Theorem eulerpartlemgs2 31040

Description: Lemma for eulerpart 31042: The 𝐺 function also preserves partition sums. (Contributed by Thierry Arnoux, 10-Sep-2017.)

Hypotheses

Ref	Expression
eulerpart.p	⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
eulerpart.o	⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d	⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
eulerpart.j	⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f	⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h	⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m	⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
eulerpart.r	⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
eulerpart.t	⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
eulerpart.g	⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
eulerpart.s	⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))

Assertion

Ref	Expression
eulerpartlemgs2	⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝐴)) = (𝑆‘𝐴))

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.

Step	Hyp	Ref	Expression
1		cnvimass 5739	. . . . . . . 8 ⊢ (◡(𝐺‘𝐴) “ ℕ) ⊆ dom (𝐺‘𝐴)
2		eulerpart.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
3		eulerpart.o	. . . . . . . . . . . . . 14 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
4		eulerpart.d	. . . . . . . . . . . . . 14 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
5		eulerpart.j	. . . . . . . . . . . . . 14 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
6		eulerpart.f	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
7		eulerpart.h	. . . . . . . . . . . . . 14 ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
8		eulerpart.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
9		eulerpart.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
10		eulerpart.t	. . . . . . . . . . . . . 14 ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
11		eulerpart.g	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
12	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	eulerpartgbij 31032	. . . . . . . . . . . . 13 ⊢ 𝐺:(𝑇 ∩ 𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅)
13		f1of 6391	. . . . . . . . . . . . 13 ⊢ (𝐺:(𝑇 ∩ 𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅) → 𝐺:(𝑇 ∩ 𝑅)⟶(({0, 1} ↑𝑚 ℕ) ∩ 𝑅))
14	12, 13	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝐺:(𝑇 ∩ 𝑅)⟶(({0, 1} ↑𝑚 ℕ) ∩ 𝑅)
15	14	ffvelrni 6622	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅))
16		elin 4019	. . . . . . . . . . 11 ⊢ ((𝐺‘𝐴) ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) ↔ ((𝐺‘𝐴) ∈ ({0, 1} ↑𝑚 ℕ) ∧ (𝐺‘𝐴) ∈ 𝑅))
17	15, 16	sylib 210	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((𝐺‘𝐴) ∈ ({0, 1} ↑𝑚 ℕ) ∧ (𝐺‘𝐴) ∈ 𝑅))
18	17	simpld 490	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) ∈ ({0, 1} ↑𝑚 ℕ))
19		elmapi 8162	. . . . . . . . 9 ⊢ ((𝐺‘𝐴) ∈ ({0, 1} ↑𝑚 ℕ) → (𝐺‘𝐴):ℕ⟶{0, 1})
20		fdm 6299	. . . . . . . . 9 ⊢ ((𝐺‘𝐴):ℕ⟶{0, 1} → dom (𝐺‘𝐴) = ℕ)
21	18, 19, 20	3syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → dom (𝐺‘𝐴) = ℕ)
22	1, 21	syl5sseq 3872	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ ℕ) ⊆ ℕ)
23	22	sselda 3821	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)) → 𝑘 ∈ ℕ)
24	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	eulerpartlemgvv 31036	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝐴)‘𝑘) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0))
25	24	oveq1d 6937	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝐴)‘𝑘) · 𝑘) = (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
26	23, 25	syldan 585	. . . . 5 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)) → (((𝐺‘𝐴)‘𝑘) · 𝑘) = (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
27	26	sumeq2dv 14841	. . . 4 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)(((𝐺‘𝐴)‘𝑘) · 𝑘) = Σ𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)(if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
