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Theorem bcthlem4 25299
Description: Lemma for bcth 25301. 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 25258 . . . . . . 7 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
31, 2syl 17 . . . . . 6 (𝜑𝐷 ∈ (Met‘𝑋))
4 metxmet 24284 . . . . . 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 25297 . . . . 5 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔𝑛)))
15 elrp 13011 . . . . . . . . 9 (𝑟 ∈ ℝ+ ↔ (𝑟 ∈ ℝ ∧ 0 < 𝑟))
16 nnrecl 12503 . . . . . . . . 9 ((𝑟 ∈ ℝ ∧ 0 < 𝑟) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
1715, 16sylbi 216 . . . . . . . 8 (𝑟 ∈ ℝ+ → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
1817adantl 480 . . . . . . 7 ((𝜑𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
19 peano2nn 12257 . . . . . . . . . 10 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
2019adantl 480 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
21 fvoveq1 7442 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
22 id 22 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚𝑘 = 𝑚)
23 fveq2 6896 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑔𝑘) = (𝑔𝑚))
2422, 23oveq12d 7437 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑘𝐹(𝑔𝑘)) = (𝑚𝐹(𝑔𝑚)))
2521, 24eleq12d 2819 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)) ↔ (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚))))
2625rspccva 3605 . . . . . . . . . . . . . 14 ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)))
2713, 26sylan 578 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔𝑚)))
286ffvelcdmda 7093 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑔𝑚) ∈ (𝑋 × ℝ+))
297, 1, 8bcthlem1 25296 . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . . 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 1140 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
3433adantlr 713 . . . . . . . . . 10 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
3532simp1d 1139 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+))
36 xp2nd 8027 . . . . . . . . . . . . . 14 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
3735, 36syl 17 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
3837rpred 13051 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
3938adantlr 713 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
40 nnrecre 12287 . . . . . . . . . . . 12 (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
4140adantl 480 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ)
42 rpre 13017 . . . . . . . . . . . 12 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
4342ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑟 ∈ ℝ)
44 lttr 11322 . . . . . . . . . . 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 693 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
47 2fveq3 6901 . . . . . . . . . . 11 (𝑛 = (𝑚 + 1) → (2nd ‘(𝑔𝑛)) = (2nd ‘(𝑔‘(𝑚 + 1))))
4847breq1d 5159 . . . . . . . . . 10 (𝑛 = (𝑚 + 1) → ((2nd ‘(𝑔𝑛)) < 𝑟 ↔ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
4948rspcev 3606 . . . . . . . . 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 3144 . . . . . . 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 3135 . . . . 5 (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝑔𝑛)) < 𝑟)
545, 6, 14, 53caubl 25280 . . . 4 (𝜑 → (1st𝑔) ∈ (Cau‘𝐷))
557cmetcau 25261 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ (1st𝑔) ∈ (Cau‘𝐷)) → (1st𝑔) ∈ dom (⇝𝑡𝐽))
561, 54, 55syl2anc 582 . . 3 (𝜑 → (1st𝑔) ∈ dom (⇝𝑡𝐽))
57 fo1st 8014 . . . . . 6 1st :V–onto→V
58 fofun 6811 . . . . . 6 (1st :V–onto→V → Fun 1st )
5957, 58ax-mp 5 . . . . 5 Fun 1st
60 vex 3465 . . . . 5 𝑔 ∈ V
61 cofunexg 7953 . . . . 5 ((Fun 1st𝑔 ∈ V) → (1st𝑔) ∈ V)
6259, 60, 61mp2an 690 . . . 4 (1st𝑔) ∈ V
6362eldm 5903 . . 3 ((1st𝑔) ∈ dom (⇝𝑡𝐽) ↔ ∃𝑥(1st𝑔)(⇝𝑡𝐽)𝑥)
6456, 63sylib 217 . 2 (𝜑 → ∃𝑥(1st𝑔)(⇝𝑡𝐽)𝑥)
65 1nn 12256 . . . . . 6 1 ∈ ℕ
667, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 25298 . . . . . 6 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥 ∧ 1 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
6765, 66mp3an3 1446 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
6812fveq2d 6900 . . . . . . 7 (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩))
69 df-ov 7422 . . . . . . 7 (𝐶(ball‘𝐷)𝑅) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩)
7068, 69eqtr4di 2783 . . . . . 6 (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
