Theorem heiborlem6 35127

Description: Lemma for heibor 35132. Since the sequence of balls connected by the function 𝑇 ensures that each ball nontrivially intersects with the next (since the empty set has a finite subcover, the intersection of any two successive balls in the sequence is nonempty), and each ball is half the size of the previous one, the distance between the centers is at most 3 / 2 times the size of the larger, and so if we expand each ball by a factor of 3 we get a nested sequence of balls. (Contributed by Jeff Madsen, 23-Jan-2014.)

Hypotheses

Ref	Expression
heibor.1	⊢ 𝐽 = (MetOpen‘𝐷)
heibor.3	⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
heibor.4	⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heibor.5	⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
heibor.6	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
heibor.7	⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
heibor.8	⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
heibor.9	⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
heibor.10	⊢ (𝜑 → 𝐶𝐺0)
heibor.11	⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
heibor.12	⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)

Assertion

Ref	Expression
heiborlem6	⊢ (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))

Distinct variable groups:   𝑥,𝑛,𝑦,𝑘,𝑢,𝐹   𝑘,𝐺,𝑥   𝜑,𝑘,𝑥   𝑘,𝑚,𝑣,𝑧,𝐷,𝑛,𝑢,𝑥,𝑦   𝑘,𝑀,𝑚,𝑢,𝑥,𝑦,𝑧   𝑇,𝑚,𝑛,𝑥,𝑦,𝑧   𝐵,𝑛,𝑢,𝑣,𝑦   𝑘,𝐽,𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑈,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑆,𝑘,𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑘,𝑋,𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝐶,𝑚,𝑛,𝑢,𝑣,𝑦   𝑛,𝐾,𝑥,𝑦,𝑧   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐵(𝑧,𝑘,𝑚)   𝐶(𝑥,𝑧,𝑘)   𝑇(𝑣,𝑢,𝑘)   𝑈(𝑘,𝑚)   𝐹(𝑧,𝑣,𝑚)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐾(𝑣,𝑢,𝑘,𝑚)   𝑀(𝑣,𝑛)

Proof of Theorem heiborlem6

Step	Hyp	Ref	Expression
1		nnnn0 11898	. . . 4 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
2		heibor.6	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
3		cmetmet 23884	. . . . . . . 8 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
5		metxmet 22939	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
7	6	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐷 ∈ (∞Met‘𝑋))
8		heibor.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
9		inss1 4198	. . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
10		fss 6520	. . . . . . . . 9 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋) → 𝐹:ℕ0⟶𝒫 𝑋)
11	8, 9, 10	sylancl 588	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ0⟶𝒫 𝑋)
12		peano2nn0 11931	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
13		ffvelrn 6842	. . . . . . . 8 ⊢ ((𝐹:ℕ0⟶𝒫 𝑋 ∧ (𝑘 + 1) ∈ ℕ0) → (𝐹‘(𝑘 + 1)) ∈ 𝒫 𝑋)
14	11, 12, 13	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) ∈ 𝒫 𝑋)
15	14	elpwid 4543	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) ⊆ 𝑋)
16		heibor.1	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷)
17		heibor.3	. . . . . . . . 9 ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
18		heibor.4	. . . . . . . . 9 ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
19		heibor.5	. . . . . . . . 9 ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
20		heibor.8	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
21		heibor.9	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
22		heibor.10	. . . . . . . . 9 ⊢ (𝜑 → 𝐶𝐺0)
23		heibor.11	. . . . . . . . 9 ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
24	16, 17, 18, 19, 2, 8, 20, 21, 22, 23	heiborlem4 35125	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ ℕ0) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))
25	12, 24	sylan2 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))
26		fvex 6676	. . . . . . . . 9 ⊢ (𝑆‘(𝑘 + 1)) ∈ V
27		ovex 7182	. . . . . . . . 9 ⊢ (𝑘 + 1) ∈ V
28	16, 17, 18, 26, 27	heiborlem2 35123	. . . . . . . 8 ⊢ ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) ↔ ((𝑘 + 1) ∈ ℕ0 ∧ (𝑆‘(𝑘 + 1)) ∈ (𝐹‘(𝑘 + 1)) ∧ ((𝑆‘(𝑘 + 1))𝐵(𝑘 + 1)) ∈ 𝐾))
29	28	simp2bi 1141	. . . . . . 7 ⊢ ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) → (𝑆‘(𝑘 + 1)) ∈ (𝐹‘(𝑘 + 1)))
30	25, 29	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) ∈ (𝐹‘(𝑘 + 1)))
31	15, 30	sseldd 3961	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) ∈ 𝑋)
32	11	ffvelrnda 6844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ 𝒫 𝑋)
33	32	elpwid 4543	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ⊆ 𝑋)
34	16, 17, 18, 19, 2, 8, 20, 21, 22, 23	heiborlem4 35125	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘)𝐺𝑘)
35		fvex 6676	. . . . . . . . 9 ⊢ (𝑆‘𝑘) ∈ V
36		vex 3494	. . . . . . . . 9 ⊢ 𝑘 ∈ V
