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Theorem bcthlem4 23910
Description: Lemma for bcth 23912. 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 23869 . . . . . . 7 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
31, 2syl 17 . . . . . 6 (𝜑𝐷 ∈ (Met‘𝑋))
4 metxmet 22920 . . . . . 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 23908 . . . . 5 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔𝑛)))
15 elrp 12369 . . . . . . . . 9 (𝑟 ∈ ℝ+ ↔ (𝑟 ∈ ℝ ∧ 0 < 𝑟))
16 nnrecl 11873 . . . . . . . . 9 ((𝑟 ∈ ℝ ∧ 0 < 𝑟) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
1715, 16sylbi 220 . . . . . . . 8 (𝑟 ∈ ℝ+ → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
1817adantl 485 . . . . . . 7 ((𝜑𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
19 peano2nn 11627 . . . . . . . . . 10 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
2019adantl 485 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
21 fvoveq1 7153 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
22 id 22 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚𝑘 = 𝑚)
23 fveq2 6643 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑔𝑘) = (𝑔𝑚))
2422, 23oveq12d 7148 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑘𝐹(𝑔𝑘)) = (𝑚𝐹(𝑔𝑚)))
2521, 24eleq12d 2906 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)) ↔ (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚))))
2625rspccva 3599 . . . . . . . . . . . . . 14 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)))
2713, 26sylan 583 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)))
286ffvelrnda 6824 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑔𝑚) ∈ (𝑋 × ℝ+))
297, 1, 8bcthlem1 23907 . . . . . . . . . . . . . . 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)))))
3029expr 460 . . . . . . . . . . . . . 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))))))
3128, 30mpd 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)))))
3227, 31mpbid 235 . . . . . . . . . . . 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))))
3332simp2d 1140 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
3433adantlr 714 . . . . . . . . . 10 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
3532simp1d 1139 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+))
36 xp2nd 7697 . . . . . . . . . . . . . 14 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
3735, 36syl 17 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
3837rpred 12409 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
3938adantlr 714 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
40 nnrecre 11657 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
4140adantl 485 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ)
42 rpre 12375 . . . . . . . . . . . 12 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
4342ad2antlr 726 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑟 ∈ ℝ)
44 lttr 10694 . . . . . . . . . . 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))
4539, 41, 43, 44syl3anc 1368 . . . . . . . . . 10 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (((2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ (1 / 𝑚) < 𝑟) → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
4634, 45mpand 694 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
47 2fveq3 6648 . . . . . . . . . . 11 (𝑛 = (𝑚 + 1) → (2nd ‘(𝑔𝑛)) = (2nd ‘(𝑔‘(𝑚 + 1))))
4847breq1d 5049 . . . . . . . . . 10 (𝑛 = (𝑚 + 1) → ((2nd ‘(𝑔𝑛)) < 𝑟 ↔ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
4948rspcev 3600 . . . . . . . . 9 (((𝑚 + 1) ∈ ℕ ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
5020, 46, 49syl6an 683 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟))
5150rexlimdva 3270 . . . . . . 7 ((𝜑𝑟 ∈ ℝ+) → (∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟))
5218, 51mpd 15 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
5352ralrimiva 3170 . . . . 5 (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
545, 6, 14, 53caubl 23891 . . . 4 (𝜑 → (1st𝑔) ∈ (Cau‘𝐷))
557cmetcau 23872 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (1st𝑔) ∈ (Cau‘𝐷)) → (1st𝑔) ∈ dom (⇝𝑡𝐽))
561, 54, 55syl2anc 587 . . 3 (𝜑 → (1st𝑔) ∈ dom (⇝𝑡𝐽))
57 fo1st 7684 . . . . . 6 1st :V–onto→V
58 fofun 6564 . . . . . 6 (1st :V–onto→V → Fun 1st )
5957, 58ax-mp 5 . . . . 5 Fun 1st
60 vex 3474 . . . . 5 𝑔 ∈ V
61 cofunexg 7625 . . . . 5 ((Fun 1st𝑔 ∈ V) → (1st𝑔) ∈ V)
6259, 60, 61mp2an 691 . . . 4 (1st𝑔) ∈ V
6362eldm 5742 . . 3 ((1st𝑔) ∈ dom (⇝𝑡𝐽) ↔ ∃𝑥(1st𝑔)(⇝𝑡𝐽)𝑥)
6456, 63sylib 221 . 2 (𝜑 → ∃𝑥(1st𝑔)(⇝𝑡𝐽)𝑥)
65 1nn 11626 . . . . . 6 1 ∈ ℕ
667, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 23909 . . . . . 6 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥 ∧ 1 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
6765, 66mp3an3 1447 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
6812fveq2d 6647 . . . . . . 7 (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩))
69 df-ov 7133 . . . . . . 7 (𝐶(ball‘𝐷)𝑅) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩)
7068, 69syl6eqr 2874 . . . . . 6 (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
7170adantr 484 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
7267, 71eleqtrd 2914 . . . 4 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ (𝐶(ball‘𝐷)𝑅))