28		eqeq2 2789	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑘))
29	28	2rexbidv 3242	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘))
30	29	elrab 3572	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ↔ (𝑘 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘))
31	30	simprbi 492	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘)
32	31	iftrued 4315	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) = 1)
33	32	oveq1d 6937	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = (1 · 𝑘))
34		elrabi 3567	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℕ)
35	34	nncnd 11392	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℂ)
36	35	mulid2d 10395	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (1 · 𝑘) = 𝑘)
37	33, 36	eqtrd 2814	. . . . . . 7 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = 𝑘)
38	37	sumeq2i 14837	. . . . . 6 ⊢ Σ𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = Σ𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}𝑘
39		id 22	. . . . . . 7 ⊢ (𝑘 = ((2↑(2nd ‘𝑤)) · (1st ‘𝑤)) → 𝑘 = ((2↑(2nd ‘𝑤)) · (1st ‘𝑤)))
40	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	eulerpartlemgf 31039	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ ℕ) ∈ Fin)
41	34	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℕ)
42	41, 24	syldan 585	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺‘𝐴)‘𝑘) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0))
43	31	adantl 475	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘)
44	43	iftrued 4315	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) = 1)
45	42, 44	eqtrd 2814	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺‘𝐴)‘𝑘) = 1)
46		1nn 11387	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
47	45, 46	syl6eqel 2867	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺‘𝐴)‘𝑘) ∈ ℕ)
48	18, 19	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴):ℕ⟶{0, 1})
49		ffn 6291	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝐴):ℕ⟶{0, 1} → (𝐺‘𝐴) Fn ℕ)
50		elpreima 6600	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝐴) Fn ℕ → (𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ) ↔ (𝑘 ∈ ℕ ∧ ((𝐺‘𝐴)‘𝑘) ∈ ℕ)))
51	48, 49, 50	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ) ↔ (𝑘 ∈ ℕ ∧ ((𝐺‘𝐴)‘𝑘) ∈ ℕ)))
52	51	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → (𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ) ↔ (𝑘 ∈ ℕ ∧ ((𝐺‘𝐴)‘𝑘) ∈ ℕ)))
53	41, 47, 52	mpbir2and 703	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ))
54	53	ex 403	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)))
55	54	ssrdv 3827	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ (◡(𝐺‘𝐴) “ ℕ))
56		ssfi 8468	. . . . . . . . 9 ⊢ (((◡(𝐺‘𝐴) “ ℕ) ∈ Fin ∧ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ (◡(𝐺‘𝐴) “ ℕ)) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ∈ Fin)
57	40, 55, 56	syl2anc 579	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ∈ Fin)
58		cnvexg 7391	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ◡𝐴 ∈ V)
59		imaexg 7382	. . . . . . . . . . 11 ⊢ (◡𝐴 ∈ V → (◡𝐴 “ ℕ) ∈ V)
60		inex1g 5038	. . . . . . . . . . 11 ⊢ ((◡𝐴 “ ℕ) ∈ V → ((◡𝐴 “ ℕ) ∩ 𝐽) ∈ V)
61	58, 59, 60	3syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡𝐴 “ ℕ) ∩ 𝐽) ∈ V)
62		snex 5140	. . . . . . . . . . . 12 ⊢ {𝑡} ∈ V
63		fvex 6459	. . . . . . . . . . . 12 ⊢ (bits‘(𝐴‘𝑡)) ∈ V
64	62, 63	xpex 7240	. . . . . . . . . . 11 ⊢ ({𝑡} × (bits‘(𝐴‘𝑡))) ∈ V
65	64	rgenw 3106	. . . . . . . . . 10 ⊢ ∀𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ∈ V
66		iunexg 7421	. . . . . . . . . 10 ⊢ ((((◡𝐴 “ ℕ) ∩ 𝐽) ∈ V ∧ ∀𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ∈ V) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ∈ V)
67	61, 65, 66	sylancl 580	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ∈ V)
68		eqid 2778	. . . . . . . . . 10 ⊢ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))