37	16, 17, 18, 35, 36	heiborlem2 35123	. . . . . . . 8 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆‘𝑘) ∈ (𝐹‘𝑘) ∧ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾))
38	37	simp2bi 1141	. . . . . . 7 ⊢ ((𝑆‘𝑘)𝐺𝑘 → (𝑆‘𝑘) ∈ (𝐹‘𝑘))
39	34, 38	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ (𝐹‘𝑘))
40	33, 39	sseldd 3961	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ 𝑋)
41		3re 11711	. . . . . 6 ⊢ 3 ∈ ℝ
42		2nn 11704	. . . . . . . . 9 ⊢ 2 ∈ ℕ
43		nnexpcl 13439	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ0) → (2↑(𝑘 + 1)) ∈ ℕ)
44	42, 12, 43	sylancr 589	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) ∈ ℕ)
45	44	nnrpd 12423	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) ∈ ℝ+)
46	45	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑(𝑘 + 1)) ∈ ℝ+)
47		rerpdivcl 12413	. . . . . 6 ⊢ ((3 ∈ ℝ ∧ (2↑(𝑘 + 1)) ∈ ℝ+) → (3 / (2↑(𝑘 + 1))) ∈ ℝ)
48	41, 46, 47	sylancr 589	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (3 / (2↑(𝑘 + 1))) ∈ ℝ)
49		nnexpcl 13439	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
50	42, 49	mpan 688	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℕ)
51	50	nnrpd 12423	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℝ+)
52	51	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℝ+)
53		rerpdivcl 12413	. . . . . 6 ⊢ ((3 ∈ ℝ ∧ (2↑𝑘) ∈ ℝ+) → (3 / (2↑𝑘)) ∈ ℝ)
54	41, 52, 53	sylancr 589	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (3 / (2↑𝑘)) ∈ ℝ)
55		oveq1 7156	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑆‘𝑘) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑚))))
56		oveq2 7157	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
57	56	oveq2d 7165	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (1 / (2↑𝑚)) = (1 / (2↑𝑘)))
58	57	oveq2d 7165	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑚))) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
59		ovex 7182	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∈ V
60	55, 58, 19, 59	ovmpo 7303	. . . . . . . . . . 11 ⊢ (((𝑆‘𝑘) ∈ 𝑋 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐵𝑘) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
61	40, 60	sylancom 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐵𝑘) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
62		df-br 5060	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ ⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺)
63		fveq2 6663	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩))
64		df-ov 7152	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆‘𝑘)𝑇𝑘) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩)
65	63, 64	syl6eqr 2873	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = ((𝑆‘𝑘)𝑇𝑘))
66	35, 36	op2ndd 7693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (2nd ‘𝑥) = 𝑘)
67	66	oveq1d 7164	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((2nd ‘𝑥) + 1) = (𝑘 + 1))
68	65, 67	breq12d 5072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
69		fveq2 6663	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩))
70		df-ov 7152	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆‘𝑘)𝐵𝑘) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩)
71	69, 70	syl6eqr 2873	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = ((𝑆‘𝑘)𝐵𝑘))
72	65, 67	oveq12d 7167	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1)) = (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)))
73	71, 72	ineq12d 4183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) = (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))))
74	73	eleq1d 2896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾 ↔ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))
75	68, 74	anbi12d 632	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) ↔ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
76	75	rspccv 3617	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) → (⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
77	21, 76	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
78	62, 77	syl5bi 244	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
79	78	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐺𝑘 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
80	34, 79	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))
81	80	simpld 497	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1))
82		ovex 7182	. . . . . . . . . . . . . . 15 ⊢ ((𝑆‘𝑘)𝑇𝑘) ∈ V
83	16, 17, 18, 82, 27	heiborlem2 35123	. . . . . . . . . . . . . 14 ⊢ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ↔ ((𝑘 + 1) ∈ ℕ0 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ (𝐹‘(𝑘 + 1)) ∧ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)) ∈ 𝐾))
