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Theorem eulerpartlemgs2 32980
Description: Lemma for eulerpart 32982: 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 6033 . . . . . . . 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 32972 . . . . . . . . . . . . 13 𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅)
13 f1of 6784 . . . . . . . . . . . . 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)
1514ffvelcdmi 7034 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅))
16 elin 3926 . . . . . . . . . . 11 ((𝐺𝐴) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
1715, 16sylib 217 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
1817simpld 495 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) ∈ ({0, 1} ↑m ℕ))
19 elmapi 8787 . . . . . . . . 9 ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) → (𝐺𝐴):ℕ⟶{0, 1})
20 fdm 6677 . . . . . . . . 9 ((𝐺𝐴):ℕ⟶{0, 1} → dom (𝐺𝐴) = ℕ)
2118, 19, 203syl 18 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → dom (𝐺𝐴) = ℕ)
221, 21sseqtrid 3996 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) “ ℕ) ⊆ ℕ)
2322sselda 3944 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ((𝐺𝐴) “ ℕ)) → 𝑘 ∈ ℕ)
242, 3, 4, 5, 6, 7, 8, 9, 10, 11eulerpartlemgvv 32976 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ℕ) → ((𝐺𝐴)‘𝑘) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0))
2524oveq1d 7372 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ℕ) → (((𝐺𝐴)‘𝑘) · 𝑘) = (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
2623, 25syldan 591 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ((𝐺𝐴) “ ℕ)) → (((𝐺𝐴)‘𝑘) · 𝑘) = (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
2726sumeq2dv 15588 . . . 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 2748 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑘))
29282rexbidv 3213 . . . . . . . . . . . 12 (𝑚 = 𝑘 → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
3029elrab 3645 . . . . . . . . . . 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 497 . . . . . . . . . 10 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘)
3231iftrued 4494 . . . . . . . . 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 7372 . . . . . . . 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 3639 . . . . . . . . . 10 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℕ)
3534nncnd 12169 . . . . . . . . 9 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℂ)
3635mulid2d 11173 . . . . . . . 8 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (1 · 𝑘) = 𝑘)
3733, 36eqtrd 2776 . . . . . . 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 15584 . . . . . 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 32979 . . . . . . . . 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 591 . . . . . . . . . . . . . 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 4494 . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺𝐴)‘𝑘) = 1)
46 1nn 12164 . . . . . . . . . . . . 13 1 ∈ ℕ
4745, 46eqeltrdi 2846 . . . . . . . . . . . 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 6668 . . . . . . . . . . . . . 14 ((𝐺𝐴):ℕ⟶{0, 1} → (𝐺𝐴) Fn ℕ)
50 elpreima 7008 . . . . . . . . . . . . . 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 711 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ((𝐺𝐴) “ ℕ))
5453ex 413 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ((𝐺𝐴) “ ℕ)))
5554ssrdv 3950 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ((𝐺𝐴) “ ℕ))
56 ssfi 9117 . . . . . . . . 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 584 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ∈ Fin)
58 cnvexg 7861 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ V)
59 imaexg 7852 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 “ ℕ) ∈ V)
60 inex1g 5276 . . . . . . . . . . 11 ((𝐴 “ ℕ) ∈ V → ((𝐴 “ ℕ) ∩ 𝐽) ∈ V)
6158, 59, 603syl 18 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ V)
62 vsnex 5386 . . . . . . . . . . . 12 {𝑡} ∈ V
63 fvex 6855 . . . . . . . . . . . 12 (bits‘(𝐴𝑡)) ∈ V
6462, 63xpex 7687 . . . . . . . . . . 11 ({𝑡} × (bits‘(𝐴𝑡))) ∈ V
6564rgenw 3068 . . . . . . . . . 10 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V
66 iunexg 7896 . . . . . . . . . 10 ((((𝐴 “ ℕ) ∩ 𝐽) ∈ V ∧ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V)
6761, 65, 66sylancl 586 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V)
68 eqid 2736 . . . . . . . . . 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 32978 . . . . . . . . 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 8911 . . . . . . . . 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 584 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ≈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
72 enfii 9133 . . . . . . . 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 584 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ Fin)
74 fvres 6861 . . . . . . . . 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 4189 . . . . . . . . . . . . . . 15 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
