝐽)))
2726fveq2d 6830 . . . . . . . . . . . . . . . 16 (𝑞 = 𝑜 → (𝑀‘(bits ∘ (𝑞𝐽))) = (𝑀‘(bits ∘ (𝑜𝐽))))
2827imaeq2d 6000 . . . . . . . . . . . . . . 15 (𝑞 = 𝑜 → (𝐹 “ (𝑀‘(bits ∘ (𝑞𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))
2928fveq2d 6830 . . . . . . . . . . . . . 14 (𝑞 = 𝑜 → ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞𝐽))))) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
3024, 29eqeq12d 2752 . . . . . . . . . . . . 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 32640 . . . . . . . . . . . . 13 (𝑞 ∈ (𝑇𝑅) → (𝐺𝑞) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑞𝐽))))))
3230, 31vtoclga 3522 . . . . . . . . . . . 12 (𝑜 ∈ (𝑇𝑅) → (𝐺𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
33323ad2ant2 1133 . . . . . . . . . . 11 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → (𝐺𝑜) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
34 simp3 1137 . . . . . . . . . . 11 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
3533, 34eqtr4d 2779 . . . . . . . . . 10 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → (𝐺𝑜) = 𝑑)
3635fveq1d 6828 . . . . . . . . 9 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → ((𝐺𝑜)‘𝑘) = (𝑑𝑘))
3736oveq1d 7353 . . . . . . . 8 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → (((𝐺𝑜)‘𝑘) · 𝑘) = ((𝑑𝑘) · 𝑘))
3837sumeq2sdv 15516 . . . . . . 7 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → Σ𝑘 ∈ ℕ (((𝐺𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘))
3924fveq2d 6830 . . . . . . . . . . 11 (𝑞 = 𝑜 → (𝑆‘(𝐺𝑞)) = (𝑆‘(𝐺𝑜)))
40 fveq2 6826 . . . . . . . . . . 11 (𝑞 = 𝑜 → (𝑆𝑞) = (𝑆𝑜))
4139, 40eqeq12d 2752 . . . . . . . . . 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 32647 . . . . . . . . . 10 (𝑞 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝑞)) = (𝑆𝑞))
4441, 43vtoclga 3522 . . . . . . . . 9 (𝑜 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝑜)) = (𝑆𝑜))
45 nn0ex 12341 . . . . . . . . . . . . 13 0 ∈ V
46 0nn0 12350 . . . . . . . . . . . . . 14 0 ∈ ℕ0
47 1nn0 12351 . . . . . . . . . . . . . 14 1 ∈ ℕ0
48 prssi 4769 . . . . . . . . . . . . . 14 ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
4946, 47, 48mp2an 689 . . . . . . . . . . . . 13 {0, 1} ⊆ ℕ0
50 mapss 8749 . . . . . . . . . . . . 13 ((ℕ0 ∈ V ∧ {0, 1} ⊆ ℕ0) → ({0, 1} ↑m ℕ) ⊆ (ℕ0m ℕ))
5145, 49, 50mp2an 689 . . . . . . . . . . . 12 ({0, 1} ↑m ℕ) ⊆ (ℕ0m ℕ)
52 ssrin 4181 . . . . . . . . . . . 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 6768 . . . . . . . . . . . . 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)
5655ffvelcdmi 7017 . . . . . . . . . . 11 (𝑜 ∈ (𝑇𝑅) → (𝐺𝑜) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅))
5753, 56sselid 3930 . . . . . . . . . 10 (𝑜 ∈ (𝑇𝑅) → (𝐺𝑜) ∈ ((ℕ0m ℕ) ∩ 𝑅))
5820, 42eulerpartlemsv1 32623 . . . . . . . . . 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 32636 . . . . . . . . . . . 12 (𝑜 ∈ (𝑇𝑅) ↔ (𝑜 ∈ (ℕ0m ℕ) ∧ (𝑜 “ ℕ) ∈ Fin ∧ (𝑜 “ ℕ) ⊆ 𝐽))
6160simp1bi 1144 . . . . . . . . . . 11 (𝑜 ∈ (𝑇𝑅) → 𝑜 ∈ (ℕ0m ℕ))
62 inss2 4177 . . . . . . . . . . . 12 (𝑇𝑅) ⊆ 𝑅
6362sseli 3928 . . . . . . . . . . 11 (𝑜 ∈ (𝑇𝑅) → 𝑜𝑅)
6461, 63elind 4142 . . . . . . . . . 10 (𝑜 ∈ (𝑇𝑅) → 𝑜 ∈ ((ℕ0m ℕ) ∩ 𝑅))
6520, 42eulerpartlemsv1 32623 . . . . . . . . . 10 (𝑜 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝑜) = Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘))
6664, 65syl 17 . . . . . . . . 9 (𝑜 ∈ (𝑇𝑅) → (𝑆𝑜) = Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘))
6744, 59, 663eqtr3d 2784 . . . . . . . 8 (𝑜 ∈ (𝑇𝑅) → Σ𝑘 ∈ ℕ (((𝐺𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘))
68673ad2ant2 1133 . . . . . . 7 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → Σ𝑘 ∈ ℕ (((𝐺𝑜)‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘))
6938, 68eqtr3d 2778 . . . . . 6 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘))
