Theorem bcthlem4 22932

Description: Lemma for bcth 22934. Given any open ball (𝐶(ball‘𝐷)𝑅) as starting point (and in particular, a ball in int(∪ ran 𝑀)), the limit point 𝑥 of the centers of the induced sequence of balls 𝑔 is outside ∪ ran 𝑀. Note that a set 𝐴 has empty interior iff every nonempty open set 𝑈 contains points outside 𝐴, i.e. (𝑈 ∖ 𝐴) ≠ ∅. (Contributed by Mario Carneiro, 7-Jan-2014.)

Hypotheses

Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)
bcthlem.4	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bcthlem.5	⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
bcthlem.6	⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
bcthlem.7	⊢ (𝜑 → 𝑅 ∈ ℝ+)
bcthlem.8	⊢ (𝜑 → 𝐶 ∈ 𝑋)
bcthlem.9	⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+))
bcthlem.10	⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
bcthlem.11	⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))

Assertion

Ref	Expression
bcthlem4	⊢ (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅)

Distinct variable groups:   𝑘,𝑟,𝑥,𝑧   𝐶,𝑟,𝑥   𝑔,𝑘,𝑟,𝑥,𝑧,𝐷   𝑔,𝐹,𝑘,𝑟,𝑥,𝑧   𝑔,𝐽,𝑘,𝑟,𝑥,𝑧   𝑔,𝑀,𝑘,𝑟,𝑥,𝑧   𝜑,𝑘,𝑟,𝑥,𝑧   𝑥,𝑅   𝑔,𝑋,𝑘,𝑟,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑔)   𝐶(𝑧,𝑔,𝑘)   𝑅(𝑧,𝑔,𝑘,𝑟)