84	83	simp2bi 1141	. . . . . . . . . . . . 13 ⊢ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) → ((𝑆‘𝑘)𝑇𝑘) ∈ (𝐹‘(𝑘 + 1)))
85	81, 84	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘) ∈ (𝐹‘(𝑘 + 1)))
86	15, 85	sseldd 3961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋)
87	12	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
88		oveq1 7156	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑆‘𝑘)𝑇𝑘) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑𝑚))))
89		oveq2 7157	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (2↑𝑚) = (2↑(𝑘 + 1)))
90	89	oveq2d 7165	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (1 / (2↑𝑚)) = (1 / (2↑(𝑘 + 1))))
91	90	oveq2d 7165	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 + 1) → (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑𝑚))) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
92		ovex 7182	. . . . . . . . . . . 12 ⊢ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))) ∈ V
93	88, 91, 19, 92	ovmpo 7303	. . . . . . . . . . 11 ⊢ ((((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋 ∧ (𝑘 + 1) ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
94	86, 87, 93	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
95	61, 94	ineq12d 4183	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) = (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))))
96	80	simprd 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)
97		0elpw 5249	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 𝒫 𝑈
98		0fin 8739	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
99		elin 4162	. . . . . . . . . . . . 13 ⊢ (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ∈ 𝒫 𝑈 ∧ ∅ ∈ Fin))
100	97, 98, 99	mpbir2an 709	. . . . . . . . . . . 12 ⊢ ∅ ∈ (𝒫 𝑈 ∩ Fin)
101		0ss 4343	. . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ ∅
102		unieq 4842	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ∅ → ∪ 𝑣 = ∪ ∅)
103	102	sseq2d 3992	. . . . . . . . . . . . 13 ⊢ (𝑣 = ∅ → (∅ ⊆ ∪ 𝑣 ↔ ∅ ⊆ ∪ ∅))
104	103	rspcev 3620	. . . . . . . . . . . 12 ⊢ ((∅ ∈ (𝒫 𝑈 ∩ Fin) ∧ ∅ ⊆ ∪ ∅) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣)
105	100, 101, 104	mp2an 690	. . . . . . . . . . 11 ⊢ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣
106		0ex 5204	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
107		sseq1 3985	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ∅ → (𝑢 ⊆ ∪ 𝑣 ↔ ∅ ⊆ ∪ 𝑣))
108	107	rexbidv 3296	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ∅ → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣))
109	108	notbid 320	. . . . . . . . . . . . 13 ⊢ (𝑢 = ∅ → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣))
110	106, 109, 17	elab2 3666	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣)
111	110	con2bii 360	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣 ↔ ¬ ∅ ∈ 𝐾)
112	105, 111	mpbi 232	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ 𝐾
113		nelne2 3114	. . . . . . . . . 10 ⊢ (((((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾 ∧ ¬ ∅ ∈ 𝐾) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ≠ ∅)
114	96, 112, 113	sylancl 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ≠ ∅)
115	95, 114	eqnetrrd 3083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) ≠ ∅)
116	51	rpreccld 12435	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑𝑘)) ∈ ℝ+)
117	116	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ+)
118	117	rpred 12425	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ)
119	45	rpreccld 12435	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑(𝑘 + 1))) ∈ ℝ+)
120	119	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ+)
121	120	rpred 12425	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ)
122		rexadd 12619	. . . . . . . . . . . 12 ⊢ (((1 / (2↑𝑘)) ∈ ℝ ∧ (1 / (2↑(𝑘 + 1))) ∈ ℝ) → ((1 / (2↑𝑘)) +𝑒 (1 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
123	118, 121, 122	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) +𝑒 (1 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
124	123	breq1d 5069	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((1 / (2↑𝑘)) +𝑒 (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ↔ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘))))
125	117	rpxrd 12426	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ*)
126	120	rpxrd 12426	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ*)
127		bldisj 23003	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑆‘𝑘) ∈ 𝑋 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋) ∧ ((1 / (2↑𝑘)) ∈ ℝ* ∧ (1 / (2↑(𝑘 + 1))) ∈ ℝ* ∧ ((1 / (2↑𝑘)) +𝑒 (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) = ∅)