69	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 68	eulerpartlemgh 31038	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 ↾ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))):∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
70		f1oeng 8260	. . . . . . . . 9 ⊢ ((∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ∈ V ∧ (𝐹 ↾ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))):∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ≈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
71	67, 69, 70	syl2anc 579	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ≈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
72		enfii 8465	. . . . . . . 8 ⊢ (({𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ∈ Fin ∧ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ≈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ∈ Fin)
73	57, 71, 72	syl2anc 579	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ∈ Fin)
74		fvres 6465	. . . . . . . . 9 ⊢ (𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) → ((𝐹 ↾ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))))‘𝑤) = (𝐹‘𝑤))
75	74	adantl 475	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))) → ((𝐹 ↾ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))))‘𝑤) = (𝐹‘𝑤))
76		inss2 4054	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
77		simpr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽))
78	76, 77	sseldi 3819	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ 𝐽)
79	78	snssd 4571	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → {𝑡} ⊆ 𝐽)
80		bitsss 15554	. . . . . . . . . . . . 13 ⊢ (bits‘(𝐴‘𝑡)) ⊆ ℕ0
81		xpss12 5370	. . . . . . . . . . . . 13 ⊢ (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴‘𝑡)) ⊆ ℕ0) → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
82	79, 80, 81	sylancl 580	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
83	82	ralrimiva 3148	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∀𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
84		iunss 4794	. . . . . . . . . . 11 ⊢ (∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
85	83, 84	sylibr 226	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
86	85	sselda 3821	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))) → 𝑤 ∈ (𝐽 × ℕ0))
87	5, 6	oddpwdcv 31015	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐽 × ℕ0) → (𝐹‘𝑤) = ((2↑(2nd ‘𝑤)) · (1st ‘𝑤)))
88	86, 87	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))) → (𝐹‘𝑤) = ((2↑(2nd ‘𝑤)) · (1st ‘𝑤)))
89	75, 88	eqtrd 2814	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))) → ((𝐹 ↾ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))))‘𝑤) = ((2↑(2nd ‘𝑤)) · (1st ‘𝑤)))
90	41	nncnd 11392	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
91	39, 73, 69, 89, 90	fsumf1o 14861	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}𝑘 = Σ𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))((2↑(2nd ‘𝑤)) · (1st ‘𝑤)))
92	38, 91	syl5eq 2826	. . . . 5 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = Σ𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))((2↑(2nd ‘𝑤)) · (1st ‘𝑤)))
93		ax-1cn 10330	. . . . . . . . 9 ⊢ 1 ∈ ℂ
94		0cn 10368	. . . . . . . . 9 ⊢ 0 ∈ ℂ
95	93, 94	ifcli 4353	. . . . . . . 8 ⊢ if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) ∈ ℂ
96	95	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) ∈ ℂ)
97		ssrab2 3908	. . . . . . . . 9 ⊢ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ℕ
98		simpr 479	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
99	97, 98	sseldi 3819	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℕ)
100	99	nncnd 11392	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
101	96, 100	mulcld 10397	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) ∈ ℂ)
102		simpr 479	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
