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Theorem eulerpartlemgs2 31712
 Description: Lemma for eulerpart 31714: The 𝐺 function also preserves partition sums. (Contributed by Thierry Arnoux, 10-Sep-2017.)
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
eulerpartlemgs2 (𝐴 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝐴)) = (𝑆𝐴))
Distinct variable groups:   𝑓,𝑔,𝑘,𝑛,𝑜,𝑥,𝑦,𝑧   𝑓,𝑟,𝐴,𝑔,𝑘,𝑛,𝑜,𝑥,𝑦   𝑓,𝐺,𝑘   𝑛,𝐹,𝑜,𝑥,𝑦   𝑜,𝐻,𝑟   𝑓,𝐽,𝑛,𝑜,𝑟,𝑥,𝑦   𝑛,𝑀,𝑜,𝑟,𝑥,𝑦   𝑓,𝑁,𝑔,𝑘,𝑛,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:   𝐴(𝑧)   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,𝑘,𝑟)   𝐺(𝑥,𝑦,𝑧,𝑔,𝑛,𝑜,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛)   𝐽(𝑧,𝑔,𝑘)   𝑀(𝑧,𝑓,𝑔,𝑘)   𝑁(𝑦,𝑧,𝑜,𝑟)   𝑂(𝑧,𝑓,𝑔,𝑘,𝑜)

Proof of Theorem eulerpartlemgs2
Dummy variables 𝑡 𝑚 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvimass 5927 . . . . . . . 8 ((𝐺𝐴) “ ℕ) ⊆ dom (𝐺𝐴)
2 eulerpart.p . . . . . . . . . . . . . 14 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
3 eulerpart.o . . . . . . . . . . . . . 14 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
4 eulerpart.d . . . . . . . . . . . . . 14 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
5 eulerpart.j . . . . . . . . . . . . . 14 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
6 eulerpart.f . . . . . . . . . . . . . 14 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
7 eulerpart.h . . . . . . . . . . . . . 14 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
8 eulerpart.m . . . . . . . . . . . . . 14 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
9 eulerpart.r . . . . . . . . . . . . . 14 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
10 eulerpart.t . . . . . . . . . . . . . 14 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
11 eulerpart.g . . . . . . . . . . . . . 14 𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
122, 3, 4, 5, 6, 7, 8, 9, 10, 11eulerpartgbij 31704 . . . . . . . . . . . . 13 𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅)
13 f1of 6597 . . . . . . . . . . . . 13 (𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅) → 𝐺:(𝑇𝑅)⟶(({0, 1} ↑m ℕ) ∩ 𝑅))
1412, 13ax-mp 5 . . . . . . . . . . . 12 𝐺:(𝑇𝑅)⟶(({0, 1} ↑m ℕ) ∩ 𝑅)
1514ffvelrni 6832 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅))
16 elin 3924 . . . . . . . . . . 11 ((𝐺𝐴) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
1715, 16sylib 221 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
1817simpld 498 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) ∈ ({0, 1} ↑m ℕ))
19 elmapi 8415 . . . . . . . . 9 ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) → (𝐺𝐴):ℕ⟶{0, 1})
20 fdm 6502 . . . . . . . . 9 ((𝐺𝐴):ℕ⟶{0, 1} → dom (𝐺𝐴) = ℕ)
2118, 19, 203syl 18 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → dom (𝐺𝐴) = ℕ)
221, 21sseqtrid 3994 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) “ ℕ) ⊆ ℕ)
2322sselda 3942 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ((𝐺𝐴) “ ℕ)) → 𝑘 ∈ ℕ)
242, 3, 4, 5, 6, 7, 8, 9, 10, 11eulerpartlemgvv 31708 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ℕ) → ((𝐺𝐴)‘𝑘) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0))
2524oveq1d 7155 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ℕ) → (((𝐺𝐴)‘𝑘) · 𝑘) = (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
2623, 25syldan 594 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ((𝐺𝐴) “ ℕ)) → (((𝐺𝐴)‘𝑘) · 𝑘) = (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
2726sumeq2dv 15051 . . . 4 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘) = Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
28 eqeq2 2834 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑘))
29282rexbidv 3286 . . . . . . . . . . . 12 (𝑚 = 𝑘 → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
3029elrab 3655 . . . . . . . . . . 11 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ↔ (𝑘 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
3130simprbi 500 . . . . . . . . . 10 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘)
3231iftrued 4447 . . . . . . . . 9 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) = 1)
