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Theorem bcthlem4 23170
Description: Lemma for bcth 23172. 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. (𝑈𝐴) ≠ ∅. (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 23130 . . . . . . 7 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
31, 2syl 17 . . . . . 6 (𝜑𝐷 ∈ (Met‘𝑋))
4 metxmet 22186 . . . . . 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 23168 . . . . 5 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔𝑛)))
15 elrp 11872 . . . . . . . . 9 (𝑟 ∈ ℝ+ ↔ (𝑟 ∈ ℝ ∧ 0 < 𝑟))
16 nnrecl 11328 . . . . . . . . 9 ((𝑟 ∈ ℝ ∧ 0 < 𝑟) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
1715, 16sylbi 207 . . . . . . . 8 (𝑟 ∈ ℝ+ → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
1817adantl 481 . . . . . . 7 ((𝜑𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
19 peano2nn 11070 . . . . . . . . . 10 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
2019adantl 481 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
21 oveq1 6697 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
2221fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
23 id 22 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚𝑘 = 𝑚)
24 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑔𝑘) = (𝑔𝑚))
2523, 24oveq12d 6708 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑘𝐹(𝑔𝑘)) = (𝑚𝐹(𝑔𝑚)))
2622, 25eleq12d 2724 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)) ↔ (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚))))
2726rspccva 3339 . . . . . . . . . . . . . 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 6399 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑔𝑚) ∈ (𝑋 × ℝ+))
307, 1, 8bcthlem1 23167 . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . 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 222 . . . . . . . . . . . 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 1094 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
3534adantlr 751 . . . . . . . . . 10 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
3633simp1d 1093 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+))
37 xp2nd 7243 . . . . . . . . . . . . . 14 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
3836, 37syl 17 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
3938rpred 11910 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
4039adantlr 751 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
41 nnrecre 11095 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
4241adantl 481 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ)
43 rpre 11877 . . . . . . . . . . . 12 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
4443ad2antlr 763 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑟 ∈ ℝ)
45 lttr 10152 . . . . . . . . . . 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 1366 . . . . . . . . . 10 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (((2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ (1 / 𝑚) < 𝑟) → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
4735, 46mpand 711 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
48 fveq2 6229 . . . . . . . . . . . 12 (𝑛 = (𝑚 + 1) → (𝑔𝑛) = (𝑔‘(𝑚 + 1)))
4948fveq2d 6233 . . . . . . . . . . 11 (𝑛 = (𝑚 + 1) → (2nd ‘(𝑔𝑛)) = (2nd ‘(𝑔‘(𝑚 + 1))))
5049breq1d 4695 . . . . . . . . . 10 (𝑛 = (𝑚 + 1) → ((2nd ‘(𝑔𝑛)) < 𝑟 ↔ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
5150rspcev 3340 . . . . . . . . 9 (((𝑚 + 1) ∈ ℕ ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
5220, 47, 51syl6an 567 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟))
5352rexlimdva 3060 . . . . . . 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 2995 . . . . 5 (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
565, 6, 14, 55caubl 23152 . . . 4 (𝜑 → (1st𝑔) ∈ (Cau‘𝐷))
577cmetcau 23133 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (1st𝑔) ∈ (Cau‘𝐷)) → (1st𝑔) ∈ dom (⇝𝑡𝐽))
581, 56, 57syl2anc 694 . . 3 (𝜑 → (1st𝑔) ∈ dom (⇝𝑡𝐽))
59 fo1st 7230 . . . . . 6 1st :V–onto→V
60 fofun 6154 . . . . . 6 (1st :V–onto→V → Fun 1st )
6159, 60ax-mp 5 . . . . 5 Fun 1st
62 vex 3234 . . . . 5 𝑔 ∈ V
63 cofunexg 7172 . . . . 5 ((Fun 1st𝑔 ∈ V) → (1st𝑔) ∈ V)
6461, 62, 63mp2an 708 . . . 4 (1st𝑔) ∈ V
6564eldm 5353 . . 3 ((1st𝑔) ∈ dom (⇝𝑡𝐽) ↔ ∃𝑥(1st𝑔)(⇝𝑡𝐽)𝑥)
6658, 65sylib 208 . 2 (𝜑 → ∃𝑥(1st𝑔)(⇝𝑡𝐽)𝑥)
67 1nn 11069 . . . . . 6 1 ∈ ℕ
687, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 23169 . . . . . 6 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥 ∧ 1 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
6967, 68mp3an3 1453 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
7012fveq2d 6233 . . . . . . 7 (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩))
71 df-ov 6693 . . . . . . 7 (𝐶(ball‘𝐷)𝑅) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩)
7270, 71syl6eqr 2703 . . . . . 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 2732 . . . 4 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ (𝐶(ball‘𝐷)𝑅))
757mopntop 22292 . . . . . . . . . . . . . 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 7242 . . . . . . . . . . . . . . 15 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
8036, 79syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
8138rpxrd 11911 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ*)
82 blssm 22270 . . . . . . . . . . . . . 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 1366 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) ⊆ 𝑋)
84 1st2nd2 7249 . . . . . . . . . . . . . . . 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 6233 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩))
87 df-ov 6693 . . . . . . . . . . . . . 14 ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
8886, 87syl6reqr 2704 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
897mopnuni 22293 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
905, 89syl 17 . . . . . . . . . . . . . 14 (𝜑𝑋 = 𝐽)
9190adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑋 = 𝐽)
9283, 88, 913sstr3d 3680 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ 𝐽)
93 eqid 2651 . . . . . . . . . . . . 13 𝐽 = 𝐽
9493sscls 20908 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
9577, 92, 94syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
9633simp3d 1095 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
9795, 96sstrd 3646 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
98973adant2 1100 . . . . . . . . 9 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
997, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 23169 . . . . . . . . . 10 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥 ∧ (𝑚 + 1) ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
10019, 99syl3an3 1401 . . . . . . . . 9 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
10198, 100sseldd 3637 . . . . . . . 8 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → 𝑥 ∈ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
102101eldifbd 3620 . . . . . . 7 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀𝑚))
1031023expa 1284 . . . . . 6 (((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀𝑚))
104103ralrimiva 2995 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚))
105 eluni2 4472 . . . . . . . . 9 (𝑥 ran 𝑀 ↔ ∃𝑦 ∈ ran 𝑀 𝑥𝑦)
106 ffn 6083 . . . . . . . . . . 11 (𝑀:ℕ⟶(Clsd‘𝐽) → 𝑀 Fn ℕ)
1079, 106syl 17 . . . . . . . . . 10 (𝜑𝑀 Fn ℕ)
108 eleq2 2719 . . . . . . . . . . 11 (𝑦 = (𝑀𝑚) → (𝑥𝑦𝑥 ∈ (𝑀𝑚)))
109108rexrn 6401 . . . . . . . . . 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 272 . . . . . . . 8 (𝜑 → (𝑥 ran 𝑀 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
112111notbid 307 . . . . . . 7 (𝜑 → (¬ 𝑥 ran 𝑀 ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
113 ralnex 3021 . . . . . . 7 (∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚) ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚))
114112, 113syl6bbr 278 . . . . . 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 3618 . . 3 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀))
118 ne0i 3954 . . 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 1903 1 (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  wss 3607  c0 3948  cop 4216   cuni 4468   class class class wbr 4685  {copab 4745   × cxp 5141  dom cdm 5143  ran crn 5144  ccom 5147  Fun wfun 5920   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  *cxr 10111   < clt 10112   / cdiv 10722  cn 11058  +crp 11870  ∞Metcxmt 19779  Metcme 19780  ballcbl 19781  MetOpencmopn 19784  Topctop 20746  Clsdccld 20868  clsccl 20870  𝑡clm 21078  Caucca 23097  CMetcms 23098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ico 12219  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lm 21081  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-cfil 23099  df-cau 23100  df-cmet 23101
This theorem is referenced by:  bcthlem5  23171
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