128	127	3exp2 1349	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑆‘𝑘) ∈ 𝑋 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋) → ((1 / (2↑𝑘)) ∈ ℝ* → ((1 / (2↑(𝑘 + 1))) ∈ ℝ* → (((1 / (2↑𝑘)) +𝑒 (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) = ∅))))
129	128	imp32 421	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑆‘𝑘) ∈ 𝑋 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋) ∧ ((1 / (2↑𝑘)) ∈ ℝ* ∧ (1 / (2↑(𝑘 + 1))) ∈ ℝ*)) → (((1 / (2↑𝑘)) +𝑒 (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) = ∅))
130	7, 40, 86, 125, 126, 129	syl32anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((1 / (2↑𝑘)) +𝑒 (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) = ∅))
131	124, 130	sylbird 262	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) = ∅))
132	131	necon3ad 3028	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) ≠ ∅ → ¬ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘))))
133	115, 132	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ¬ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
134	117, 120	rpaddcld 12440	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ∈ ℝ+)
135	134	rpred 12425	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ∈ ℝ)
136	4	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐷 ∈ (Met‘𝑋))
137		metcl 22937	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝑆‘𝑘) ∈ 𝑋 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ∈ ℝ)
138	136, 40, 86, 137	syl3anc 1366	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ∈ ℝ)
139	135, 138	letrid 10785	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ∨ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ≤ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1))))))
140	139	ord 860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ≤ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1))))))
141	133, 140	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ≤ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
142		seqp1 13381	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘0) → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
143		nn0uz 12274	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
144	142, 143	eleq2s 2930	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
145	23	fveq1i 6664	. . . . . . . . . . 11 ⊢ (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1))
146	23	fveq1i 6664	. . . . . . . . . . . 12 ⊢ (𝑆‘𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)
147	146	oveq1i 7159	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)))
148	144, 145, 147	3eqtr4g 2880	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
149		eqid 2820	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))
150		eqeq1 2824	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0))
151		oveq1 7156	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 − 1) = ((𝑘 + 1) − 1))
152	150, 151	ifbieq2d 4485	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
153		nn0p1nn 11930	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
154		nnne0 11665	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑘 + 1) ≠ 0)
155	154	neneqd 3020	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 1) ∈ ℕ → ¬ (𝑘 + 1) = 0)
156	153, 155	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → ¬ (𝑘 + 1) = 0)
157	156	iffalsed 4471	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
158		ovex 7182	. . . . . . . . . . . . . 14 ⊢ ((𝑘 + 1) − 1) ∈ V
159	157, 158	eqeltrdi 2920	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) ∈ V)
160	149, 152, 12, 159	fvmptd3 6784	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
161		nn0cn 11901	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
162		ax-1cn 10588	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
163		pncan 10885	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
164	161, 162, 163	sylancl 588	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 + 1) − 1) = 𝑘)
165	160, 157, 164	3eqtrd 2859	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘)
166	165	oveq2d 7165	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆‘𝑘)𝑇𝑘))
167	148, 166	eqtrd 2855	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
168	167	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
169	168	oveq1d 7164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) = (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)))
170		metsym 22955	. . . . . . . 8 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋 ∧ (𝑆‘𝑘) ∈ 𝑋) → (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
171	136, 86, 40, 170	syl3anc 1366	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
172	169, 171	eqtrd 2855	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
173		3cn 11712	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℂ
174	173	2timesi 11769	. . . . . . . . . . . 12 ⊢ (2 · 3) = (3 + 3)