7069eqeq1d 2738 . . . . 5 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅) ∧ 𝑑 = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → (Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁 ↔ Σ𝑘 ∈ ℕ ((𝑜𝑘) · 𝑘) = 𝑁))
7112, 23, 70f1oresrab 7056 . . . 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 2001 . . 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 5816 . . . . . . . . . 10 (𝑔 = 𝑞𝑔 = 𝑞)
7473imaeq1d 5999 . . . . . . . . 9 (𝑔 = 𝑞 → (𝑔 “ ℕ) = (𝑞 “ ℕ))
7574raleqdv 3309 . . . . . . . 8 (𝑔 = 𝑞 → (∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
7675cbvrabv 3413 . . . . . . 7 {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞𝑃 ∣ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
77 nfrab1 3422 . . . . . . . 8 𝑞{𝑞𝑃 ∣ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛}
78 nfrab1 3422 . . . . . . . 8 𝑞{𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}
79 df-3an 1088 . . . . . . . . . . . 12 ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
8079anbi1i 624 . . . . . . . . . . 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 32630 . . . . . . . . . . . 12 (𝑞𝑃 ↔ (𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
8281anbi1i 624 . . . . . . . . . . 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 643 . . . . . . . . . . 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 302 . . . . . . . . . 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 32636 . . . . . . . . . . . . 13 (𝑞 ∈ (𝑇𝑅) ↔ (𝑞 ∈ (ℕ0m ℕ) ∧ (𝑞 “ ℕ) ∈ Fin ∧ (𝑞 “ ℕ) ⊆ 𝐽))
86 nnex 12081 . . . . . . . . . . . . . . 15 ℕ ∈ V
8745, 86elmap 8731 . . . . . . . . . . . . . 14 (𝑞 ∈ (ℕ0m ℕ) ↔ 𝑞:ℕ⟶ℕ0)
88873anbi1i 1156 . . . . . . . . . . . . 13 ((𝑞 ∈ (ℕ0m ℕ) ∧ (𝑞 “ ℕ) ∈ Fin ∧ (𝑞 “ ℕ) ⊆ 𝐽) ↔ (𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin ∧ (𝑞 “ ℕ) ⊆ 𝐽))
8985, 88bitri 274 . . . . . . . . . . . 12 (𝑞 ∈ (𝑇𝑅) ↔ (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)𝑛𝐽)
92 breq2 5097 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
9392notbid 317 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
9493, 16elrab2 3637 . . . . . . . . . . . . . . . . 17 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
9594ralbii 3092 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (𝑞 “ ℕ)𝑛𝐽 ↔ ∀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))
9791, 95, 963bitri 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)
10098, 99sseqtrid 3984 . . . . . . . . . . . . . . . . 17 (𝑞:ℕ⟶ℕ0 → (𝑞 “ ℕ) ⊆ ℕ)
101 dfss3 3920 . . . . . . . . . . . . . . . . 17 ((𝑞 “ ℕ) ⊆ ℕ ↔ ∀𝑛 ∈ (𝑞 “ ℕ)𝑛 ∈ ℕ)
102100, 101sylib 217 . . . . . . . . . . . . . . . 16 (𝑞:ℕ⟶ℕ0 → ∀𝑛 ∈ (𝑞 “ ℕ)𝑛 ∈ ℕ)
103102biantrurd 533 . . . . . . . . . . . . . . 15 (𝑞:ℕ⟶ℕ0 → (∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛 ↔ (∀𝑛 ∈ (𝑞 “ ℕ)𝑛 ∈ ℕ ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛)))
10497, 103bitr4id 289 . . . . . . . . . . . . . 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 296 . . . . . . . . . . 11 (𝑞 ∈ (𝑇𝑅) ↔ ((𝑞:ℕ⟶ℕ0 ∧ (𝑞 “ ℕ) ∈ Fin) ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
108107anbi1i 624 . . . . . . . . . 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 277 . . . . . . . . 9 ((𝑞𝑃 ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛) ↔ (𝑞 ∈ (𝑇𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
110 rabid 3423 . . . . . . . . 9 (𝑞 ∈ {𝑞𝑃 ∣ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ (𝑞𝑃 ∧ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛))
111 rabid 3423 . . . . . . . . 9 (𝑞 ∈ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁} ↔ (𝑞 ∈ (𝑇𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁))
112109, 110, 1113bitr4i 302 . . . . . . . 8 (𝑞 ∈ {𝑞𝑃 ∣ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛} ↔ 𝑞 ∈ {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁})
11377, 78, 112eqri 3952 . . . . . . 7 {𝑞𝑃 ∣ ∀𝑛 ∈ (𝑞 “ ℕ) ¬ 2 ∥ 𝑛} = {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}