Proof of Theorem bcthlem4

Dummy variables 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bcthlem.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
2		cmetmet 22892	. . . . . . 7 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		metxmet 21949	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
6		bcthlem.9	. . . . 5 ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+))
7		bcth.2	. . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷)
8		bcthlem.5	. . . . . 6 ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
9		bcthlem.6	. . . . . 6 ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
10		bcthlem.7	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
11		bcthlem.8	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
12		bcthlem.10	. . . . . 6 ⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
13		bcthlem.11	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))
14	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem2 22930	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
15		elrp 11710	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ+ ↔ (𝑟 ∈ ℝ ∧ 0 < 𝑟))
16		nnrecl 11167	. . . . . . . . 9 ⊢ ((𝑟 ∈ ℝ ∧ 0 < 𝑟) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
17	15, 16	sylbi 206	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ+ → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
18	17	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
19		peano2nn 10909	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
20	19	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
21		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
22	21	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
23		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → 𝑘 = 𝑚)
24		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (𝑔‘𝑘) = (𝑔‘𝑚))
25	23, 24	oveq12d 6567	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝑘𝐹(𝑔‘𝑘)) = (𝑚𝐹(𝑔‘𝑚)))
26	22, 25	eleq12d 2682	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚))))
27	26	rspccva 3281	. . . . . . . . . . . . . 14 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)))
28	13, 27	sylan 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)))
29	6	ffvelrnda 6267	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘𝑚) ∈ (𝑋 × ℝ+))
30	7, 1, 8	bcthlem1 22929	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔‘𝑚) ∈ (𝑋 × ℝ+))) → ((𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)) ↔ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))))
31	30	expr 641	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑔‘𝑚) ∈ (𝑋 × ℝ+) → ((𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)) ↔ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚))))))
32	29, 31	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)) ↔ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))))
33	28, 32	mpbid 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚))))
34	33	simp2d 1067	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
35	34	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
36	33	simp1d 1066	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+))
37		xp2nd 7090	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
38	36, 37	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
39	38	rpred 11748	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
40	39	adantlr 747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
41		nnrecre 10934	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
42	41	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ)
43		rpre 11715	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ)
44	43	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑟 ∈ ℝ)
45		lttr 9993	. . . . . . . . . . 11 ⊢ (((2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ ∧ (1 / 𝑚) ∈ ℝ ∧ 𝑟 ∈ ℝ) → (((2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ (1 / 𝑚) < 𝑟) → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
46	40, 42, 44, 45	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (((2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ (1 / 𝑚) < 𝑟) → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
47	35, 46	mpand 707	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
48		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 + 1) → (𝑔‘𝑛) = (𝑔‘(𝑚 + 1)))
49	48	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑚 + 1) → (2nd ‘(𝑔‘𝑛)) = (2nd ‘(𝑔‘(𝑚 + 1))))
50	49	breq1d 4593	. . . . . . . . . 10 ⊢ (𝑛 = (𝑚 + 1) → ((2nd ‘(𝑔‘𝑛)) < 𝑟 ↔ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
51	50	rspcev 3282	. . . . . . . . 9 ⊢ (((𝑚 + 1) ∈ ℕ ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟)
52	20, 47, 51	syl6an 566	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟))
53	52	rexlimdva 3013	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → (∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟))
54	18, 53	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟)
55	54	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟)
56	5, 6, 14, 55	caubl 22914	. . . 4 ⊢ (𝜑 → (1st ∘ 𝑔) ∈ (Cau‘𝐷))
57	7	cmetcau 22895	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (1st ∘ 𝑔) ∈ (Cau‘𝐷)) → (1st ∘ 𝑔) ∈ dom (⇝𝑡‘𝐽))
58	1, 56, 57	syl2anc 691	. . 3 ⊢ (𝜑 → (1st ∘ 𝑔) ∈ dom (⇝𝑡‘𝐽))
59		fo1st 7079	. . . . . 6 ⊢ 1st :V–onto→V
60		fofun 6029	. . . . . 6 ⊢ (1st :V–onto→V → Fun 1st )
61	59, 60	ax-mp 5	. . . . 5 ⊢ Fun 1st
62		vex 3176	. . . . 5 ⊢ 𝑔 ∈ V
63		cofunexg 7023	. . . . 5 ⊢ ((Fun 1st ∧ 𝑔 ∈ V) → (1st ∘ 𝑔) ∈ V)
64	61, 62, 63	mp2an 704	. . . 4 ⊢ (1st ∘ 𝑔) ∈ V
65	64	eldm 5243	. . 3 ⊢ ((1st ∘ 𝑔) ∈ dom (⇝𝑡‘𝐽) ↔ ∃𝑥(1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥)
66	58, 65	sylib 207	. 2 ⊢ (𝜑 → ∃𝑥(1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥)
67		1nn 10908	. . . . . 6 ⊢ 1 ∈ ℕ
68	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 22931	. . . . . 6 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 1 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
69	67, 68	mp3an3 1405	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
70	12	fveq2d 6107	. . . . . . 7 ⊢ (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩))
71		df-ov 6552	. . . . . . 7 ⊢ (𝐶(ball‘𝐷)𝑅) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩)
72	70, 71	syl6eqr 2662	. . . . . 6 ⊢ (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
73	72	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
74	69, 73	eleqtrd 2690	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ (𝐶(ball‘𝐷)𝑅))
75	7	mopntop 22055	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
76	5, 75	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
77	76	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐽 ∈ Top)
78	5	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
79		xp1st 7089	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
80	36, 79	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
81	38	rpxrd 11749	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ*)
82		blssm 22033	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋 ∧ (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ*) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) ⊆ 𝑋)
83	78, 80, 81, 82	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) ⊆ 𝑋)
84		1st2nd2 7096	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
85	36, 84	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
86	85	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩))
87		df-ov 6552	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
88	86, 87	syl6reqr 2663	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
89	7	mopnuni 22056	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
90	5, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
91	90	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 = ∪ 𝐽)
92	83, 88, 91	3sstr3d 3610	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ∪ 𝐽)
93		eqid 2610	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
94	93	sscls 20670	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ∪ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
95	77, 92, 94	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
96	33	simp3d 1068	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
97	95, 96	sstrd 3578	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
98	97	3adant2 1073	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
99	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 22931	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ (𝑚 + 1) ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
100	19, 99	syl3an3 1353	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
101	98, 100	sseldd 3569	. . . . . . . 8 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → 𝑥 ∈ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
102	101	eldifbd 3553	. . . . . . 7 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀‘𝑚))
103	102	3expa 1257	. . . . . 6 ⊢ (((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀‘𝑚))
104	103	ralrimiva 2949	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚))
105		eluni2 4376	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ ran 𝑀 ↔ ∃𝑦 ∈ ran 𝑀 𝑥 ∈ 𝑦)
106		ffn 5958	. . . . . . . . . . 11 ⊢ (𝑀:ℕ⟶(Clsd‘𝐽) → 𝑀 Fn ℕ)
107	9, 106	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 Fn ℕ)
108		eleq2 2677	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑀‘𝑚) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑀‘𝑚)))
109	108	rexrn 6269	. . . . . . . . . 10 ⊢ (𝑀 Fn ℕ → (∃𝑦 ∈ ran 𝑀 𝑥 ∈ 𝑦 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚)))
110	107, 109	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑦 ∈ ran 𝑀 𝑥 ∈ 𝑦 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚)))
111	105, 110	syl5bb 271	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ∪ ran 𝑀 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚)))
112	111	notbid 307	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑥 ∈ ∪ ran 𝑀 ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚)))
113		ralnex 2975	. . . . . . 7 ⊢ (∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚) ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚))
114	112, 113	syl6bbr 277	. . . . . 6 ⊢ (𝜑 → (¬ 𝑥 ∈ ∪ ran 𝑀 ↔ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚)))
115	114	biimpar 501	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚)) → ¬ 𝑥 ∈ ∪ ran 𝑀)
116	104, 115	syldan 486	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ¬ 𝑥 ∈ ∪ ran 𝑀)
117	74, 116	eldifd 3551	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀))
118		ne0i 3880	. . 3 ⊢ (𝑥 ∈ ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅)
119	117, 118	syl 17	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅)
120	66, 119	exlimddv 1850	1 ⊢ (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583  {copab 4642   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  ℝ^*cxr 9952   < clt 9953   / cdiv 10563  ℕcn 10897  ℝ⁺crp 11708  ∞Metcxmt 19552  Metcme 19553  ballcbl 19554  MetOpencmopn 19557  Topctop 20517  Clsdccld 20630  clsccl 20632  ⇝_𝑡clm 20840  Caucca 22859  CMetcms 22860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lm 20843  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-cfil 22861  df-cau 22862  df-cmet 22863

This theorem is referenced by:  bcthlem5  22933