3332oveq1d 7155 . . . . . . . 8 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = (1 · 𝑘))
34 elrabi 3650 . . . . . . . . . 10 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℕ)
3534nncnd 11641 . . . . . . . . 9 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℂ)
3635mulid2d 10648 . . . . . . . 8 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (1 · 𝑘) = 𝑘)
3733, 36eqtrd 2857 . . . . . . 7 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = 𝑘)
3837sumeq2i 15047 . . . . . 6 Σ𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = Σ𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}𝑘
39 id 22 . . . . . . 7 (𝑘 = ((2↑(2nd𝑤)) · (1st𝑤)) → 𝑘 = ((2↑(2nd𝑤)) · (1st𝑤)))
402, 3, 4, 5, 6, 7, 8, 9, 10, 11eulerpartlemgf 31711 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) “ ℕ) ∈ Fin)
4134adantl 485 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℕ)
4241, 24syldan 594 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺𝐴)‘𝑘) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0))
4331adantl 485 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘)
4443iftrued 4447 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) = 1)
4542, 44eqtrd 2857 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺𝐴)‘𝑘) = 1)
46 1nn 11636 . . . . . . . . . . . . 13 1 ∈ ℕ
4745, 46eqeltrdi 2922 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺𝐴)‘𝑘) ∈ ℕ)
4818, 19syl 17 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴):ℕ⟶{0, 1})
49 ffn 6494 . . . . . . . . . . . . . 14 ((𝐺𝐴):ℕ⟶{0, 1} → (𝐺𝐴) Fn ℕ)
50 elpreima 6810 . . . . . . . . . . . . . 14 ((𝐺𝐴) Fn ℕ → (𝑘 ∈ ((𝐺𝐴) “ ℕ) ↔ (𝑘 ∈ ℕ ∧ ((𝐺𝐴)‘𝑘) ∈ ℕ)))
5148, 49, 503syl 18 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑘 ∈ ((𝐺𝐴) “ ℕ) ↔ (𝑘 ∈ ℕ ∧ ((𝐺𝐴)‘𝑘) ∈ ℕ)))
5251adantr 484 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → (𝑘 ∈ ((𝐺𝐴) “ ℕ) ↔ (𝑘 ∈ ℕ ∧ ((𝐺𝐴)‘𝑘) ∈ ℕ)))
5341, 47, 52mpbir2and 712 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ((𝐺𝐴) “ ℕ))
5453ex 416 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ((𝐺𝐴) “ ℕ)))
5554ssrdv 3948 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ((𝐺𝐴) “ ℕ))
56 ssfi 8726 . . . . . . . . 9 ((((𝐺𝐴) “ ℕ) ∈ Fin ∧ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ((𝐺𝐴) “ ℕ)) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ∈ Fin)
5740, 55, 56syl2anc 587 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ∈ Fin)
58 cnvexg 7615 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ V)
59 imaexg 7606 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 “ ℕ) ∈ V)
60 inex1g 5199 . . . . . . . . . . 11 ((𝐴 “ ℕ) ∈ V → ((𝐴 “ ℕ) ∩ 𝐽) ∈ V)
6158, 59, 603syl 18 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ V)
62 snex 5309 . . . . . . . . . . . 12 {𝑡} ∈ V
63 fvex 6665 . . . . . . . . . . . 12 (bits‘(𝐴𝑡)) ∈ V
6462, 63xpex 7461 . . . . . . . . . . 11 ({𝑡} × (bits‘(𝐴𝑡))) ∈ V
6564rgenw 3142 . . . . . . . . . 10 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V
66 iunexg 7650 . . . . . . . . . 10 ((((𝐴 “ ℕ) ∩ 𝐽) ∈ V ∧ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V)
6761, 65, 66sylancl 589 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V)
68 eqid 2822 . . . . . . . . . 10 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))
692, 3, 4, 5, 6, 7, 8, 9, 10, 11, 68eulerpartlemgh 31710 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))): 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
70 f1oeng 8515 . . . . . . . . 9 (( 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V ∧ (𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))): 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ≈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
7167, 69, 70syl2anc 587 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ≈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
72 enfii 8723 . . . . . . . 8 (({𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ∈ Fin ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ≈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ Fin)
7357, 71, 72syl2anc 587 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ Fin)
