Theorem eulerpartlemgs2 32343

Description: Lemma for eulerpart 32345: 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 5988	. . . . . . . 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 32335	. . . . . . . . . . . . 13 ⊢ 𝐺:(𝑇 ∩ 𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅)
13		f1of 6714	. . . . . . . . . . . . 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 6957	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅))
16		elin 3908	. . . . . . . . . . 11 ⊢ ((𝐺‘𝐴) ∈ (({0, 1} ↑m ℕ) ∩ 𝑅) ↔ ((𝐺‘𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺‘𝐴) ∈ 𝑅))
17	15, 16	sylib 217	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((𝐺‘𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺‘𝐴) ∈ 𝑅))
18	17	simpld 495	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) ∈ ({0, 1} ↑m ℕ))
19		elmapi 8620	. . . . . . . . 9 ⊢ ((𝐺‘𝐴) ∈ ({0, 1} ↑m ℕ) → (𝐺‘𝐴):ℕ⟶{0, 1})
20		fdm 6607	. . . . . . . . 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	sseqtrid 3978	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ ℕ) ⊆ ℕ)
23	22	sselda 3926	. . . . . 6 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)) → 𝑘 ∈ ℕ)
24	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	eulerpartlemgvv 32339	. . . . . . 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 7286	. . . . . 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 591	. . . . 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 15413	. . . 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 2752	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑘))
29	28	2rexbidv 3231	. . . . . . . . . . . 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 3626	. . . . . . . . . . 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 497	. . . . . . . . . 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 4473	. . . . . . . . 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 7286	. . . . . . . 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 3620	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℕ)
35	34	nncnd 11989	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ ℂ)
36	35	mulid2d 10994	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → (1 · 𝑘) = 𝑘)
37	33, 36	eqtrd 2780	. . . . . . 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 15409	. . . . . 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 32342	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ ℕ) ∈ Fin)
41	34	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℕ)
42	41, 24	syldan 591	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 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 4473	. . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → ((𝐺‘𝐴)‘𝑘) = 1)
46		1nn 11984	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
47	45, 46	eqeltrdi 2849	. . . . . . . . . . . 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 6598	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝐴):ℕ⟶{0, 1} → (𝐺‘𝐴) Fn ℕ)
50		elpreima 6932	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 710	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ))
54	53	ex 413	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} → 𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)))
55	54	ssrdv 3932	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ (◡(𝐺‘𝐴) “ ℕ))
56		ssfi 8938	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ∈ Fin)
58		cnvexg 7765	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ◡𝐴 ∈ V)
59		imaexg 7756	. . . . . . . . . . 11 ⊢ (◡𝐴 ∈ V → (◡𝐴 “ ℕ) ∈ V)
60		inex1g 5247	. . . . . . . . . . 11 ⊢ ((◡𝐴 “ ℕ) ∈ V → ((◡𝐴 “ ℕ) ∩ 𝐽) ∈ V)
61	58, 59, 60	3syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡𝐴 “ ℕ) ∩ 𝐽) ∈ V)
62		snex 5358	. . . . . . . . . . . 12 ⊢ {𝑡} ∈ V
63		fvex 6784	. . . . . . . . . . . 12 ⊢ (bits‘(𝐴‘𝑡)) ∈ V
64	62, 63	xpex 7597	. . . . . . . . . . 11 ⊢ ({𝑡} × (bits‘(𝐴‘𝑡))) ∈ V
65	64	rgenw 3078	. . . . . . . . . 10 ⊢ ∀𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ∈ V
66		iunexg 7799	. . . . . . . . . 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 586	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ∈ V)