175	174	oveq1i 7159	. . . . . . . . . . 11 ⊢ ((2 · 3) − 3) = ((3 + 3) − 3)
176	173, 173	pncan3oi 10895	. . . . . . . . . . 11 ⊢ ((3 + 3) − 3) = 3
177		df-3 11695	. . . . . . . . . . 11 ⊢ 3 = (2 + 1)
178	175, 176, 177	3eqtri 2847	. . . . . . . . . 10 ⊢ ((2 · 3) − 3) = (2 + 1)
179	178	oveq1i 7159	. . . . . . . . 9 ⊢ (((2 · 3) − 3) / (2↑(𝑘 + 1))) = ((2 + 1) / (2↑(𝑘 + 1)))
180		rpcn 12393	. . . . . . . . . . 11 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (2↑(𝑘 + 1)) ∈ ℂ)
181		rpne0 12399	. . . . . . . . . . 11 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (2↑(𝑘 + 1)) ≠ 0)
182		2cn 11706	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
183	182, 173	mulcli 10641	. . . . . . . . . . . 12 ⊢ (2 · 3) ∈ ℂ
184		divsubdir 11327	. . . . . . . . . . . 12 ⊢ (((2 · 3) ∈ ℂ ∧ 3 ∈ ℂ ∧ ((2↑(𝑘 + 1)) ∈ ℂ ∧ (2↑(𝑘 + 1)) ≠ 0)) → (((2 · 3) − 3) / (2↑(𝑘 + 1))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
185	183, 173, 184	mp3an12 1446	. . . . . . . . . . 11 ⊢ (((2↑(𝑘 + 1)) ∈ ℂ ∧ (2↑(𝑘 + 1)) ≠ 0) → (((2 · 3) − 3) / (2↑(𝑘 + 1))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
186	180, 181, 185	syl2anc 586	. . . . . . . . . 10 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (((2 · 3) − 3) / (2↑(𝑘 + 1))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
187	45, 186	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (((2 · 3) − 3) / (2↑(𝑘 + 1))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
188		divdir 11316	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ ((2↑(𝑘 + 1)) ∈ ℂ ∧ (2↑(𝑘 + 1)) ≠ 0)) → ((2 + 1) / (2↑(𝑘 + 1))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
189	182, 162, 188	mp3an12 1446	. . . . . . . . . . 11 ⊢ (((2↑(𝑘 + 1)) ∈ ℂ ∧ (2↑(𝑘 + 1)) ≠ 0) → ((2 + 1) / (2↑(𝑘 + 1))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
190	180, 181, 189	syl2anc 586	. . . . . . . . . 10 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → ((2 + 1) / (2↑(𝑘 + 1))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
191	45, 190	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → ((2 + 1) / (2↑(𝑘 + 1))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
192	179, 187, 191	3eqtr3a 2879	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
193		rpcn 12393	. . . . . . . . . . . 12 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ∈ ℂ)
194		rpne0 12399	. . . . . . . . . . . 12 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ≠ 0)
195		2cnne0 11841	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
196		divcan5 11335	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℂ ∧ ((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
197	173, 195, 196	mp3an13 1447	. . . . . . . . . . . 12 ⊢ (((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
198	193, 194, 197	syl2anc 586	. . . . . . . . . . 11 ⊢ ((2↑𝑘) ∈ ℝ+ → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
199	51, 198	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
200	51, 193	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℂ)
201		mulcom 10616	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ (2↑𝑘) ∈ ℂ) → (2 · (2↑𝑘)) = ((2↑𝑘) · 2))
202	182, 200, 201	sylancr 589	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (2 · (2↑𝑘)) = ((2↑𝑘) · 2))
203		expp1 13433	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (2↑(𝑘 + 1)) = ((2↑𝑘) · 2))
204	182, 203	mpan 688	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) = ((2↑𝑘) · 2))
205	202, 204	eqtr4d 2858	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (2 · (2↑𝑘)) = (2↑(𝑘 + 1)))
206	205	oveq2d 7165	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 3) / (2 · (2↑𝑘))) = ((2 · 3) / (2↑(𝑘 + 1))))
207	199, 206	eqtr3d 2857	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (3 / (2↑𝑘)) = ((2 · 3) / (2↑(𝑘 + 1))))
208	207	oveq1d 7164	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
209		divcan5 11335	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ ((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
210	162, 195, 209	mp3an13 1447	. . . . . . . . . . . 12 ⊢ (((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
211	193, 194, 210	syl2anc 586	. . . . . . . . . . 11 ⊢ ((2↑𝑘) ∈ ℝ+ → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
212	51, 211	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
213		2t1e2 11794	. . . . . . . . . . . 12 ⊢ (2 · 1) = 2
214	213	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (2 · 1) = 2)
215	214, 205	oveq12d 7167	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 1) / (2 · (2↑𝑘))) = (2 / (2↑(𝑘 + 1))))
216	212, 215	eqtr3d 2857	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑𝑘)) = (2 / (2↑(𝑘 + 1))))
217	216	oveq1d 7164	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