74 fvres 6671 . . . . . . . . 9 (𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) → ((𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))‘𝑤) = (𝐹𝑤))
7574adantl 485 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → ((𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))‘𝑤) = (𝐹𝑤))
76 inss2 4180 . . . . . . . . . . . . . . 15 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
77 simpr 488 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽))
7876, 77sseldi 3940 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡𝐽)
7978snssd 4715 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → {𝑡} ⊆ 𝐽)
80 bitsss 15764 . . . . . . . . . . . . 13 (bits‘(𝐴𝑡)) ⊆ ℕ0
81 xpss12 5547 . . . . . . . . . . . . 13 (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴𝑡)) ⊆ ℕ0) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8279, 80, 81sylancl 589 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8382ralrimiva 3174 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
84 iunss 4944 . . . . . . . . . . 11 ( 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8583, 84sylibr 237 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8685sselda 3942 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → 𝑤 ∈ (𝐽 × ℕ0))
875, 6oddpwdcv 31687 . . . . . . . . 9 (𝑤 ∈ (𝐽 × ℕ0) → (𝐹𝑤) = ((2↑(2nd𝑤)) · (1st𝑤)))
8886, 87syl 17 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → (𝐹𝑤) = ((2↑(2nd𝑤)) · (1st𝑤)))
8975, 88eqtrd 2857 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → ((𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))‘𝑤) = ((2↑(2nd𝑤)) · (1st𝑤)))
9041nncnd 11641 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
9139, 73, 69, 89, 90fsumf1o 15071 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}𝑘 = Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)))
9238, 91syl5eq 2869 . . . . 5 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)))
93 ax-1cn 10584 . . . . . . . . 9 1 ∈ ℂ
94 0cn 10622 . . . . . . . . 9 0 ∈ ℂ
9593, 94ifcli 4485 . . . . . . . 8 if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) ∈ ℂ
9695a1i 11 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) ∈ ℂ)
97 ssrab2 4031 . . . . . . . . 9 {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ℕ
98 simpr 488 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
9997, 98sseldi 3940 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℕ)
10099nncnd 11641 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
10196, 100mulcld 10650 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) ∈ ℂ)
102 simpr 488 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
103102eldifbd 3921 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → ¬ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
10422ssdifssd 4094 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℕ)
105104sselda 3942 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ℕ)
10630notbii 323 . . . . . . . . . . 11 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ↔ ¬ (𝑘 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
107 imnan 403 . . . . . . . . . . 11 ((𝑘 ∈ ℕ → ¬ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘) ↔ ¬ (𝑘 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
108106, 107sylbb2 241 . . . . . . . . . 10 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (𝑘 ∈ ℕ → ¬ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
109103, 105, 108sylc 65 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → ¬ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘)
110109iffalsed 4450 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) = 0)
111110oveq1d 7155 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = (0 · 𝑘))
112 nnsscn 11630 . . . . . . . . . 10 ℕ ⊆ ℂ
113104, 112sstrdi 3954 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℂ)
114113sselda 3942 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ℂ)
115114mul02d 10827 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → (0 · 𝑘) = 0)
116111, 115eqtrd 2857 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = 0)
11755, 101, 116, 40fsumss 15073 . . . . 5 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
