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Theorem bcthlem4 24691
Description: Lemma for bcth 24693. 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 24650 . . . . . . 7 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
31, 2syl 17 . . . . . 6 (𝜑𝐷 ∈ (Met‘𝑋))
4 metxmet 23687 . . . . . 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 24689 . . . . 5 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔𝑛)))
15 elrp 12917 . . . . . . . . 9 (𝑟 ∈ ℝ+ ↔ (𝑟 ∈ ℝ ∧ 0 < 𝑟))
16 nnrecl 12411 . . . . . . . . 9 ((𝑟 ∈ ℝ ∧ 0 < 𝑟) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
1715, 16sylbi 216 . . . . . . . 8 (𝑟 ∈ ℝ+ → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
1817adantl 482 . . . . . . 7 ((𝜑𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
19 peano2nn 12165 . . . . . . . . . 10 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
2019adantl 482 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
21 fvoveq1 7380 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
22 id 22 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚𝑘 = 𝑚)
23 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑔𝑘) = (𝑔𝑚))
2422, 23oveq12d 7375 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑘𝐹(𝑔𝑘)) = (𝑚𝐹(𝑔𝑚)))
2521, 24eleq12d 2832 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)) ↔ (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚))))
2625rspccva 3580 . . . . . . . . . . . . . 14 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)))
2713, 26sylan 580 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)))
286ffvelcdmda 7035 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑔𝑚) ∈ (𝑋 × ℝ+))
297, 1, 8bcthlem1 24688 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 231 . . . . . . . . . . . 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 1143 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
3433adantlr 713 . . . . . . . . . 10 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
3532simp1d 1142 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+))
36 xp2nd 7954 . . . . . . . . . . . . . 14 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
3735, 36syl 17 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
3837rpred 12957 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
3938adantlr 713 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
40 nnrecre 12195 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
4140adantl 482 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ)
42 rpre 12923 . . . . . . . . . . . 12 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
4342ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑟 ∈ ℝ)
44 lttr 11231 . . . . . . . . . . 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 1371 . . . . . . . . . 10 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (((2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ (1 / 𝑚) < 𝑟) → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
4634, 45mpand 693 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
47 2fveq3 6847 . . . . . . . . . . 11 (𝑛 = (𝑚 + 1) → (2nd ‘(𝑔𝑛)) = (2nd ‘(𝑔‘(𝑚 + 1))))
4847breq1d 5115 . . . . . . . . . 10 (𝑛 = (𝑚 + 1) → ((2nd ‘(𝑔𝑛)) < 𝑟 ↔ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
4948rspcev 3581 . . . . . . . . 9 (((𝑚 + 1) ∈ ℕ ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
5020, 46, 49syl6an 682 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟))
5150rexlimdva 3152 . . . . . . 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 3143 . . . . 5 (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
545, 6, 14, 53caubl 24672 . . . 4 (𝜑 → (1st𝑔) ∈ (Cau‘𝐷))
557cmetcau 24653 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (1st𝑔) ∈ (Cau‘𝐷)) → (1st𝑔) ∈ dom (⇝𝑡𝐽))
561, 54, 55syl2anc 584 . . 3 (𝜑 → (1st𝑔) ∈ dom (⇝𝑡𝐽))
57 fo1st 7941 . . . . . 6 1st :V–onto→V
58 fofun 6757 . . . . . 6 (1st :V–onto→V → Fun 1st )
5957, 58ax-mp 5 . . . . 5 Fun 1st
60 vex 3449 . . . . 5 𝑔 ∈ V
61 cofunexg 7881 . . . . 5 ((Fun 1st𝑔 ∈ V) → (1st𝑔) ∈ V)
6259, 60, 61mp2an 690 . . . 4 (1st𝑔) ∈ V
6362eldm 5856 . . 3 ((1st𝑔) ∈ dom (⇝𝑡𝐽) ↔ ∃𝑥(1st𝑔)(⇝𝑡𝐽)𝑥)
6456, 63sylib 217 . 2 (𝜑 → ∃𝑥(1st𝑔)(⇝𝑡𝐽)𝑥)
65 1nn 12164 . . . . . 6 1 ∈ ℕ
667, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 24690 . . . . . 6 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥 ∧ 1 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
6765, 66mp3an3 1450 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
6812fveq2d 6846 . . . . . . 7 (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩))
69 df-ov 7360 . . . . . . 7 (𝐶(ball‘𝐷)𝑅) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩)