737mopntop 23026 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
745, 73syl 17 . . . . . . . . . . . . 13 (𝜑𝐽 ∈ Top)
7574adantr 484 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐽 ∈ Top)
765adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
77 xp1st 7696 . . . . . . . . . . . . . . 15 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
7835, 77syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
7937rpxrd 12410 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ*)
80 blssm 23004 . . . . . . . . . . . . . 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)
8176, 78, 79, 80syl3anc 1368 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) ⊆ 𝑋)
82 1st2nd2 7703 . . . . . . . . . . . . . . . 16 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
8335, 82syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
8483fveq2d 6647 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩))
85 df-ov 7133 . . . . . . . . . . . . . 14 ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
8684, 85syl6reqr 2875 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
877mopnuni 23027 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
885, 87syl 17 . . . . . . . . . . . . . 14 (𝜑𝑋 = 𝐽)
8988adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑋 = 𝐽)
9081, 86, 893sstr3d 3989 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ 𝐽)
91 eqid 2821 . . . . . . . . . . . . 13 𝐽 = 𝐽
9291sscls 21640 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
9375, 90, 92syl2anc 587 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
9432simp3d 1141 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
9593, 94sstrd 3953 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
96953adant2 1128 . . . . . . . . 9 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
977, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 23909 . . . . . . . . . 10 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥 ∧ (𝑚 + 1) ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
9819, 97syl3an3 1162 . . . . . . . . 9 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
9996, 98sseldd 3944 . . . . . . . 8 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → 𝑥 ∈ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
10099eldifbd 3923 . . . . . . 7 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀𝑚))
1011003expa 1115 . . . . . 6 (((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀𝑚))
102101ralrimiva 3170 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚))
103 eluni2 4815 . . . . . . . . 9 (𝑥 ran 𝑀 ↔ ∃𝑦 ∈ ran 𝑀 𝑥𝑦)
1049ffnd 6488 . . . . . . . . . 10 (𝜑𝑀 Fn ℕ)
105 eleq2 2900 . . . . . . . . . . 11 (𝑦 = (𝑀𝑚) → (𝑥𝑦𝑥 ∈ (𝑀𝑚)))
106105rexrn 6826 . . . . . . . . . 10 (𝑀 Fn ℕ → (∃𝑦 ∈ ran 𝑀 𝑥𝑦 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
107104, 106syl 17 . . . . . . . . 9 (𝜑 → (∃𝑦 ∈ ran 𝑀 𝑥𝑦 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
108103, 107syl5bb 286 . . . . . . . 8 (𝜑 → (𝑥 ran 𝑀 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
109108notbid 321 . . . . . . 7 (𝜑 → (¬ 𝑥 ran 𝑀 ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
110 ralnex 3224 . . . . . . 7 (∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚) ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚))
111109, 110syl6bbr 292 . . . . . 6 (𝜑 → (¬ 𝑥 ran 𝑀 ↔ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚)))
112111biimpar 481 . . . . 5 ((𝜑 ∧ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚)) → ¬ 𝑥 ran 𝑀)
113102, 112syldan 594 . . . 4 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ¬ 𝑥 ran 𝑀)
11472, 113eldifd 3921 . . 3 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀))
115114ne0d 4274 . 2 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)
11664, 115exlimddv 1937 1 (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  wne 3007  wral 3126  wrex 3127  Vcvv 3471  cdif 3907  wss 3910  c0 4266  cop 4546   cuni 4811   class class class wbr 5039  {copab 5101   × cxp 5526  dom cdm 5528  ran crn 5529  ccom 5532  Fun wfun 6322   Fn wfn 6323  wf 6324  ontowfo 6326  cfv 6328  (class class class)co 7130  cmpo 7132  1st c1st 7662  2nd c2nd 7663  cr 10513  0cc0 10514  1c1 10515   + caddc 10517  *cxr 10651   < clt 10652   / cdiv 11274  cn 11615  +crp 12367  ∞Metcxmet 20506  Metcmet 20507  ballcbl 20508  MetOpencmopn 20511  Topctop 21477  Clsdccld 21600  clsccl 21602  𝑡clm 21810  Cauccau 23836  CMetccmet 23837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-map 8383  df-pm 8384  df-en 8485  df-dom 8486  df-sdom 8487  df-sup 8882  df-inf 8883  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-2 11678  df-n0 11876  df-z 11960  df-uz 12222  df-q 12327  df-rp 12368  df-xneg 12485  df-xadd 12486  df-xmul 12487  df-ico 12722  df-rest 16675  df-topgen 16696  df-psmet 20513  df-xmet 20514  df-met 20515  df-bl 20516  df-mopn 20517  df-fbas 20518  df-fg 20519  df-top 21478  df-topon 21495  df-bases 21530  df-cld 21603  df-ntr 21604  df-cls 21605  df-nei 21682  df-lm 21813  df-fil 22430  df-fm 22522  df-flim 22523  df-flf 22524  df-cfil 23838  df-cau 23839  df-cmet 23840
This theorem is referenced by:  bcthlem5  23911
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