11892, 117eqtr3d 2859 . . . 4 (𝐴 ∈ (𝑇𝑅) → Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)) = Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
1192, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemt0 31701 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
120119simp1bi 1142 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0m ℕ))
121 elmapi 8415 . . . . . . . . . . . 12 (𝐴 ∈ (ℕ0m ℕ) → 𝐴:ℕ⟶ℕ0)
122120, 121syl 17 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
123122adantr 484 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝐴:ℕ⟶ℕ0)
124 cnvimass 5927 . . . . . . . . . . . . 13 (𝐴 “ ℕ) ⊆ dom 𝐴
125124, 122fssdm 6511 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ ℕ)
126125adantr 484 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (𝐴 “ ℕ) ⊆ ℕ)
127 inss1 4179 . . . . . . . . . . . 12 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ (𝐴 “ ℕ)
128127, 77sseldi 3940 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ (𝐴 “ ℕ))
129126, 128sseldd 3943 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℕ)
130123, 129ffvelrnd 6834 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (𝐴𝑡) ∈ ℕ0)
131 bitsfi 15775 . . . . . . . . 9 ((𝐴𝑡) ∈ ℕ0 → (bits‘(𝐴𝑡)) ∈ Fin)
132130, 131syl 17 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (bits‘(𝐴𝑡)) ∈ Fin)
133129nncnd 11641 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℂ)
134 2cnd 11703 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 2 ∈ ℂ)
135 simprr 772 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ (bits‘(𝐴𝑡)))
13680, 135sseldi 3940 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ ℕ0)
137134, 136expcld 13506 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℂ)
138137anassrs 471 . . . . . . . 8 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → (2↑𝑛) ∈ ℂ)
139132, 133, 138fsummulc1 15131 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡))
140139sumeq2dv 15051 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡))
141 bitsinv1 15780 . . . . . . . . 9 ((𝐴𝑡) ∈ ℕ0 → Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) = (𝐴𝑡))
142141oveq1d 7155 . . . . . . . 8 ((𝐴𝑡) ∈ ℕ0 → (Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = ((𝐴𝑡) · 𝑡))
143130, 142syl 17 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = ((𝐴𝑡) · 𝑡))
144143sumeq2dv 15051 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡))
145 vex 3472 . . . . . . . . . 10 𝑡 ∈ V
146 vex 3472 . . . . . . . . . 10 𝑛 ∈ V
147145, 146op2ndd 7686 . . . . . . . . 9 (𝑤 = ⟨𝑡, 𝑛⟩ → (2nd𝑤) = 𝑛)
148147oveq2d 7156 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑛⟩ → (2↑(2nd𝑤)) = (2↑𝑛))
149145, 146op1std 7685 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑛⟩ → (1st𝑤) = 𝑡)
150148, 149oveq12d 7158 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑(2nd𝑤)) · (1st𝑤)) = ((2↑𝑛) · 𝑡))
151 inss2 4180 . . . . . . . . . 10 (𝑇𝑅) ⊆ 𝑅
152151sseli 3938 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝐴𝑅)
153 cnveq 5721 . . . . . . . . . . . 12 (𝑓 = 𝐴𝑓 = 𝐴)
154153imaeq1d 5906 . . . . . . . . . . 11 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
155154eleq1d 2898 . . . . . . . . . 10 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
156155, 9elab2g 3643 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (𝐴𝑅 ↔ (𝐴 “ ℕ) ∈ Fin))
157152, 156mpbid 235 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ∈ Fin)
158 ssfi 8726 . . . . . . . 8 (((𝐴 “ ℕ) ∈ Fin ∧ ((𝐴 “ ℕ) ∩ 𝐽) ⊆ (𝐴 “ ℕ)) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
159157, 127, 158sylancl 589 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
160133adantrr 716 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℂ)
161137, 160mulcld 10650 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ∈ ℂ)
162150, 159, 132, 161fsum2d 15117 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)))
163140, 144, 1623eqtr3d 2865 . . . . 5 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡) = Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)))
164 inss1 4179 . . . . . . . . 9 (𝑇𝑅) ⊆ 𝑇
165164sseli 3938 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝐴𝑇)
