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Theorem eulerpartlemgs2 34382
Description: Lemma for eulerpart 34384: 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 6100 . . . . . . . 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 34374 . . . . . . . . . . . . 13 𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅)
13 f1of 6848 . . . . . . . . . . . . 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 7103 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅))
16 elin 3967 . . . . . . . . . . 11 ((𝐺𝐴) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
1715, 16sylib 218 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
1817simpld 494 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) ∈ ({0, 1} ↑m ℕ))
19 elmapi 8889 . . . . . . . . 9 ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) → (𝐺𝐴):ℕ⟶{0, 1})
20 fdm 6745 . . . . . . . . 9 ((𝐺𝐴):ℕ⟶{0, 1} → dom (𝐺𝐴) = ℕ)
2118, 19, 203syl 18 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → dom (𝐺𝐴) = ℕ)
221, 21sseqtrid 4026 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) “ ℕ) ⊆ ℕ)
2322sselda 3983 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ((𝐺𝐴) “ ℕ)) → 𝑘 ∈ ℕ)
242, 3, 4, 5, 6, 7, 8, 9, 10, 11eulerpartlemgvv 34378 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ ℕ) → ((𝐺𝐴)‘𝑘) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0))
2524oveq1d 7446 . . . . . 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 15738 . . . 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 2749 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑘))
29282rexbidv 3222 . . . . . . . . . . . 12 (𝑚 = 𝑘 → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
3029elrab 3692 . . . . . . . . . . 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 496 . . . . . . . . . 10 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘)
3231iftrued 4533 . . . . . . . . 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 7446 . . . . . . . 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 3687 . . . . . . . . . 10 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℕ)
3534nncnd 12282 . . . . . . . . 9 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℂ)
3635mullidd 11279 . . . . . . . 8 (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (1 · 𝑘) = 𝑘)
3733, 36eqtrd 2777 . . . . . . 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 15734 . . . . . 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 34381 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) “ ℕ) ∈ Fin)
4134adantl 481 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 4533 . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺𝐴)‘𝑘) = 1)
46 1nn 12277 . . . . . . . . . . . . 13 1 ∈ ℕ
4745, 46eqeltrdi 2849 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺𝐴)‘𝑘) ∈ ℕ)
48 ffn 6736 . . . . . . . . . . . . . 14 ((𝐺𝐴):ℕ⟶{0, 1} → (𝐺𝐴) Fn ℕ)
49 elpreima 7078 . . . . . . . . . . . . . 14 ((𝐺𝐴) Fn ℕ → (𝑘 ∈ ((𝐺𝐴) “ ℕ) ↔ (𝑘 ∈ ℕ ∧ ((𝐺𝐴)‘𝑘) ∈ ℕ)))
5018, 19, 48, 494syl 19 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑘 ∈ ((𝐺𝐴) “ ℕ) ↔ (𝑘 ∈ ℕ ∧ ((𝐺𝐴)‘𝑘) ∈ ℕ)))
5150adantr 480 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → (𝑘 ∈ ((𝐺𝐴) “ ℕ) ↔ (𝑘 ∈ ℕ ∧ ((𝐺𝐴)‘𝑘) ∈ ℕ)))
5241, 47, 51mpbir2and 713 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ((𝐺𝐴) “ ℕ))
5352ex 412 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ((𝐺𝐴) “ ℕ)))
5453ssrdv 3989 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ((𝐺𝐴) “ ℕ))
55 ssfi 9213 . . . . . . . . 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)
5640, 54, 55syl2anc 584 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ∈ Fin)
57 cnvexg 7946 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ V)
58 imaexg 7935 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 “ ℕ) ∈ V)
59 inex1g 5319 . . . . . . . . . . 11 ((𝐴 “ ℕ) ∈ V → ((𝐴 “ ℕ) ∩ 𝐽) ∈ V)
6057, 58, 593syl 18 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ V)
61 vsnex 5434 . . . . . . . . . . . 12 {𝑡} ∈ V
62 fvex 6919 . . . . . . . . . . . 12 (bits‘(𝐴𝑡)) ∈ V
6361, 62xpex 7773 . . . . . . . . . . 11 ({𝑡} × (bits‘(𝐴𝑡))) ∈ V
6463rgenw 3065 . . . . . . . . . 10 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V
65 iunexg 7988 . . . . . . . . . 10 ((((𝐴 “ ℕ) ∩ 𝐽) ∈ V ∧ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V)