103	102	eldifbd 3805	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → ¬ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
104	22	ssdifssd 3971	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℕ)
105	104	sselda 3821	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ℕ)
106	30	notbii 312	. . . . . . . . . . 11 ⊢ (¬ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ↔ ¬ (𝑘 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘))
107		imnan 390	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ → ¬ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘) ↔ ¬ (𝑘 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘))
108	106, 107	sylbb2 230	. . . . . . . . . 10 ⊢ (¬ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (𝑘 ∈ ℕ → ¬ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘))
109	103, 105, 108	sylc 65	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → ¬ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘)
110	109	iffalsed 4318	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) = 0)
111	110	oveq1d 6937	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = (0 · 𝑘))
112		nnsscn 11379	. . . . . . . . . 10 ⊢ ℕ ⊆ ℂ
113	104, 112	syl6ss 3833	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℂ)
114	113	sselda 3821	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → 𝑘 ∈ ℂ)
115	114	mul02d 10574	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → (0 · 𝑘) = 0)
116	111, 115	eqtrd 2814	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})) → (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = 0)
117	55, 101, 116, 40	fsumss 14863	. . . . 5 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} (if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘) = Σ𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)(if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
118	92, 117	eqtr3d 2816	. . . 4 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))((2↑(2nd ‘𝑤)) · (1st ‘𝑤)) = Σ𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)(if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑘, 1, 0) · 𝑘))
119	2, 3, 4, 5, 6, 7, 8, 9, 10	eulerpartlemt0 31029	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
120	119	simp1bi 1136	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ (ℕ0 ↑𝑚 ℕ))
121		elmapi 8162	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) → 𝐴:ℕ⟶ℕ0)
122	120, 121	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
123	122	adantr 474	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝐴:ℕ⟶ℕ0)
124		cnvimass 5739	. . . . . . . . . . . . 13 ⊢ (◡𝐴 “ ℕ) ⊆ dom 𝐴
125	124, 122	fssdm 6307	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡𝐴 “ ℕ) ⊆ ℕ)
126	125	adantr 474	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → (◡𝐴 “ ℕ) ⊆ ℕ)
127		inss1 4053	. . . . . . . . . . . 12 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ (◡𝐴 “ ℕ)
128	127, 77	sseldi 3819	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ (◡𝐴 “ ℕ))
129	126, 128	sseldd 3822	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℕ)
130	123, 129	ffvelrnd 6624	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → (𝐴‘𝑡) ∈ ℕ0)
131		bitsfi 15565	. . . . . . . . 9 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (bits‘(𝐴‘𝑡)) ∈ Fin)
132	130, 131	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → (bits‘(𝐴‘𝑡)) ∈ Fin)
133	129	nncnd 11392	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℂ)
134		2cnd 11453	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 2 ∈ ℂ)
135		simprr 763	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ (bits‘(𝐴‘𝑡)))
136	80, 135	sseldi 3819	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ ℕ0)
137	134, 136	expcld 13327	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℂ)
138	137	anassrs 461	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑛) ∈ ℂ)
139	132, 133, 138	fsummulc1 14921	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → (Σ𝑛 ∈ (bits‘(𝐴‘𝑡))(2↑𝑛) · 𝑡) = Σ𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡))