77 simpr 485 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽))
7876, 77sselid 3942 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡𝐽)
7978snssd 4769 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → {𝑡} ⊆ 𝐽)
80 bitsss 16306 . . . . . . . . . . . . 13 (bits‘(𝐴𝑡)) ⊆ ℕ0
81 xpss12 5648 . . . . . . . . . . . . 13 (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴𝑡)) ⊆ ℕ0) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8279, 80, 81sylancl 586 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8382ralrimiva 3143 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
84 iunss 5005 . . . . . . . . . . 11 ( 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8583, 84sylibr 233 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8685sselda 3944 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → 𝑤 ∈ (𝐽 × ℕ0))
875, 6oddpwdcv 32955 . . . . . . . . 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 2776 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → ((𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))‘𝑤) = ((2↑(2nd𝑤)) · (1st𝑤)))
9041nncnd 12169 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
9139, 73, 69, 89, 90fsumf1o 15608 . . . . . 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, 91eqtrid 2788 . . . . 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 11109 . . . . . . . . 9 1 ∈ ℂ
94 0cn 11147 . . . . . . . . 9 0 ∈ ℂ
9593, 94ifcli 4533 . . . . . . . 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 4037 . . . . . . . . 9 {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ℕ
98 simpr 485 . . . . . . . . 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, 98sselid 3942 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℕ)
10099nncnd 12169 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
10196, 100mulcld 11175 . . . . . 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 485 . . . . . . . . . . 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 3923 . . . . . . . . . 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 4102 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℕ)
105104sselda 3944 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ℕ)
10630notbii 319 . . . . . . . . . . 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 400 . . . . . . . . . . 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 237 . . . . . . . . . 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 4497 . . . . . . . 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 7372 . . . . . . 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 12158 . . . . . . . . . 10 ℕ ⊆ ℂ
113104, 112sstrdi 3956 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℂ)
114113sselda 3944 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ℂ)
115114mul02d 11353 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → (0 · 𝑘) = 0)
116111, 115eqtrd 2776 . . . . . 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 15610 . . . . 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 2778 . . . 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 32969 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
120119simp1bi 1145 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0m ℕ))
121 elmapi 8787 . . . . . . . . . . . 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 6033 . . . . . . . . . . . . 13 (𝐴 “ ℕ) ⊆ dom 𝐴
125124, 122fssdm 6688 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ ℕ)
126125adantr 481 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (𝐴 “ ℕ) ⊆ ℕ)
127 inss1 4188 . . . . . . . . . . . 12 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ (𝐴 “ ℕ)
128127, 77sselid 3942 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ (𝐴 “ ℕ))
129126, 128sseldd 3945 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℕ)
130123, 129ffvelcdmd 7036 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (𝐴𝑡) ∈ ℕ0)
131 bitsfi 16317 . . . . . . . . 9 ((𝐴𝑡) ∈ ℕ0 → (bits‘(𝐴𝑡)) ∈ Fin)
132130, 131syl 17 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (bits‘(𝐴𝑡)) ∈ Fin)
133129nncnd 12169 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℂ)
134 2cnd 12231 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 2 ∈ ℂ)
135 simprr 771 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ (bits‘(𝐴𝑡)))
13680, 135sselid 3942 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ ℕ0)
137134, 136expcld 14051 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℂ)
138137anassrs 468 . . . . . . . 8 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → (2↑𝑛) ∈ ℂ)
139132, 133, 138fsummulc1 15670 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡))
140139sumeq2dv 15588 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡))
141 bitsinv1 16322 . . . . . . . . 9 ((𝐴𝑡) ∈ ℕ0 → Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) = (𝐴𝑡))
