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Theorem eulerpartlemgs2 30416
Description: Lemma for eulerpart 30418: 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 5473 . . . . . . . 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 30408 . . . . . . . . . . . . 13 𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅)
13 f1of 6124 . . . . . . . . . . . . 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 6344 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅))
16 elin 3788 . . . . . . . . . . 11 ((𝐺𝐴) ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) ↔ ((𝐺𝐴) ∈ ({0, 1} ↑𝑚 ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
1715, 16sylib 208 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) ∈ ({0, 1} ↑𝑚 ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
1817simpld 475 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) ∈ ({0, 1} ↑𝑚 ℕ))
19 elmapi 7864 . . . . . . . . 9 ((𝐺𝐴) ∈ ({0, 1} ↑𝑚 ℕ) → (𝐺𝐴):ℕ⟶{0, 1})
20 fdm 6038 . . . . . . . . 9 ((𝐺𝐴):ℕ⟶{0, 1} → dom (𝐺𝐴) = ℕ)
2118, 19, 203syl 18 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → dom (𝐺𝐴) = ℕ)
221, 21syl5sseq 3645 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) “ ℕ) ⊆ ℕ)
2322sselda 3595 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ((𝐺𝐴) “ ℕ)) → 𝑘 ∈ ℕ)
242, 3, 4, 5, 6, 7, 8, 9, 10, 11eulerpartlemgvv 30412 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ℕ) → ((𝐺𝐴)‘𝑘) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0))
2524oveq1d 6650 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ℕ) → (((𝐺𝐴)‘𝑘) · 𝑘) = (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
2623, 25syldan 487 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ((𝐺𝐴) “ ℕ)) → (((𝐺𝐴)‘𝑘) · 𝑘) = (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
2726sumeq2dv 14414 . . . 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 2631 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑘))
29282rexbidv 3053 . . . . . . . . . . . 12 (𝑚 = 𝑘 → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
3029elrab 3357 . . . . . . . . . . 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 480 . . . . . . . . . 10 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘)
3231iftrued 4085 . . . . . . . . 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 6650 . . . . . . . 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 3353 . . . . . . . . . 10 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℕ)
3534nncnd 11021 . . . . . . . . 9 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℂ)
3635mulid2d 10043 . . . . . . . 8 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (1 · 𝑘) = 𝑘)
3733, 36eqtrd 2654 . . . . . . 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 14410 . . . . . 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 30415 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) “ ℕ) ∈ Fin)
4134adantl 482 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℕ)
4241, 24syldan 487 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 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 4085 . . . . . . . . . . . . . 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 2654 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺𝐴)‘𝑘) = 1)
46 1nn 11016 . . . . . . . . . . . . 13 1 ∈ ℕ
4745, 46syl6eqel 2707 . . . . . . . . . . . 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 6032 . . . . . . . . . . . . . 14 ((𝐺𝐴):ℕ⟶{0, 1} → (𝐺𝐴) Fn ℕ)
50 elpreima 6323 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 956 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ((𝐺𝐴) “ ℕ))
5453ex 450 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ((𝐺𝐴) “ ℕ)))
5554ssrdv 3601 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ((𝐺𝐴) “ ℕ))
56 ssfi 8165 . . . . . . . . 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 692 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ∈ Fin)
58 cnvexg 7097 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ V)
59 imaexg 7088 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 “ ℕ) ∈ V)
60 inex1g 4792 . . . . . . . . . . 11 ((𝐴 “ ℕ) ∈ V → ((𝐴 “ ℕ) ∩ 𝐽) ∈ V)
6158, 59, 603syl 18 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ V)