11414, 76, 1133eqtri 2768 . . . . . 6 𝑂 = {𝑞 ∈ (𝑇𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑞𝑘) · 𝑘) = 𝑁}
115114reseq2i 5921 . . . . 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 2904 . . . . . 6 𝑑𝐷
119 nfrab1 3422 . . . . . 6 𝑑{𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁}
120 fnima 6615 . . . . . . . . . . . . . . . . 17 (𝑑 Fn ℕ → (𝑑 “ ℕ) = ran 𝑑)
121120sseq1d 3963 . . . . . . . . . . . . . . . 16 (𝑑 Fn ℕ → ((𝑑 “ ℕ) ⊆ {0, 1} ↔ ran 𝑑 ⊆ {0, 1}))
122121anbi2d 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)
12449, 123mpan2 688 . . . . . . . . . . . . . . . 16 (ran 𝑑 ⊆ {0, 1} → ran 𝑑 ⊆ ℕ0)
125124pm4.71ri 561 . . . . . . . . . . . . . . 15 (ran 𝑑 ⊆ {0, 1} ↔ (ran 𝑑 ⊆ ℕ0 ∧ ran 𝑑 ⊆ {0, 1}))
126122, 125bitr4di 288 . . . . . . . . . . . . . 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 6484 . . . . . . . . . . . . 13 (𝑑:ℕ⟶{0, 1} ↔ (𝑑 Fn ℕ ∧ ran 𝑑 ⊆ {0, 1}))
130127, 128, 1293bitr4ri 303 . . . . . . . . . . . 12 (𝑑:ℕ⟶{0, 1} ↔ ((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
131 prex 5378 . . . . . . . . . . . . 13 {0, 1} ∈ V
132131, 86elmap 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))
134133anbi1i 624 . . . . . . . . . . . 12 ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑 Fn ℕ ∧ ran 𝑑 ⊆ ℕ0) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
135130, 132, 1343bitr4i 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 = 𝑑𝑓 = 𝑑)
138137imaeq1d 5999 . . . . . . . . . . . . 13 (𝑓 = 𝑑 → (𝑓 “ ℕ) = (𝑑 “ ℕ))
139138eleq1d 2821 . . . . . . . . . . . 12 (𝑓 = 𝑑 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝑑 “ ℕ) ∈ Fin))
140136, 139, 20elab2 3623 . . . . . . . . . . 11 (𝑑𝑅 ↔ (𝑑 “ ℕ) ∈ Fin)
141135, 140anbi12i 627 . . . . . . . . . 10 ((𝑑 ∈ ({0, 1} ↑m ℕ) ∧ 𝑑𝑅) ↔ ((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) ∧ 𝑑𝑅))
143 an32 643 . . . . . . . . . 10 (((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ (𝑑 “ ℕ) ∈ Fin))
144141, 142, 1433bitr4i 302 . . . . . . . . 9 (𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
145144anbi1i 624 . . . . . . . 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 32630 . . . . . . . . . 10 (𝑑𝑃 ↔ (𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁))
147146anbi1i 624 . . . . . . . . 9 ((𝑑𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁) ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
148 df-3an 1088 . . . . . . . . . 10 ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁) ↔ ((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁))
149148anbi1i 624 . . . . . . . . 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 643 . . . . . . . . 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 296 . . . . . . . 8 ((𝑑𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ↔ (((𝑑:ℕ⟶ℕ0 ∧ (𝑑 “ ℕ) ∈ Fin) ∧ (𝑑 “ ℕ) ⊆ {0, 1}) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁))
152145, 151bitr4i 277 . . . . . . 7 ((𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∧ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁) ↔ (𝑑𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
153 rabid 3423 . . . . . . 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 32633 . . . . . . 7 (𝑑𝐷 ↔ (𝑑𝑃 ∧ (𝑑 “ ℕ) ⊆ {0, 1}))
155152, 153, 1543bitr4ri 303 . . . . . 6 (𝑑𝐷𝑑 ∈ {𝑑 ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ∣ Σ𝑘 ∈ ℕ ((𝑑𝑘) · 𝑘) = 𝑁})
156118, 119, 155eqri 3952 . . . . 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 6762 . . 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 256 . 2 (⊤ → (𝐺𝑂):𝑂1-1-onto𝐷)
160159mptru 1547 1 (𝐺𝑂):𝑂1-1-onto𝐷
Colors of variables: wff setvar class
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-ontowf1o 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  cn 12075  2c2 12130  0cn0 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
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