6660, 64, 65sylancl 586 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ V)
67 eqid 2737 . . . . . . . . . 10 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))
682, 3, 4, 5, 6, 7, 8, 9, 10, 11, 67eulerpartlemgh 34380 . . . . . . . . 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})
69 f1oeng 9011 . . . . . . . . 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})
7066, 68, 69syl2anc 584 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ≈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
71 enfii 9226 . . . . . . . 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)
7256, 70, 71syl2anc 584 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ∈ Fin)
73 fvres 6925 . . . . . . . . 9 (𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) → ((𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))‘𝑤) = (𝐹𝑤))
7473adantl 481 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → ((𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))‘𝑤) = (𝐹𝑤))
75 inss2 4238 . . . . . . . . . . . . . . 15 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
76 simpr 484 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽))
7775, 76sselid 3981 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡𝐽)
7877snssd 4809 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → {𝑡} ⊆ 𝐽)
79 bitsss 16463 . . . . . . . . . . . . 13 (bits‘(𝐴𝑡)) ⊆ ℕ0
80 xpss12 5700 . . . . . . . . . . . . 13 (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴𝑡)) ⊆ ℕ0) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8178, 79, 80sylancl 586 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8281ralrimiva 3146 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
83 iunss 5045 . . . . . . . . . . 11 ( 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8482, 83sylibr 234 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8584sselda 3983 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → 𝑤 ∈ (𝐽 × ℕ0))
865, 6oddpwdcv 34357 . . . . . . . . 9 (𝑤 ∈ (𝐽 × ℕ0) → (𝐹𝑤) = ((2↑(2nd𝑤)) · (1st𝑤)))
8785, 86syl 17 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → (𝐹𝑤) = ((2↑(2nd𝑤)) · (1st𝑤)))
8874, 87eqtrd 2777 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) → ((𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))‘𝑤) = ((2↑(2nd𝑤)) · (1st𝑤)))
8941nncnd 12282 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
9039, 72, 68, 88, 89fsumf1o 15759 . . . . . 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)))
9138, 90eqtrid 2789 . . . . 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)))
92 ax-1cn 11213 . . . . . . . . 9 1 ∈ ℂ
93 0cn 11253 . . . . . . . . 9 0 ∈ ℂ
9492, 93ifcli 4573 . . . . . . . 8 if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) ∈ ℂ
9594a1i 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)
96 ssrab2 4080 . . . . . . . . 9 {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ℕ
97 simpr 484 . . . . . . . . 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})
9896, 97sselid 3981 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℕ)
9998nncnd 12282 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
10095, 99mulcld 11281 . . . . . 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)
101 simpr 484 . . . . . . . . . . 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}))
102101eldifbd 3964 . . . . . . . . . 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})
10322ssdifssd 4147 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℕ)
104103sselda 3983 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ℕ)
10530notbii 320 . . . . . . . . . . 11 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ↔ ¬ (𝑘 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
106 imnan 399 . . . . . . . . . . 11 ((𝑘 ∈ ℕ → ¬ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘) ↔ ¬ (𝑘 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
107105, 106sylbb2 238 . . . . . . . . . 10 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (𝑘 ∈ ℕ → ¬ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑘))
108102, 104, 107sylc 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)
109108iffalsed 4536 . . . . . . . 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)