62 snex 4899 . . . . . . . . . . . 12 {𝑡} ∈ V
63 fvex 6188 . . . . . . . . . . . 12 (bits‘(𝐴𝑡)) ∈ V
6462, 63xpex 6947 . . . . . . . . . . 11 ({𝑡} × (bits‘(𝐴𝑡))) ∈ V
6564rgenw 2921 . . . . . . . . . 10 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V
66 iunexg 7128 . . . . . . . . . 10 ((((𝐴 “ ℕ) ∩ 𝐽) ∈ V ∧ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V)
6761, 65, 66sylancl 693 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V)
68 eqid 2620 . . . . . . . . . 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 30414 . . . . . . . . 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 7959 . . . . . . . . 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 692 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ≈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
72 enfii 8162 . . . . . . . 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 692 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ Fin)
74 fvres 6194 . . . . . . . . 9 (𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) → ((𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))‘𝑤) = (𝐹𝑤))
7574adantl 482 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → ((𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))‘𝑤) = (𝐹𝑤))
76 inss2 3826 . . . . . . . . . . . . . . 15 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
77 simpr 477 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽))
7876, 77sseldi 3593 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡𝐽)
7978snssd 4331 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → {𝑡} ⊆ 𝐽)
80 bitsss 15129 . . . . . . . . . . . . 13 (bits‘(𝐴𝑡)) ⊆ ℕ0
81 xpss12 5215 . . . . . . . . . . . . 13 (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴𝑡)) ⊆ ℕ0) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8279, 80, 81sylancl 693 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8382ralrimiva 2963 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
84 iunss 4552 . . . . . . . . . . 11 ( 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8583, 84sylibr 224 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8685sselda 3595 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → 𝑤 ∈ (𝐽 × ℕ0))
875, 6oddpwdcv 30391 . . . . . . . . 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 2654 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → ((𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))‘𝑤) = ((2↑(2nd𝑤)) · (1st𝑤)))
9041nncnd 11021 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
9139, 73, 69, 89, 90fsumf1o 14435 . . . . . 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 2666 . . . . 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 9979 . . . . . . . . 9 1 ∈ ℂ
94 0cn 10017 . . . . . . . . 9 0 ∈ ℂ
9593, 94keepel 4146 . . . . . . . 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 3679 . . . . . . . . 9 {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ℕ
98 simpr 477 . . . . . . . . 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 3593 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℕ)
10099nncnd 11021 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
10196, 100mulcld 10045 . . . . . 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 477 . . . . . . . . . . 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 3580 . . . . . . . . . 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 3740 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℕ)
105104sselda 3595 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ℕ)
10630notbii 310 . . . . . . . . . . 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 438 . . . . . . . . . . 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 228 . . . . . . . . . 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 4088 . . . . . . . 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 6650 . . . . . . 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 11010 . . . . . . . . . 10 ℕ ⊆ ℂ
113104, 112syl6ss 3607 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℂ)
114113sselda 3595 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ℂ)
115114mul02d 10219 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → (0 · 𝑘) = 0)