68		eqid 2740	. . . . . . . . . 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 32341	. . . . . . . . 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 8742	. . . . . . . . 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 584	. . . . . . . 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 8954	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ∈ Fin)
74		fvres 6790	. . . . . . . . 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 482	. . . . . . . 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 4169	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
77		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽))
78	76, 77	sselid 3924	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ 𝐽)
79	78	snssd 4748	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → {𝑡} ⊆ 𝐽)
80		bitsss 16131	. . . . . . . . . . . . 13 ⊢ (bits‘(𝐴‘𝑡)) ⊆ ℕ0
81		xpss12 5605	. . . . . . . . . . . . 13 ⊢ (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴‘𝑡)) ⊆ ℕ0) → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
82	79, 80, 81	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → ({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
83	82	ralrimiva 3110	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∀𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
84		iunss 4980	. . . . . . . . . . 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 233	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⊆ (𝐽 × ℕ0))
86	85	sselda 3926	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑤 ∈ ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))) → 𝑤 ∈ (𝐽 × ℕ0))
87	5, 6	oddpwdcv 32318	. . . . . . . . 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 2780	. . . . . . 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 11989	. . . . . . 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 15433	. . . . . 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	eqtrid 2792	. . . . 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 10930	. . . . . . . . 9 ⊢ 1 ∈ ℂ
94		0cn 10968	. . . . . . . . 9 ⊢ 0 ∈ ℂ
95	93, 94	ifcli 4512	. . . . . . . 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 4018	. . . . . . . . 9 ⊢ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚} ⊆ ℕ
98		simpr 485	. . . . . . . . 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	sselid 3924	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℕ)
100	99	nncnd 11989	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑘 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) → 𝑘 ∈ ℂ)
101	96, 100	mulcld 10996	. . . . . 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 485	. . . . . . . . . . 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 3905	. . . . . . . . . 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 4082	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℕ)
105	104	sselda 3926	. . . . . . . . . 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 320	. . . . . . . . . . 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 400	. . . . . . . . . . 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 237	. . . . . . . . . 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 4476	. . . . . . . 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 7286	. . . . . . 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 11978	. . . . . . . . . 10 ⊢ ℕ ⊆ ℂ
113	104, 112	sstrdi 3938	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡(𝐺‘𝐴) “ ℕ) ∖ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) ⊆ ℂ)
114	113	sselda 3926	. . . . . . . 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 11173	. . . . . . 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 2780	. . . . . 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 15435	. . . . 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 2782	. . . 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 32332	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
120	119	simp1bi 1144	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ (ℕ0 ↑m ℕ))
121		elmapi 8620	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴:ℕ⟶ℕ0)