218	192, 208, 217	3eqtr4d 2865	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
219	218	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
220	141, 172, 219	3brtr4d 5091	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) ≤ ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))))
221		blss2 23009	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑆‘(𝑘 + 1)) ∈ 𝑋 ∧ (𝑆‘𝑘) ∈ 𝑋) ∧ ((3 / (2↑(𝑘 + 1))) ∈ ℝ ∧ (3 / (2↑𝑘)) ∈ ℝ ∧ ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) ≤ ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))))) → ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) ⊆ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
222	7, 31, 40, 48, 54, 220, 221	syl33anc 1380	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) ⊆ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
223	1, 222	sylan2 594	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) ⊆ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
224		peano2nn 11643	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
225		fveq2 6663	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (𝑆‘𝑛) = (𝑆‘(𝑘 + 1)))
226		oveq2 7157	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → (2↑𝑛) = (2↑(𝑘 + 1)))
227	226	oveq2d 7165	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (3 / (2↑𝑛)) = (3 / (2↑(𝑘 + 1))))
228	225, 227	opeq12d 4804	. . . . . . . 8 ⊢ (𝑛 = (𝑘 + 1) → ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
229		heibor.12	. . . . . . . 8 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)
230		opex 5349	. . . . . . . 8 ⊢ ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩ ∈ V
231	228, 229, 230	fvmpt 6761	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑀‘(𝑘 + 1)) = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
232	224, 231	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑀‘(𝑘 + 1)) = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
233	232	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑘 + 1)) = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
234	233	fveq2d 6667	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) = ((ball‘𝐷)‘⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩))
235		df-ov 7152	. . . 4 ⊢ ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) = ((ball‘𝐷)‘⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
236	234, 235	syl6eqr 2873	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) = ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))))
237		fveq2 6663	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑆‘𝑛) = (𝑆‘𝑘))
238		oveq2 7157	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
239	238	oveq2d 7165	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘)))
240	237, 239	opeq12d 4804	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
241		opex 5349	. . . . . . 7 ⊢ ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩ ∈ V
242	240, 229, 241	fvmpt 6761	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑀‘𝑘) = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
243	242	fveq2d 6667	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((ball‘𝐷)‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩))
244		df-ov 7152	. . . . 5 ⊢ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))) = ((ball‘𝐷)‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
245	243, 244	syl6eqr 2873	. . . 4 ⊢ (𝑘 ∈ ℕ → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
246	245	adantl 484	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
247	223, 236, 246	3sstr4d 4007	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))
248	247	ralrimiva 3181	1 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113  {cab 2798   ≠ wne 3015  ∀wral 3137  ∃wrex 3138  Vcvv 3491   ∩ cin 3928   ⊆ wss 3929  ∅c0 4284  ifcif 4460  𝒫 cpw 4532  ⟨cop 4566  ∪ cuni 4831  ∪ ciun 4912   class class class wbr 5059  {copab 5121   ↦ cmpt 5139  ⟶wf 6344  ‘cfv 6348  (class class class)co 7149   ∈ cmpo 7151  2^nd c2nd 7681  Fincfn 8502  ℂcc 10528  ℝcr 10529  0cc0 10530  1c1 10531   + caddc 10533   · cmul 10535  ℝ^*cxr 10667   ≤ cle 10669   − cmin 10863   / cdiv 11290  ℕcn 11631  2c2 11686  3c3 11687  ℕ₀cn0 11891  ℤ_≥cuz 12237  ℝ⁺crp 12383   +_𝑒 cxad 12499  seqcseq 13366  ↑cexp 13426  ∞Metcxmet 20525  Metcmet 20526  ballcbl 20527  MetOpencmopn 20530  CMetccmet 23852

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-er 8282  df-map 8401  df-en 8503  df-dom 8504  df-sdom 8505  df-fin 8506  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11632  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-seq 13367  df-exp 13427  df-psmet 20532  df-xmet 20533  df-met 20534  df-bl 20535  df-cmet 23855

This theorem is referenced by:  heiborlem8  35129  heiborlem9  35130