142141oveq1d 7372 . . . . . . . 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 15588 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡))
145 vex 3449 . . . . . . . . . 10 𝑡 ∈ V
146 vex 3449 . . . . . . . . . 10 𝑛 ∈ V
147145, 146op2ndd 7932 . . . . . . . . 9 (𝑤 = ⟨𝑡, 𝑛⟩ → (2nd𝑤) = 𝑛)
148147oveq2d 7373 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑛⟩ → (2↑(2nd𝑤)) = (2↑𝑛))
149145, 146op1std 7931 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑛⟩ → (1st𝑤) = 𝑡)
150148, 149oveq12d 7375 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑(2nd𝑤)) · (1st𝑤)) = ((2↑𝑛) · 𝑡))
151 inss2 4189 . . . . . . . . . 10 (𝑇𝑅) ⊆ 𝑅
152151sseli 3940 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝐴𝑅)
153 cnveq 5829 . . . . . . . . . . . 12 (𝑓 = 𝐴𝑓 = 𝐴)
154153imaeq1d 6012 . . . . . . . . . . 11 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
155154eleq1d 2822 . . . . . . . . . 10 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
156155, 9elab2g 3632 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (𝐴𝑅 ↔ (𝐴 “ ℕ) ∈ Fin))
157152, 156mpbid 231 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ∈ Fin)
158 ssfi 9117 . . . . . . . 8 (((𝐴 “ ℕ) ∈ Fin ∧ ((𝐴 “ ℕ) ∩ 𝐽) ⊆ (𝐴 “ ℕ)) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
159157, 127, 158sylancl 586 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
160133adantrr 715 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℂ)
161137, 160mulcld 11175 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ∈ ℂ)
162150, 159, 132, 161fsum2d 15656 . . . . . 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 2784 . . . . 5 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡) = Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)))
164 inss1 4188 . . . . . . . . 9 (𝑇𝑅) ⊆ 𝑇
165164sseli 3940 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝐴𝑇)
166154sseq1d 3975 . . . . . . . . . 10 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ (𝐴 “ ℕ) ⊆ 𝐽))
167166, 10elrab2 3648 . . . . . . . . 9 (𝐴𝑇 ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽))
168167simprbi 497 . . . . . . . 8 (𝐴𝑇 → (𝐴 “ ℕ) ⊆ 𝐽)
169165, 168syl 17 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
170 df-ss 3927 . . . . . . 7 ((𝐴 “ ℕ) ⊆ 𝐽 ↔ ((𝐴 “ ℕ) ∩ 𝐽) = (𝐴 “ ℕ))
171169, 170sylib 217 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) = (𝐴 “ ℕ))
172171sumeq1d 15586 . . . . 5 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
173163, 172eqtr3d 2778 . . . 4 (𝐴 ∈ (𝑇𝑅) → Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
17427, 118, 1733eqtr2d 2782 . . 3 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
175 fveq2 6842 . . . . 5 (𝑘 = 𝑡 → (𝐴𝑘) = (𝐴𝑡))
176 id 22 . . . . 5 (𝑘 = 𝑡𝑘 = 𝑡)
177175, 176oveq12d 7375 . . . 4 (𝑘 = 𝑡 → ((𝐴𝑘) · 𝑘) = ((𝐴𝑡) · 𝑡))
178177cbvsumv 15581 . . 3 Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡)
179174, 178eqtr4di 2794 . 2 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
180 0nn0 12428 . . . . . . . 8 0 ∈ ℕ0
181 1nn0 12429 . . . . . . . 8 1 ∈ ℕ0
182 prssi 4781 . . . . . . . 8 ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
183180, 181, 182mp2an 690 . . . . . . 7 {0, 1} ⊆ ℕ0
184 fss 6685 . . . . . . 7 (((𝐺𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℕ0) → (𝐺𝐴):ℕ⟶ℕ0)
185183, 184mpan2 689 . . . . . 6 ((𝐺𝐴):ℕ⟶{0, 1} → (𝐺𝐴):ℕ⟶ℕ0)
186 nn0ex 12419 . . . . . . . 8 0 ∈ V
187 nnex 12159 . . . . . . . 8 ℕ ∈ V
188186, 187elmap 8809 . . . . . . 7 ((𝐺𝐴) ∈ (ℕ0m ℕ) ↔ (𝐺𝐴):ℕ⟶ℕ0)
189188biimpri 227 . . . . . 6 ((𝐺𝐴):ℕ⟶ℕ0 → (𝐺𝐴) ∈ (ℕ0m ℕ))
19019, 185, 1893syl 18 . . . . 5 ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) → (𝐺𝐴) ∈ (ℕ0m ℕ))
191190anim1i 615 . . . 4 (((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺𝐴) ∈ 𝑅) → ((𝐺𝐴) ∈ (ℕ0m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
192 elin 3926 . . . 4 ((𝐺𝐴) ∈ ((ℕ0m ℕ) ∩ 𝑅) ↔ ((𝐺𝐴) ∈ (ℕ0m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
193191, 16, 1923imtr4i 291 . . 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 32958 . . 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 4154 . . 3 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅))
1989, 194eulerpartlemsv2 32958 . . 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 2786 1 (𝐴 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝐴)) = (𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  cin 3909  wss 3910  c0 4282  ifcif 4486  𝒫 cpw 4560  {csn 4586  {cpr 4588  cop 4592   ciun 4954   class class class wbr 5105  {copab 5167  cmpt 5188   × cxp 5631  ccnv 5632  dom cdm 5633  cres 5635  cima 5636  ccom 5637   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920   supp csupp 8092  m cmap 8765  cen 8880  Fincfn 8883  cc 11049  0cc0 11051  1c1 11052   · cmul 11056  cle 11190  cn 12153  2c2 12208  0cn0 12413  cexp 13967  Σcsu 15570  cdvds 16136  bitscbits 16299  𝟭cind 32609
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-dvds 16137  df-bits 16302  df-ind 32610
This theorem is referenced by:  eulerpartlemn  32981
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