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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 (𝜑𝐷 ∈ (CMet‘𝑋))
bcthlem.5 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})
bcthlem.6 (𝜑𝑀:ℕ⟶(Clsd‘𝐽))
bcthlem.7 (𝜑𝑅 ∈ ℝ+)
bcthlem.8 (𝜑𝐶𝑋)
bcthlem.9 (𝜑𝑔:ℕ⟶(𝑋 × ℝ+))
bcthlem.10 (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
bcthlem.11 (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))
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.
StepHypRef Expression
1 bcthlem.4 . . . 4 (𝜑𝐷 ∈ (CMet‘𝑋))
2 cmetmet 22892 . . . . . . 7 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
31, 2syl 17 . . . . . 6 (𝜑𝐷 ∈ (Met‘𝑋))
4 metxmet 21949 . . . . . 6 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
53, 4syl 17 . . . . 5 (𝜑𝐷 ∈ (∞Met‘𝑋))
6 bcthlem.9 . . . . 5 (𝜑𝑔:ℕ⟶(𝑋 × ℝ+))
7 bcth.2 . . . . . 6 𝐽 = (MetOpen‘𝐷)
8 bcthlem.5 . . . . . 6 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})
9 bcthlem.6 . . . . . 6 (𝜑𝑀:ℕ⟶(Clsd‘𝐽))
10 bcthlem.7 . . . . . 6 (𝜑𝑅 ∈ ℝ+)
11 bcthlem.8 . . . . . 6 (𝜑𝐶𝑋)
12 bcthlem.10 . . . . . 6 (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
13 bcthlem.11 . . . . . 6 (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))
147, 1, 8, 9, 10, 11, 6, 12, 13bcthlem2 22930 . . . . 5 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔𝑛)))
15 elrp 11710 . . . . . . . . 9 (𝑟 ∈ ℝ+ ↔ (𝑟 ∈ ℝ ∧ 0 < 𝑟))
16 nnrecl 11167 . . . . . . . . 9 ((𝑟 ∈ ℝ ∧ 0 < 𝑟) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
1715, 16sylbi 206 . . . . . . . 8 (𝑟 ∈ ℝ+ → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
1817adantl 481 . . . . . . 7 ((𝜑𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
19 peano2nn 10909 . . . . . . . . . 10 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
2019adantl 481 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
21 oveq1 6556 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
2221fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
23 id 22 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚𝑘 = 𝑚)
24 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑔𝑘) = (𝑔𝑚))
2523, 24oveq12d 6567 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑘𝐹(𝑔𝑘)) = (𝑚𝐹(𝑔𝑚)))
2622, 25eleq12d 2682 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)) ↔ (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚))))
2726rspccva 3281 . . . . . . . . . . . . . 14 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)))
2813, 27sylan 487 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)))
296ffvelrnda 6267 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑔𝑚) ∈ (𝑋 × ℝ+))
307, 1, 8bcthlem1 22929 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔𝑚) ∈ (𝑋 × ℝ+))) → ((𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)) ↔ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))))
3130expr 641 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((𝑔𝑚) ∈ (𝑋 × ℝ+) → ((𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)) ↔ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚))))))
3229, 31mpd 15 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)) ↔ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))))
3328, 32mpbid 221 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚))))
3433simp2d 1067 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
3534adantlr 747 . . . . . . . . . 10 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
3633simp1d 1066 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+))
37 xp2nd 7090 . . . . . . . . . . . . . 14 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
3836, 37syl 17 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
3938rpred 11748 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
4039adantlr 747 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
41 nnrecre 10934 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
4241adantl 481 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ)
43 rpre 11715 . . . . . . . . . . . 12 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
4443ad2antlr 759 . . . . . . . . . . 11 (((𝜑𝑟 ∈ 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))
4640, 42, 44, 45syl3anc 1318 . . . . . . . . . 10 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (((2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ (1 / 𝑚) < 𝑟) → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
4735, 46mpand 707 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
48 fveq2 6103 . . . . . . . . . . . 12 (𝑛 = (𝑚 + 1) → (𝑔𝑛) = (𝑔‘(𝑚 + 1)))
4948fveq2d 6107 . . . . . . . . . . 11 (𝑛 = (𝑚 + 1) → (2nd ‘(𝑔𝑛)) = (2nd ‘(𝑔‘(𝑚 + 1))))
5049breq1d 4593 . . . . . . . . . 10 (𝑛 = (𝑚 + 1) → ((2nd ‘(𝑔𝑛)) < 𝑟 ↔ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
5150rspcev 3282 . . . . . . . . 9 (((𝑚 + 1) ∈ ℕ ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
5220, 47, 51syl6an 566 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟))
5352rexlimdva 3013 . . . . . . 7 ((𝜑𝑟 ∈ ℝ+) → (∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟))
5418, 53mpd 15 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
5554ralrimiva 2949 . . . . 5 (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
565, 6, 14, 55caubl 22914 . . . 4 (𝜑 → (1st𝑔) ∈ (Cau‘𝐷))