110109oveq1d 7446 . . . . . . 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))
111 nnsscn 12271 . . . . . . . . . 10 ℕ ⊆ ℂ
112103, 111sstrdi 3996 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℂ)
113112sselda 3983 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ℂ)
114113mul02d 11459 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑘 ∈ (((𝐺𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → (0 · 𝑘) = 0)
115110, 114eqtrd 2777 . . . . . 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)
11654, 100, 115, 40fsumss 15761 . . . . 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))
11791, 116eqtr3d 2779 . . . 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))
1182, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemt0 34371 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
119118simp1bi 1146 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0m ℕ))
120 elmapi 8889 . . . . . . . . . . . 12 (𝐴 ∈ (ℕ0m ℕ) → 𝐴:ℕ⟶ℕ0)
121119, 120syl 17 . . . . . . . . . . 11 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
122121adantr 480 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝐴:ℕ⟶ℕ0)
123 cnvimass 6100 . . . . . . . . . . . . 13 (𝐴 “ ℕ) ⊆ dom 𝐴
124123, 121fssdm 6755 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ ℕ)
125124adantr 480 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (𝐴 “ ℕ) ⊆ ℕ)
126 inss1 4237 . . . . . . . . . . . 12 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ (𝐴 “ ℕ)
127126, 76sselid 3981 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ (𝐴 “ ℕ))
128125, 127sseldd 3984 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℕ)
129122, 128ffvelcdmd 7105 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (𝐴𝑡) ∈ ℕ0)
130 bitsfi 16474 . . . . . . . . 9 ((𝐴𝑡) ∈ ℕ0 → (bits‘(𝐴𝑡)) ∈ Fin)
131129, 130syl 17 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (bits‘(𝐴𝑡)) ∈ Fin)
132128nncnd 12282 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℂ)
133 2cnd 12344 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 2 ∈ ℂ)
134 simprr 773 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ (bits‘(𝐴𝑡)))
13579, 134sselid 3981 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ ℕ0)
136133, 135expcld 14186 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℂ)
137136anassrs 467 . . . . . . . 8 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → (2↑𝑛) ∈ ℂ)
138131, 132, 137fsummulc1 15821 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡))
139138sumeq2dv 15738 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡))
140 bitsinv1 16479 . . . . . . . . 9 ((𝐴𝑡) ∈ ℕ0 → Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) = (𝐴𝑡))
141140oveq1d 7446 . . . . . . . 8 ((𝐴𝑡) ∈ ℕ0 → (Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = ((𝐴𝑡) · 𝑡))
142129, 141syl 17 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)) → (Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = ((𝐴𝑡) · 𝑡))
143142sumeq2dv 15738 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡))
144 vex 3484 . . . . . . . . . 10 𝑡 ∈ V
145 vex 3484 . . . . . . . . . 10 𝑛 ∈ V
146144, 145op2ndd 8025 . . . . . . . . 9 (𝑤 = ⟨𝑡, 𝑛⟩ → (2nd𝑤) = 𝑛)
147146oveq2d 7447 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑛⟩ → (2↑(2nd𝑤)) = (2↑𝑛))
148144, 145op1std 8024 . . . . . . . 8 (𝑤 = ⟨𝑡, 𝑛⟩ → (1st𝑤) = 𝑡)
149147, 148oveq12d 7449 . . . . . . 7 (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑(2nd𝑤)) · (1st𝑤)) = ((2↑𝑛) · 𝑡))
150 inss2 4238 . . . . . . . . . 10 (𝑇𝑅) ⊆ 𝑅
151150sseli 3979 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → 𝐴𝑅)
152 cnveq 5884 . . . . . . . . . . . 12 (𝑓 = 𝐴𝑓 = 𝐴)
153152imaeq1d 6077 . . . . . . . . . . 11 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
154153eleq1d 2826 . . . . . . . . . 10 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
155154, 9elab2g 3680 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (𝐴𝑅 ↔ (𝐴 “ ℕ) ∈ Fin))
156151, 155mpbid 232 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ∈ Fin)
157 ssfi 9213 . . . . . . . 8 (((𝐴 “ ℕ) ∈ Fin ∧ ((𝐴 “ ℕ) ∩ 𝐽) ⊆ (𝐴 “ ℕ)) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
158156, 126, 157sylancl 586 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
159132adantrr 717 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℂ)