122	120, 121	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
123	122	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝐴:ℕ⟶ℕ0)
124		cnvimass 5988	. . . . . . . . . . . . 13 ⊢ (◡𝐴 “ ℕ) ⊆ dom 𝐴
125	124, 122	fssdm 6618	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡𝐴 “ ℕ) ⊆ ℕ)
126	125	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → (◡𝐴 “ ℕ) ⊆ ℕ)
127		inss1 4168	. . . . . . . . . . . 12 ⊢ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ (◡𝐴 “ ℕ)
128	127, 77	sselid 3924	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ (◡𝐴 “ ℕ))
129	126, 128	sseldd 3927	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℕ)
130	123, 129	ffvelrnd 6959	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → (𝐴‘𝑡) ∈ ℕ0)
131		bitsfi 16142	. . . . . . . . 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 11989	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) → 𝑡 ∈ ℂ)
134		2cnd 12051	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 2 ∈ ℂ)
135		simprr 770	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ (bits‘(𝐴‘𝑡)))
136	80, 135	sselid 3924	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ ℕ0)
137	134, 136	expcld 13862	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℂ)
138	137	anassrs 468	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑛) ∈ ℂ)
139	132, 133, 138	fsummulc1 15495	. . . . . . 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 15413	. . . . . 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 16147	. . . . . . . . 9 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → Σ𝑛 ∈ (bits‘(𝐴‘𝑡))(2↑𝑛) = (𝐴‘𝑡))
142	141	oveq1d 7286	. . . . . . . 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 15413	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)(Σ𝑛 ∈ (bits‘(𝐴‘𝑡))(2↑𝑛) · 𝑡) = Σ𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)((𝐴‘𝑡) · 𝑡))
145		vex 3435	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
146		vex 3435	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
147	145, 146	op2ndd 7835	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑡, 𝑛⟩ → (2nd ‘𝑤) = 𝑛)
148	147	oveq2d 7287	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑛⟩ → (2↑(2nd ‘𝑤)) = (2↑𝑛))
149	145, 146	op1std 7834	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑡, 𝑛⟩ → (1st ‘𝑤) = 𝑡)
150	148, 149	oveq12d 7289	. . . . . . 7 ⊢ (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑(2nd ‘𝑤)) · (1st ‘𝑤)) = ((2↑𝑛) · 𝑡))
151		inss2 4169	. . . . . . . . . 10 ⊢ (𝑇 ∩ 𝑅) ⊆ 𝑅
152	151	sseli 3922	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ 𝑅)
153		cnveq 5781	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐴 → ◡𝑓 = ◡𝐴)
154	153	imaeq1d 5967	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐴 → (◡𝑓 “ ℕ) = (◡𝐴 “ ℕ))
155	154	eleq1d 2825	. . . . . . . . . 10 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐴 “ ℕ) ∈ Fin))
156	155, 9	elab2g 3613	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐴 ∈ 𝑅 ↔ (◡𝐴 “ ℕ) ∈ Fin))
157	152, 156	mpbid 231	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡𝐴 “ ℕ) ∈ Fin)
158		ssfi 8938	. . . . . . . 8 ⊢ (((◡𝐴 “ ℕ) ∈ Fin ∧ ((◡𝐴 “ ℕ) ∩ 𝐽) ⊆ (◡𝐴 “ ℕ)) → ((◡𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
159	157, 127, 158	sylancl 586	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡𝐴 “ ℕ) ∩ 𝐽) ∈ Fin)
160	133	adantrr 714	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℂ)
161	137, 160	mulcld 10996	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ∈ ℂ)
162	150, 159, 132, 161	fsum2d 15481	. . . . . 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 2788	. . . . 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 4168	. . . . . . . . 9 ⊢ (𝑇 ∩ 𝑅) ⊆ 𝑇
165	164	sseli 3922	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ 𝑇)
166	154	sseq1d 3957	. . . . . . . . . 10 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ⊆ 𝐽 ↔ (◡𝐴 “ ℕ) ⊆ 𝐽))
167	166, 10	elrab2 3629	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑇 ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
168	167	simprbi 497	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑇 → (◡𝐴 “ ℕ) ⊆ 𝐽)