7170adantr 479 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
7267, 71eleqtrd 2827 . . . 4 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ (𝐶(ball‘𝐷)𝑅))
737mopntop 24390 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
745, 73syl 17 . . . . . . . . . . . . 13 (𝜑𝐽 ∈ Top)
7574adantr 479 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐽 ∈ Top)
765adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
77 xp1st 8026 . . . . . . . . . . . . . . 15 ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
7835, 77syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
7937rpxrd 13052 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ*)
80 blssm 24368 . . . . . . . . . . . . . 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 df-ov 7422 . . . . . . . . . . . . . 14 ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
83 1st2nd2 8033 . . . . . . . . . . . . . . . 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 6900 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩))
8682, 85eqtr4id 2784 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
877mopnuni 24391 . . . . . . . . . . . . . . 15 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
885, 87syl 17 . . . . . . . . . . . . . 14 (𝜑𝑋 = 𝐽)
8988adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑋 = 𝐽)
9081, 86, 893sstr3d 4023 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ 𝐽)
91 eqid 2725 . . . . . . . . . . . . 13 𝐽 = 𝐽
9291sscls 23004 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
9375, 90, 92syl2anc 582 . . . . . . . . . . 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 3987 . . . . . . . . . 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 25298 . . . . . . . . . 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 3977 . . . . . . . 8 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → 𝑥 ∈ (((ball‘𝐷)‘(𝑔𝑚)) ∖ (𝑀𝑚)))
10099eldifbd 3957 . . . . . . 7 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀𝑚))
1011003expa 1115 . . . . . 6 (((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀𝑚))
102101ralrimiva 3135 . . . . 5 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚))
103 eluni2 4913 . . . . . . . . 9 (𝑥 ran 𝑀 ↔ ∃𝑦 ∈ ran 𝑀 𝑥𝑦)
1049ffnd 6724 . . . . . . . . . 10 (𝜑𝑀 Fn ℕ)
105 eleq2 2814 . . . . . . . . . . 11 (𝑦 = (𝑀𝑚) → (𝑥𝑦𝑥 ∈ (𝑀𝑚)))
106105rexrn 7096 . . . . . . . . . 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 3061 . . . . . . 7 (∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚) ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀𝑚))
111109, 110bitr4di 288 . . . . . 6 (𝜑 → (¬ 𝑥 ran 𝑀 ↔ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚)))
112111biimpar 476 . . . . 5 ((𝜑 ∧ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀𝑚)) → ¬ 𝑥 ran 𝑀)
113102, 112syldan 589 . . . 4 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ¬ 𝑥 ran 𝑀)
11472, 113eldifd 3955 . . 3 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → 𝑥 ∈ ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀))
115114ne0d 4335 . 2 ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥) → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)
11664, 115exlimddv 1930 1 (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wne 2929  wral 3050  wrex 3059  Vcvv 3461  cdif 3941  wss 3944  c0 4322  cop 4636   cuni 4909   class class class wbr 5149  {copab 5211   × cxp 5676  dom cdm 5678  ran crn 5679  ccom 5682  Fun wfun 6543   Fn wfn 6544  wf 6545  ontowfo 6547  cfv 6549  (class class class)co 7419  cmpo 7421  1st c1st 7992  2nd c2nd 7993  cr 11139  0cc0 11140  1c1 11141   + caddc 11143  *cxr 11279   < clt 11280   / cdiv 11903  cn 12245  +crp 13009  ∞Metcxmet 21281  Metcmet 21282  ballcbl 21283  MetOpencmopn 21286  Topctop 22839  Clsdccld 22964  clsccl 22966  𝑡clm 23174  Cauccau 25225  CMetccmet 25226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-n0 12506  df-z 12592  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-ico 13365  df-rest 17407  df-topgen 17428  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-fbas 21293  df-fg 21294  df-top 22840  df-topon 22857  df-bases 22893  df-cld 22967  df-ntr 22968  df-cls 22969  df-nei 23046  df-lm 23177  df-fil 23794  df-fm 23886  df-flim 23887  df-flf 23888  df-cfil 25227  df-cau 25228  df-cmet 25229
This theorem is referenced by:  bcthlem5  25300
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