160136, 159mulcld 11281 . . . . . . 7 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ∈ ℂ)
161149, 158, 131, 160fsum2d 15807 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)))
162139, 143, 1613eqtr3d 2785 . . . . 5 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡) = Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)))
163 inss1 4237 . . . . . . . . 9 (𝑇𝑅) ⊆ 𝑇
164163sseli 3979 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → 𝐴𝑇)
165153sseq1d 4015 . . . . . . . . . 10 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ (𝐴 “ ℕ) ⊆ 𝐽))
166165, 10elrab2 3695 . . . . . . . . 9 (𝐴𝑇 ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽))
167166simprbi 496 . . . . . . . 8 (𝐴𝑇 → (𝐴 “ ℕ) ⊆ 𝐽)
168164, 167syl 17 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
169 dfss2 3969 . . . . . . 7 ((𝐴 “ ℕ) ⊆ 𝐽 ↔ ((𝐴 “ ℕ) ∩ 𝐽) = (𝐴 “ ℕ))
170168, 169sylib 218 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → ((𝐴 “ ℕ) ∩ 𝐽) = (𝐴 “ ℕ))
171170sumeq1d 15736 . . . . 5 (𝐴 ∈ (𝑇𝑅) → Σ𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)((𝐴𝑡) · 𝑡) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
172162, 171eqtr3d 2779 . . . 4 (𝐴 ∈ (𝑇𝑅) → Σ𝑤 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))((2↑(2nd𝑤)) · (1st𝑤)) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
17327, 117, 1723eqtr2d 2783 . . 3 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡))
174 fveq2 6906 . . . . 5 (𝑘 = 𝑡 → (𝐴𝑘) = (𝐴𝑡))
175 id 22 . . . . 5 (𝑘 = 𝑡𝑘 = 𝑡)
176174, 175oveq12d 7449 . . . 4 (𝑘 = 𝑡 → ((𝐴𝑘) · 𝑘) = ((𝐴𝑡) · 𝑡))
177176cbvsumv 15732 . . 3 Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘) = Σ𝑡 ∈ (𝐴 “ ℕ)((𝐴𝑡) · 𝑡)
178173, 177eqtr4di 2795 . 2 (𝐴 ∈ (𝑇𝑅) → Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
179 0nn0 12541 . . . . . . . 8 0 ∈ ℕ0
180 1nn0 12542 . . . . . . . 8 1 ∈ ℕ0
181 prssi 4821 . . . . . . . 8 ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
182179, 180, 181mp2an 692 . . . . . . 7 {0, 1} ⊆ ℕ0
183 fss 6752 . . . . . . 7 (((𝐺𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℕ0) → (𝐺𝐴):ℕ⟶ℕ0)
184182, 183mpan2 691 . . . . . 6 ((𝐺𝐴):ℕ⟶{0, 1} → (𝐺𝐴):ℕ⟶ℕ0)
185 nn0ex 12532 . . . . . . . 8 0 ∈ V
186 nnex 12272 . . . . . . . 8 ℕ ∈ V
187185, 186elmap 8911 . . . . . . 7 ((𝐺𝐴) ∈ (ℕ0m ℕ) ↔ (𝐺𝐴):ℕ⟶ℕ0)
188187biimpri 228 . . . . . 6 ((𝐺𝐴):ℕ⟶ℕ0 → (𝐺𝐴) ∈ (ℕ0m ℕ))
18919, 184, 1883syl 18 . . . . 5 ((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) → (𝐺𝐴) ∈ (ℕ0m ℕ))
190189anim1i 615 . . . 4 (((𝐺𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺𝐴) ∈ 𝑅) → ((𝐺𝐴) ∈ (ℕ0m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
191 elin 3967 . . . 4 ((𝐺𝐴) ∈ ((ℕ0m ℕ) ∩ 𝑅) ↔ ((𝐺𝐴) ∈ (ℕ0m ℕ) ∧ (𝐺𝐴) ∈ 𝑅))
192190, 16, 1913imtr4i 292 . . 3 ((𝐺𝐴) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) → (𝐺𝐴) ∈ ((ℕ0m ℕ) ∩ 𝑅))
193 eulerpart.s . . . 4 𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))
1949, 193eulerpartlemsv2 34360 . . 3 ((𝐺𝐴) ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆‘(𝐺𝐴)) = Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘))
19515, 192, 1943syl 18 . 2 (𝐴 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝐴)) = Σ𝑘 ∈ ((𝐺𝐴) “ ℕ)(((𝐺𝐴)‘𝑘) · 𝑘))
196119, 151elind 4200 . . 3 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅))
1979, 193eulerpartlemsv2 34360 . . 3 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
198196, 197syl 17 . 2 (𝐴 ∈ (𝑇𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
199178, 195, 1983eqtr4d 2787 1 (𝐴 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝐴)) = (𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  cin 3950  wss 3951  c0 4333  ifcif 4525  𝒫 cpw 4600  {csn 4626  {cpr 4628  cop 4632   ciun 4991   class class class wbr 5143  {copab 5205  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  cres 5687  cima 5688  ccom 5689   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013   supp csupp 8185  m cmap 8866  cen 8982  Fincfn 8985  cc 11153  0cc0 11155  1c1 11156   · cmul 11160  cle 11296  cn 12266  2c2 12321  0cn0 12526  cexp 14102  Σcsu 15722  cdvds 16290  bitscbits 16456  𝟭cind 32835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-dvds 16291  df-bits 16459  df-ind 32836
This theorem is referenced by:  eulerpartlemn  34383
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