166154sseq1d 3973 . . . . . . . . . 10 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ (𝐴 “ ℕ) ⊆ 𝐽))
167166, 10elrab2 3658 . . . . . . . . 9 (𝐴𝑇 ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽))
168167simprbi 500 . . . . . . . 8 (𝐴𝑇 → (𝐴 “ ℕ) ⊆ 𝐽)
169165, 168syl 17 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
170 df-ss 3925 . . . . . . 7 ((𝐴 “ ℕ) ⊆ 𝐽 ↔ ((𝐴 “ ℕ) ∩ 𝐽) = (𝐴 “ ℕ))
171169, 170sylib 221 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) = (𝐴 “ ℕ))
172171sumeq1d 15049 . . . . 5 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
173163, 172eqtr3d 2859 . . . 4 (𝐴 ∈ (𝑇𝑅) → Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
17427, 118, 1733eqtr2d 2863 . . 3 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
175 fveq2 6652 . . . . 5 (𝑘 = 𝑡 → (𝐴𝑘) = (𝐴𝑡))
176 id 22 . . . . 5 (𝑘 = 𝑡𝑘 = 𝑡)
177175, 176oveq12d 7158 . . . 4 (𝑘 = 𝑡 → ((𝐴𝑘) · 𝑘) = ((𝐴𝑡) · 𝑡))
178177cbvsumv 15044 . . 3 Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡)
179174, 178eqtr4di 2875 . 2 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
180 0nn0 11900 . . . . . . . 8 0 ∈ ℕ0
181 1nn0 11901 . . . . . . . 8 1 ∈ ℕ0
182 prssi 4727 . . . . . . . 8 ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
183180, 181, 182mp2an 691 . . . . . . 7 {0, 1} ⊆ ℕ0
184 fss 6508 . . . . . . 7 (((𝐺𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℕ0) → (𝐺𝐴):ℕ⟶ℕ0)
185183, 184mpan2 690 . . . . . 6 ((𝐺𝐴):ℕ⟶{0, 1} → (𝐺𝐴):ℕ⟶ℕ0)
186 nn0ex 11891 . . . . . . . 8 0 ∈ V
187 nnex 11631 . . . . . . . 8 ℕ ∈ V
188186, 187elmap 8422 . . . . . . 7 ((𝐺𝐴) ∈ (ℕ0m ℕ) ↔ (𝐺𝐴):ℕ⟶ℕ0)
189188biimpri 231 . . . . . 6 ((𝐺𝐴):ℕ⟶ℕ0 → (𝐺𝐴) ∈ (ℕ0m ℕ))
19019, 185, 1893syl 18 . . . . 5 ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) → (𝐺𝐴) ∈ (ℕ0m ℕ))
191190anim1i 617 . . . 4 (((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺𝐴) ∈ 𝑅) → ((𝐺𝐴) ∈ (ℕ0m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
192 elin 3924 . . . 4 ((𝐺𝐴) ∈ ((ℕ0m ℕ) ∩ 𝑅) ↔ ((𝐺𝐴) ∈ (ℕ0m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
193191, 16, 1923imtr4i 295 . . 3 ((𝐺𝐴) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) → (𝐺𝐴) ∈ ((ℕ0m ℕ) ∩ 𝑅))
194 eulerpart.s . . . 4 𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))
1959, 194eulerpartlemsv2 31690 . . 3 ((𝐺𝐴) ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆‘(𝐺𝐴)) = Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘))
19615, 193, 1953syl 18 . 2 (𝐴 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝐴)) = Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘))
197120, 152elind 4145 . . 3 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅))
1989, 194eulerpartlemsv2 31690 . . 3 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
199197, 198syl 17 . 2 (𝐴 ∈ (𝑇𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
200179, 196, 1993eqtr4d 2867 1 (𝐴 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝐴)) = (𝑆𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114  {cab 2800  ∀wral 3130  ∃wrex 3131  {crab 3134  Vcvv 3469   ∖ cdif 3905   ∩ cin 3907   ⊆ wss 3908  ∅c0 4265  ifcif 4439  𝒫 cpw 4511  {csn 4539  {cpr 4541  ⟨cop 4545  ∪ ciun 4894   class class class wbr 5042  {copab 5104   ↦ cmpt 5122   × cxp 5530  ◡ccnv 5531  dom cdm 5532   ↾ cres 5534   “ cima 5535   ∘ ccom 5536   Fn wfn 6329  ⟶wf 6330  –1-1-onto→wf1o 6333  ‘cfv 6334  (class class class)co 7140   ∈ cmpo 7142  1st c1st 7673  2nd c2nd 7674   supp csupp 7817   ↑m cmap 8393   ≈ cen 8493  Fincfn 8496  ℂcc 10524  0cc0 10526  1c1 10527   · cmul 10531   ≤ cle 10665  ℕcn 11625  2c2 11680  ℕ0cn0 11885  ↑cexp 13425  Σcsu 15033   ∥ cdvds 15598  bitscbits 15757  𝟭cind 31343 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  ax-ac2 9874  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-disj 5008  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-ac 9531  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-clim 14836  df-sum 15034  df-dvds 15599  df-bits 15760  df-ind 31344 This theorem is referenced by:  eulerpartlemn  31713
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