7068, 69eqtr4di 2794 . . . . . 6 (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
7170adantr 481 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
7267, 71eleqtrd 2840 . . . 4 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ (𝐶(ball‘𝐷)𝑅))
737mopntop 23793 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
745, 73syl 17 . . . . . . . . . . . . 13 (𝜑𝐽 ∈ Top)
7574adantr 481 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐽 ∈ Top)
765adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
77 xp1st 7953 . . . . . . . . . . . . . . 15 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
7835, 77syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
7937rpxrd 12958 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ*)
80 blssm 23771 . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) ⊆ 𝑋)
82 df-ov 7360 . . . . . . . . . . . . . 14 ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
83 1st2nd2 7960 . . . . . . . . . . . . . . . 16 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
8435, 83syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
8584fveq2d 6846 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩))
8682, 85eqtr4id 2795 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
877mopnuni 23794 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
885, 87syl 17 . . . . . . . . . . . . . 14 (𝜑𝑋 = 𝐽)
8988adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑋 = 𝐽)
9081, 86, 893sstr3d 3990 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ 𝐽)
91 eqid 2736 . . . . . . . . . . . . 13 𝐽 = 𝐽
9291sscls 22407 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
9375, 90, 92syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
9432simp3d 1144 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
9593, 94sstrd 3954 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
96953adant2 1131 . . . . . . . . 9 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
977, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 24690 . . . . . . . . . 10 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥 ∧ (𝑚 + 1) ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
9819, 97syl3an3 1165 . . . . . . . . 9 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
9996, 98sseldd 3945 . . . . . . . 8 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → 𝑥 ∈ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
10099eldifbd 3923 . . . . . . 7 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀𝑚))
1011003expa 1118 . . . . . 6 (((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀𝑚))
102101ralrimiva 3143 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚))
103 eluni2 4869 . . . . . . . . 9 (𝑥 ran 𝑀 ↔ ∃𝑦 ∈ ran 𝑀 𝑥𝑦)
1049ffnd 6669 . . . . . . . . . 10 (𝜑𝑀 Fn ℕ)
105 eleq2 2826 . . . . . . . . . . 11 (𝑦 = (𝑀𝑚) → (𝑥𝑦𝑥 ∈ (𝑀𝑚)))
106105rexrn 7037 . . . . . . . . . 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, 107bitrid 282 . . . . . . . 8 (𝜑 → (𝑥 ran 𝑀 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
109108notbid 317 . . . . . . 7 (𝜑 → (¬ 𝑥 ran 𝑀 ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚)))
110 ralnex 3075 . . . . . . 7 (∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚) ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚))
111109, 110bitr4di 288 . . . . . 6 (𝜑 → (¬ 𝑥 ran 𝑀 ↔ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚)))
112111biimpar 478 . . . . 5 ((𝜑 ∧ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚)) → ¬ 𝑥 ran 𝑀)
113102, 112syldan 591 . . . 4 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ¬ 𝑥 ran 𝑀)
11472, 113eldifd 3921 . . 3 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀))
115114ne0d 4295 . 2 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)
11664, 115exlimddv 1938 1 (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  wss 3910  c0 4282  cop 4592   cuni 4865   class class class wbr 5105  {copab 5167   × cxp 5631  dom cdm 5633  ran crn 5634  ccom 5637  Fun wfun 6490   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  *cxr 11188   < clt 11189   / cdiv 11812  cn 12153  +crp 12915  ∞Metcxmet 20781  Metcmet 20782  ballcbl 20783  MetOpencmopn 20786  Topctop 22242  Clsdccld 22367  clsccl 22369  𝑡clm 22577  Cauccau 24617  CMetccmet 24618
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ico 13270  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-top 22243  df-topon 22260  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lm 22580  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-cfil 24619  df-cau 24620  df-cmet 24621
This theorem is referenced by:  bcthlem5  24692
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