140	139	sumeq2dv 14841	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴‘𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)Σ𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡))
141		bitsinv1 15570	. . . . . . . . 9 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → Σ𝑛 ∈ (bits‘(𝐴‘𝑡))(2↑𝑛) = (𝐴‘𝑡))
142	141	oveq1d 6937	. . . . . . . 8 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (Σ𝑛 ∈ (bits‘(𝐴‘𝑡))(2↑𝑛) · 𝑡) = ((𝐴‘𝑡) · 𝑡))
143	130, 142	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → (Σ𝑛 ∈ (bits‘(𝐴‘𝑡))(2↑𝑛) · 𝑡) = ((𝐴‘𝑡) · 𝑡))
144	143	sumeq2dv 14841	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴‘𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)((𝐴‘𝑡) · 𝑡))
145		vex 3401	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
146		vex 3401	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
147	145, 146	op2ndd 7456	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑡, 𝑛⟩ → (2nd ‘𝑤) = 𝑛)
148	147	oveq2d 6938	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑛⟩ → (2↑(2nd ‘𝑤)) = (2↑𝑛))
149	145, 146	op1std 7455	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑛⟩ → (1st ‘𝑤) = 𝑡)
150	148, 149	oveq12d 6940	. . . . . . 7 ⊢ (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑(2nd ‘𝑤)) · (1st ‘𝑤)) = ((2↑𝑛) · 𝑡))
151		inss2 4054	. . . . . . . . . 10 ⊢ (𝑇 ∩ 𝑅) ⊆ 𝑅
152	151	sseli 3817	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ 𝑅)
153		cnveq 5541	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐴 → ◡𝑓 = ◡𝐴)
154	153	imaeq1d 5719	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐴 → (◡𝑓 “ ℕ) = (◡𝐴 “ ℕ))
155	154	eleq1d 2844	. . . . . . . . . 10 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐴 “ ℕ) ∈ Fin))
156	155, 9	elab2g 3561	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐴 ∈ 𝑅 ↔ (◡𝐴 “ ℕ) ∈ Fin))
157	152, 156	mpbid 224	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡𝐴 “ ℕ) ∈ Fin)
158		ssfi 8468	. . . . . . . 8 ⊢ (((◡𝐴 “ ℕ) ∈ Fin ∧ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ (◡𝐴 “ ℕ)) → ((◡𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
159	157, 127, 158	sylancl 580	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
160	133	adantrr 707	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℂ)
161	137, 160	mulcld 10397	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ∈ ℂ)
162	150, 159, 132, 161	fsum2d 14907	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)Σ𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = Σ𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))((2↑(2nd ‘𝑤)) · (1st ‘𝑤)))
163	140, 144, 162	3eqtr3d 2822	. . . . 5 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)((𝐴‘𝑡) · 𝑡) = Σ𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))((2↑(2nd ‘𝑤)) · (1st ‘𝑤)))
164		inss1 4053	. . . . . . . . 9 ⊢ (𝑇 ∩ 𝑅) ⊆ 𝑇
165	164	sseli 3817	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ 𝑇)
166	154	sseq1d 3851	. . . . . . . . . 10 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ⊆ 𝐽 ↔ (◡𝐴 “ ℕ) ⊆ 𝐽))
167	166, 10	elrab2 3576	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑇 ↔ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
168	167	simprbi 492	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑇 → (◡𝐴 “ ℕ) ⊆ 𝐽)
169	165, 168	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡𝐴 “ ℕ) ⊆ 𝐽)
170		df-ss 3806	. . . . . . 7 ⊢ ((◡𝐴 “ ℕ) ⊆ 𝐽 ↔ ((◡𝐴 “ ℕ) ∩ 𝐽) = (◡𝐴 “ ℕ))
171	169, 170	sylib 210	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡𝐴 “ ℕ) ∩ 𝐽) = (◡𝐴 “ ℕ))
172	171	sumeq1d 14839	. . . . 5 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)((𝐴‘𝑡) · 𝑡) = Σ𝑡 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑡) · 𝑡))
173	163, 172	eqtr3d 2816	. . . 4 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))((2↑(2nd ‘𝑤)) · (1st ‘𝑤)) = Σ𝑡 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑡) · 𝑡))
174	27, 118, 173	3eqtr2d 2820	. . 3 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)(((𝐺‘𝐴)‘𝑘) · 𝑘) = Σ𝑡 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑡) · 𝑡))
175		fveq2 6446	. . . . 5 ⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡))
176		id 22	. . . . 5 ⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡)
177	175, 176	oveq12d 6940	. . . 4 ⊢ (𝑘 = 𝑡 → ((𝐴‘𝑘) · 𝑘) = ((𝐴‘𝑡) · 𝑡))
178	177	cbvsumv 14834	. . 3 ⊢ Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘) = Σ𝑡 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑡) · 𝑡)
179	174, 178	syl6eqr 2832	. 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)(((𝐺‘𝐴)‘𝑘) · 𝑘) = Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘))
180		0nn0 11659	. . . . . . . 8 ⊢ 0 ∈ ℕ0
181		1nn0 11660	. . . . . . . 8 ⊢ 1 ∈ ℕ0
182		prssi 4583	. . . . . . . 8 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
183	180, 181, 182	mp2an 682	. . . . . . 7 ⊢ {0, 1} ⊆ ℕ0
184		fss 6304	. . . . . . 7 ⊢ (((𝐺‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℕ0) → (𝐺‘𝐴):ℕ⟶ℕ0)
185	183, 184	mpan2 681	. . . . . 6 ⊢ ((𝐺‘𝐴):ℕ⟶{0, 1} → (𝐺‘𝐴):ℕ⟶ℕ0)
186		nn0ex 11649	. . . . . . . 8 ⊢ ℕ0 ∈ V
187		nnex 11381	. . . . . . . 8 ⊢ ℕ ∈ V
188	186, 187	elmap 8169	. . . . . . 7 ⊢ ((𝐺‘𝐴) ∈ (ℕ0 ↑𝑚 ℕ) ↔ (𝐺‘𝐴):ℕ⟶ℕ0)
189	188	biimpri 220	. . . . . 6 ⊢ ((𝐺‘𝐴):ℕ⟶ℕ0 → (𝐺‘𝐴) ∈ (ℕ0 ↑𝑚 ℕ))
190	19, 185, 189	3syl 18	. . . . 5 ⊢ ((𝐺‘𝐴) ∈ ({0, 1} ↑𝑚 ℕ) → (𝐺‘𝐴) ∈ (ℕ0 ↑𝑚 ℕ))
191	190	anim1i 608	. . . 4 ⊢ (((𝐺‘𝐴) ∈ ({0, 1} ↑𝑚 ℕ) ∧ (𝐺‘𝐴) ∈ 𝑅) → ((𝐺‘𝐴) ∈ (ℕ0 ↑𝑚 ℕ) ∧ (𝐺‘𝐴) ∈ 𝑅))
192		elin 4019	. . . 4 ⊢ ((𝐺‘𝐴) ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↔ ((𝐺‘𝐴) ∈ (ℕ0 ↑𝑚 ℕ) ∧ (𝐺‘𝐴) ∈ 𝑅))
193	191, 16, 192	3imtr4i 284	. . 3 ⊢ ((𝐺‘𝐴) ∈ (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) → (𝐺‘𝐴) ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅))
194		eulerpart.s	. . . 4 ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))
195	9, 194	eulerpartlemsv2 31018	. . 3 ⊢ ((𝐺‘𝐴) ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘(𝐺‘𝐴)) = Σ𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)(((𝐺‘𝐴)‘𝑘) · 𝑘))
196	15, 193, 195	3syl 18	. 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝐴)) = Σ𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)(((𝐺‘𝐴)‘𝑘) · 𝑘))
197	120, 152	elind 4021	. . 3 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅))
198	9, 194	eulerpartlemsv2 31018	. . 3 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘))
199	197, 198	syl 17	. 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘))
200	179, 196, 199	3eqtr4d 2824	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝐴)) = (𝑆‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {cab 2763  ∀wral 3090  ∃wrex 3091  {crab 3094  Vcvv 3398   ∖ cdif 3789   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  ifcif 4307  𝒫 cpw 4379  {csn 4398  {cpr 4400  ⟨cop 4404  ∪ ciun 4753   class class class wbr 4886  {copab 4948   ↦ cmpt 4965   × cxp 5353  ^◡ccnv 5354  dom cdm 5355   ↾ cres 5357   “ cima 5358   ∘ ccom 5359   Fn wfn 6130  ⟶wf 6131  –1-1-onto→wf1o 6134  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  1^st c1st 7443  2^nd c2nd 7444   supp csupp 7576   ↑_𝑚 cmap 8140   ≈ cen 8238  Fincfn 8241  ℂcc 10270  0cc0 10272  1c1 10273   · cmul 10277   ≤ cle 10412  ℕcn 11374  2c2 11430  ℕ₀cn0 11642  ↑cexp 13178  Σcsu 14824   ∥ cdvds 15387  bitscbits 15547  𝟭cind 30670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-ac2 9620  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-disj 4855  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-acn 9101  df-ac 9272  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-rp 12138  df-fz 12644  df-fzo 12785  df-fl 12912  df-mod 12988  df-seq 13120  df-exp 13179  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-sum 14825  df-dvds 15388  df-bits 15550  df-ind 30671

This theorem is referenced by:  eulerpartlemn  31041