116111, 115eqtrd 2654 . . . . . 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 14437 . . . . 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 2656 . . . 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 30405 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
120119simp1bi 1074 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
121 elmapi 7864 . . . . . . . . . . . 12 (𝐴 ∈ (ℕ0𝑚 ℕ) → 𝐴:ℕ⟶ℕ0)
122120, 121syl 17 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
123122adantr 481 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝐴:ℕ⟶ℕ0)
124 cnvimass 5473 . . . . . . . . . . . . 13 (𝐴 “ ℕ) ⊆ dom 𝐴
125 fdm 6038 . . . . . . . . . . . . . 14 (𝐴:ℕ⟶ℕ0 → dom 𝐴 = ℕ)
126122, 125syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → dom 𝐴 = ℕ)
127124, 126syl5sseq 3645 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ ℕ)
128127adantr 481 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (𝐴 “ ℕ) ⊆ ℕ)
129 inss1 3825 . . . . . . . . . . . 12 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ (𝐴 “ ℕ)
130129, 77sseldi 3593 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ (𝐴 “ ℕ))
131128, 130sseldd 3596 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℕ)
132123, 131ffvelrnd 6346 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (𝐴𝑡) ∈ ℕ0)
133 bitsfi 15140 . . . . . . . . 9 ((𝐴𝑡) ∈ ℕ0 → (bits‘(𝐴𝑡)) ∈ Fin)
134132, 133syl 17 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (bits‘(𝐴𝑡)) ∈ Fin)
135131nncnd 11021 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℂ)
136 2cnd 11078 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 2 ∈ ℂ)
137 simprr 795 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ (bits‘(𝐴𝑡)))
13880, 137sseldi 3593 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ ℕ0)
139136, 138expcld 12991 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℂ)
140139anassrs 679 . . . . . . . 8 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → (2↑𝑛) ∈ ℂ)
141134, 135, 140fsummulc1 14498 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡))
142141sumeq2dv 14414 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡))
143 bitsinv1 15145 . . . . . . . . 9 ((𝐴𝑡) ∈ ℕ0 → Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) = (𝐴𝑡))
144143oveq1d 6650 . . . . . . . 8 ((𝐴𝑡) ∈ ℕ0 → (Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = ((𝐴𝑡) · 𝑡))
145132, 144syl 17 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = ((𝐴𝑡) · 𝑡))
146145sumeq2dv 14414 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡))
147 vex 3198 . . . . . . . . . 10 𝑡 ∈ V
148 vex 3198 . . . . . . . . . 10 𝑛 ∈ V
149147, 148op2ndd 7164 . . . . . . . . 9 (𝑤 = ⟨𝑡, 𝑛⟩ → (2nd𝑤) = 𝑛)
150149oveq2d 6651 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑛⟩ → (2↑(2nd𝑤)) = (2↑𝑛))
151147, 148op1std 7163 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑛⟩ → (1st𝑤) = 𝑡)
152150, 151oveq12d 6653 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑(2nd𝑤)) · (1st𝑤)) = ((2↑𝑛) · 𝑡))
153 inss2 3826 . . . . . . . . . 10 (𝑇𝑅) ⊆ 𝑅
154153sseli 3591 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝐴𝑅)
155 cnveq 5285 . . . . . . . . . . . 12 (𝑓 = 𝐴𝑓 = 𝐴)
156155imaeq1d 5453 . . . . . . . . . . 11 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
157156eleq1d 2684 . . . . . . . . . 10 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
158157, 9elab2g 3347 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (𝐴𝑅 ↔ (𝐴 “ ℕ) ∈ Fin))
159154, 158mpbid 222 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ∈ Fin)
160 ssfi 8165 . . . . . . . 8 (((𝐴 “ ℕ) ∈ Fin ∧ ((𝐴 “ ℕ) ∩ 𝐽) ⊆ (𝐴 “ ℕ)) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
161159, 129, 160sylancl 693 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
162135adantrr 752 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℂ)
163139, 162mulcld 10045 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ∈ ℂ)
164152, 161, 134, 163fsum2d 14483 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)))
165142, 146, 1643eqtr3d 2662 . . . . 5 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡) = Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)))