577cmetcau 22895 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (1st𝑔) ∈ (Cau‘𝐷)) → (1st𝑔) ∈ dom (⇝𝑡𝐽))
581, 56, 57syl2anc 691 . . 3 (𝜑 → (1st𝑔) ∈ dom (⇝𝑡𝐽))
59 fo1st 7079 . . . . . 6 1st :V–onto→V
60 fofun 6029 . . . . . 6 (1st :V–onto→V → Fun 1st )
6159, 60ax-mp 5 . . . . 5 Fun 1st
62 vex 3176 . . . . 5 𝑔 ∈ V
63 cofunexg 7023 . . . . 5 ((Fun 1st𝑔 ∈ V) → (1st𝑔) ∈ V)
6461, 62, 63mp2an 704 . . . 4 (1st𝑔) ∈ V
6564eldm 5243 . . 3 ((1st𝑔) ∈ dom (⇝𝑡𝐽) ↔ ∃𝑥(1st𝑔)(⇝𝑡𝐽)𝑥)
6658, 65sylib 207 . 2 (𝜑 → ∃𝑥(1st𝑔)(⇝𝑡𝐽)𝑥)
67 1nn 10908 . . . . . 6 1 ∈ ℕ
687, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 22931 . . . . . 6 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥 ∧ 1 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
6967, 68mp3an3 1405 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
7012fveq2d 6107 . . . . . . 7 (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩))
71 df-ov 6552 . . . . . . 7 (𝐶(ball‘𝐷)𝑅) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩)
7270, 71syl6eqr 2662 . . . . . 6 (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
7372adantr 480 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
7469, 73eleqtrd 2690 . . . 4 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ (𝐶(ball‘𝐷)𝑅))
757mopntop 22055 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
765, 75syl 17 . . . . . . . . . . . . 13 (𝜑𝐽 ∈ Top)
7776adantr 480 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐽 ∈ Top)
785adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
79 xp1st 7089 . . . . . . . . . . . . . . 15 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
8036, 79syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
8138rpxrd 11749 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ 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)
8378, 80, 81, 82syl3anc 1318 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ 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)))⟩)
8536, 84syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
8685fveq2d 6107 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ 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)))⟩)
8886, 87syl6reqr 2663 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
897mopnuni 22056 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
905, 89syl 17 . . . . . . . . . . . . . 14 (𝜑𝑋 = 𝐽)
9190adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑋 = 𝐽)
9283, 88, 913sstr3d 3610 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ 𝐽)
93 eqid 2610 . . . . . . . . . . . . 13 𝐽 = 𝐽
9493sscls 20670 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
9577, 92, 94syl2anc 691 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
9633simp3d 1068 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
9795, 96sstrd 3578 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
98973adant2 1073 . . . . . . . . 9 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
997, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 22931 . . . . . . . . . 10 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥 ∧ (𝑚 + 1) ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
10019, 99syl3an3 1353 . . . . . . . . 9 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
10198, 100sseldd 3569 . . . . . . . 8 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → 𝑥 ∈ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
102101eldifbd 3553 . . . . . . 7 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀𝑚))
1031023expa 1257 . . . . . 6 (((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀𝑚))
104103ralrimiva 2949 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚))
105 eluni2 4376 . . . . . . . . 9 (𝑥 ran 𝑀 ↔ ∃𝑦 ∈ ran 𝑀 𝑥𝑦)
106 ffn 5958 . . . . . . . . . . 11 (𝑀:ℕ⟶(Clsd‘𝐽) → 𝑀 Fn ℕ)
1079, 106syl 17 . . . . . . . . . 10 (𝜑𝑀 Fn ℕ)
108 eleq2 2677 . . . . . . . . . . 11 (𝑦 = (𝑀𝑚) → (𝑥𝑦𝑥 ∈ (𝑀𝑚)))
109108rexrn 6269 . . . . . . . . . 10 (𝑀 Fn ℕ → (∃𝑦 ∈ ran 𝑀 𝑥𝑦 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
110107, 109syl 17 . . . . . . . . 9 (𝜑 → (∃𝑦 ∈ ran 𝑀 𝑥𝑦 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
111105, 110syl5bb 271 . . . . . . . 8 (𝜑 → (𝑥 ran 𝑀 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
112111notbid 307 . . . . . . 7 (𝜑 → (¬ 𝑥 ran 𝑀 ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
113 ralnex 2975 . . . . . . 7 (∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚) ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚))
114112, 113syl6bbr 277 . . . . . 6 (𝜑 → (¬ 𝑥 ran 𝑀 ↔ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚)))
115114biimpar 501 . . . . 5 ((𝜑 ∧ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚)) → ¬ 𝑥 ran 𝑀)
116104, 115syldan 486 . . . 4 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ¬ 𝑥 ran 𝑀)
11774, 116eldifd 3551 . . 3 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)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) ≠ ∅)
119117, 118syl 17 . 2 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)
12066, 119exlimddv 1850 1 (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)
 Colors of variables: wff setvar class 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  1st c1st 7057  2nd 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
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