169	165, 168	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡𝐴 “ ℕ) ⊆ 𝐽)
170		df-ss 3909	. . . . . . 7 ⊢ ((◡𝐴 “ ℕ) ⊆ 𝐽 ↔ ((◡𝐴 “ ℕ) ∩ 𝐽) = (◡𝐴 “ ℕ))
171	169, 170	sylib 217	. . . . . 6 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → ((◡𝐴 “ ℕ) ∩ 𝐽) = (◡𝐴 “ ℕ))
172	171	sumeq1d 15411	. . . . 5 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)((𝐴‘𝑡) · 𝑡) = Σ𝑡 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑡) · 𝑡))
173	163, 172	eqtr3d 2782	. . . 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 2786	. . 3 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)(((𝐺‘𝐴)‘𝑘) · 𝑘) = Σ𝑡 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑡) · 𝑡))
175		fveq2 6771	. . . . 5 ⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡))
176		id 22	. . . . 5 ⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡)
177	175, 176	oveq12d 7289	. . . 4 ⊢ (𝑘 = 𝑡 → ((𝐴‘𝑘) · 𝑘) = ((𝐴‘𝑡) · 𝑡))
178	177	cbvsumv 15406	. . 3 ⊢ Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘) = Σ𝑡 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑡) · 𝑡)
179	174, 178	eqtr4di 2798	. 2 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → Σ𝑘 ∈ (◡(𝐺‘𝐴) “ ℕ)(((𝐺‘𝐴)‘𝑘) · 𝑘) = Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘))
180		0nn0 12248	. . . . . . . 8 ⊢ 0 ∈ ℕ0
181		1nn0 12249	. . . . . . . 8 ⊢ 1 ∈ ℕ0
182		prssi 4760	. . . . . . . 8 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
183	180, 181, 182	mp2an 689	. . . . . . 7 ⊢ {0, 1} ⊆ ℕ0
184		fss 6615	. . . . . . 7 ⊢ (((𝐺‘𝐴):ℕ⟶{0, 1} ∧ {0, 1} ⊆ ℕ0) → (𝐺‘𝐴):ℕ⟶ℕ0)
185	183, 184	mpan2 688	. . . . . 6 ⊢ ((𝐺‘𝐴):ℕ⟶{0, 1} → (𝐺‘𝐴):ℕ⟶ℕ0)
186		nn0ex 12239	. . . . . . . 8 ⊢ ℕ0 ∈ V
187		nnex 11979	. . . . . . . 8 ⊢ ℕ ∈ V
188	186, 187	elmap 8642	. . . . . . 7 ⊢ ((𝐺‘𝐴) ∈ (ℕ0 ↑m ℕ) ↔ (𝐺‘𝐴):ℕ⟶ℕ0)
189	188	biimpri 227	. . . . . 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 615	. . . 4 ⊢ (((𝐺‘𝐴) ∈ ({0, 1} ↑m ℕ) ∧ (𝐺‘𝐴) ∈ 𝑅) → ((𝐺‘𝐴) ∈ (ℕ0 ↑m ℕ) ∧ (𝐺‘𝐴) ∈ 𝑅))
192		elin 3908	. . . 4 ⊢ ((𝐺‘𝐴) ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↔ ((𝐺‘𝐴) ∈ (ℕ0 ↑m ℕ) ∧ (𝐺‘𝐴) ∈ 𝑅))
193	191, 16, 192	3imtr4i 292	. . 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 32321	. . 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 4133	. . 3 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅))
198	9, 194	eulerpartlemsv2 32321	. . 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 2790	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝐴)) = (𝑆‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  {cab 2717  ∀wral 3066  ∃wrex 3067  {crab 3070  Vcvv 3431   ∖ cdif 3889   ∩ cin 3891   ⊆ wss 3892  ∅c0 4262  ifcif 4465  𝒫 cpw 4539  {csn 4567  {cpr 4569  ⟨cop 4573  ∪ ciun 4930   class class class wbr 5079  {copab 5141   ↦ cmpt 5162   × cxp 5588  ^◡ccnv 5589  dom cdm 5590   ↾ cres 5592   “ cima 5593   ∘ ccom 5594   Fn wfn 6427  ⟶wf 6428  –1-1-onto→wf1o 6431  ‘cfv 6432  (class class class)co 7271   ∈ cmpo 7273  1^st c1st 7822  2^nd c2nd 7823   supp csupp 7968   ↑_m cmap 8598   ≈ cen 8713  Fincfn 8716  ℂcc 10870  0cc0 10872  1c1 10873   · cmul 10877   ≤ cle 11011  ℕcn 11973  2c2 12028  ℕ₀cn0 12233  ↑cexp 13780  Σcsu 15395   ∥ cdvds 15961  bitscbits 16124  𝟭cind 31974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-inf2 9377  ax-ac2 10220  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949  ax-pre-sup 10950

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-disj 5045  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-supp 7969  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-2o 8289  df-oadd 8292  df-er 8481  df-map 8600  df-pm 8601  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-fsupp 9107  df-sup 9179  df-inf 9180  df-oi 9247  df-dju 9660  df-card 9698  df-acn 9701  df-ac 9873  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12582  df-rp 12730  df-fz 13239  df-fzo 13382  df-fl 13510  df-mod 13588  df-seq 13720  df-exp 13781  df-hash 14043  df-cj 14808  df-re 14809  df-im 14810  df-sqrt 14944  df-abs 14945  df-clim 15195  df-sum 15396  df-dvds 15962  df-bits 16127  df-ind 31975

This theorem is referenced by:  eulerpartlemn  32344