166 inss1 3825 . . . . . . . . 9 (𝑇𝑅) ⊆ 𝑇
167166sseli 3591 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝐴𝑇)
168156sseq1d 3624 . . . . . . . . . 10 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ (𝐴 “ ℕ) ⊆ 𝐽))
169168, 10elrab2 3360 . . . . . . . . 9 (𝐴𝑇 ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽))
170169simprbi 480 . . . . . . . 8 (𝐴𝑇 → (𝐴 “ ℕ) ⊆ 𝐽)
171167, 170syl 17 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
172 df-ss 3581 . . . . . . 7 ((𝐴 “ ℕ) ⊆ 𝐽 ↔ ((𝐴 “ ℕ) ∩ 𝐽) = (𝐴 “ ℕ))
173171, 172sylib 208 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) = (𝐴 “ ℕ))
174173sumeq1d 14412 . . . . 5 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
175165, 174eqtr3d 2656 . . . 4 (𝐴 ∈ (𝑇𝑅) → Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
17627, 118, 1753eqtr2d 2660 . . 3 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
177 fveq2 6178 . . . . 5 (𝑘 = 𝑡 → (𝐴𝑘) = (𝐴𝑡))
178 id 22 . . . . 5 (𝑘 = 𝑡𝑘 = 𝑡)
179177, 178oveq12d 6653 . . . 4 (𝑘 = 𝑡 → ((𝐴𝑘) · 𝑘) = ((𝐴𝑡) · 𝑡))
180179cbvsumv 14407 . . 3 Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡)
181176, 180syl6eqr 2672 . 2 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
182 0nn0 11292 . . . . . . . 8 0 ∈ ℕ0
183 1nn0 11293 . . . . . . . 8 1 ∈ ℕ0
184 prssi 4344 . . . . . . . 8 ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
185182, 183, 184mp2an 707 . . . . . . 7 {0, 1} ⊆ ℕ0
186 fss 6043 . . . . . . 7 (((𝐺𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℕ0) → (𝐺𝐴):ℕ⟶ℕ0)
187185, 186mpan2 706 . . . . . 6 ((𝐺𝐴):ℕ⟶{0, 1} → (𝐺𝐴):ℕ⟶ℕ0)
188 nn0ex 11283 . . . . . . . 8 0 ∈ V
189 nnex 11011 . . . . . . . 8 ℕ ∈ V
190188, 189elmap 7871 . . . . . . 7 ((𝐺𝐴) ∈ (ℕ0𝑚 ℕ) ↔ (𝐺𝐴):ℕ⟶ℕ0)
191190biimpri 218 . . . . . 6 ((𝐺𝐴):ℕ⟶ℕ0 → (𝐺𝐴) ∈ (ℕ0𝑚 ℕ))
19219, 187, 1913syl 18 . . . . 5 ((𝐺𝐴) ∈ ({0, 1} ↑𝑚 ℕ) → (𝐺𝐴) ∈ (ℕ0𝑚 ℕ))
193192anim1i 591 . . . 4 (((𝐺𝐴) ∈ ({0, 1} ↑𝑚 ℕ) ∧ (𝐺𝐴) ∈ 𝑅) → ((𝐺𝐴) ∈ (ℕ0𝑚 ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
194 elin 3788 . . . 4 ((𝐺𝐴) ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ↔ ((𝐺𝐴) ∈ (ℕ0𝑚 ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
195193, 16, 1943imtr4i 281 . . 3 ((𝐺𝐴) ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) → (𝐺𝐴) ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅))
196 eulerpart.s . . . 4 𝑆 = (𝑓 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))
1979, 196eulerpartlemsv2 30394 . . 3 ((𝐺𝐴) ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝑆‘(𝐺𝐴)) = Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘))
19815, 195, 1973syl 18 . 2 (𝐴 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝐴)) = Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘))
199120, 154elind 3790 . . 3 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅))
2009, 196eulerpartlemsv2 30394 . . 3 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
201199, 200syl 17 . 2 (𝐴 ∈ (𝑇𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
202181, 198, 2013eqtr4d 2664 1 (𝐴 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝐴)) = (𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  {cab 2606  wral 2909  wrex 2910  {crab 2913  Vcvv 3195  cdif 3564  cin 3566  wss 3567  c0 3907  ifcif 4077  𝒫 cpw 4149  {csn 4168  {cpr 4170  cop 4174   ciun 4511   class class class wbr 4644  {copab 4703  cmpt 4720   × cxp 5102  ccnv 5103  dom cdm 5104  cres 5106  cima 5107  ccom 5108   Fn wfn 5871  wf 5872  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152   supp csupp 7280  𝑚 cmap 7842  cen 7937  Fincfn 7940  cc 9919  0cc0 9921  1c1 9922   · cmul 9926  cle 10060  cn 11005  2c2 11055  0cn0 11277  cexp 12843  Σcsu 14397  cdvds 14964  bitscbits 15122  𝟭cind 30046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-disj 4612  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-acn 8753  df-ac 8924  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-rp 11818  df-fz 12312  df-fzo 12450  df-fl 12576  df-mod 12652  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-sum 14398  df-dvds 14965  df-bits 15125  df-ind 30047
This theorem is referenced by:  eulerpartlemn  30417
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