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Theorem bcthlem5 24692
Description: Lemma for bcth 24693. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 13889, which in the form used here accepts a "selection" function 𝐹 from each element of 𝐾 to a nonempty subset of 𝐾, and the result function 𝑔 maps 𝑔(𝑛 + 1) to an element of 𝐹(𝑛, 𝑔(𝑛)). The trick here is thus in the choice of 𝐹 and 𝐾: we let 𝐾 be the set of all tagged nonempty open sets (tagged here meaning that we have a point and an open set, in an ordered pair), and 𝐹(𝑘, ⟨𝑥, 𝑧⟩) gives the set of all balls of size less than 1 / 𝑘, tagged by their centers, whose closures fit within the given open set 𝑧 and miss 𝑀(𝑘).

Since 𝑀(𝑘) is closed, 𝑧𝑀(𝑘) is open and also nonempty, since 𝑧 is nonempty and 𝑀(𝑘) has empty interior. Then there is some ball contained in it, and hence our function 𝐹 is valid (it never maps to the empty set). Now starting at a point in the interior of ran 𝑀, DC gives us the function 𝑔 all whose elements are constrained by 𝐹 acting on the previous value. (This is all proven in this lemma.) Now 𝑔 is a sequence of tagged open balls, forming an inclusion chain (see bcthlem2 24689) and whose sizes tend to zero, since they are bounded above by 1 / 𝑘. Thus, the centers of these balls form a Cauchy sequence, and converge to a point 𝑥 (see bcthlem4 24691). Since the inclusion chain also ensures the closure of each ball is in the previous ball, the point 𝑥 must be in all these balls (see bcthlem3 24690) and hence misses each 𝑀(𝑘), contradicting the fact that 𝑥 is in the interior of ran 𝑀 (which was the starting point). (Contributed by Mario Carneiro, 6-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))
bcthlem5.7 (𝜑 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
Assertion
Ref Expression
bcthlem5 (𝜑 → ((int‘𝐽)‘ ran 𝑀) = ∅)
Distinct variable groups:   𝑘,𝑟,𝑥,𝑧,𝐷   𝑘,𝐹,𝑟,𝑥,𝑧   𝑘,𝐽,𝑟,𝑥,𝑧   𝑘,𝑀,𝑟,𝑥,𝑧   𝜑,𝑘,𝑟,𝑥,𝑧   𝑘,𝑋,𝑟,𝑥,𝑧

Proof of Theorem bcthlem5
Dummy variables 𝑛 𝑔 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bcthlem.4 . . . . . 6 (𝜑𝐷 ∈ (CMet‘𝑋))
2 cmetmet 24650 . . . . . 6 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3 metxmet 23687 . . . . . 6 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
41, 2, 33syl 18 . . . . 5 (𝜑𝐷 ∈ (∞Met‘𝑋))
5 bcth.2 . . . . . . . 8 𝐽 = (MetOpen‘𝐷)
65mopntop 23793 . . . . . . 7 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
74, 6syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
8 bcthlem.6 . . . . . . . . 9 (𝜑𝑀:ℕ⟶(Clsd‘𝐽))
98frnd 6676 . . . . . . . 8 (𝜑 → ran 𝑀 ⊆ (Clsd‘𝐽))
10 eqid 2736 . . . . . . . . 9 𝐽 = 𝐽
1110cldss2 22381 . . . . . . . 8 (Clsd‘𝐽) ⊆ 𝒫 𝐽
129, 11sstrdi 3956 . . . . . . 7 (𝜑 → ran 𝑀 ⊆ 𝒫 𝐽)
13 sspwuni 5060 . . . . . . 7 (ran 𝑀 ⊆ 𝒫 𝐽 ran 𝑀 𝐽)
1412, 13sylib 217 . . . . . 6 (𝜑 ran 𝑀 𝐽)
1510ntropn 22400 . . . . . 6 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((int‘𝐽)‘ ran 𝑀) ∈ 𝐽)
167, 14, 15syl2anc 584 . . . . 5 (𝜑 → ((int‘𝐽)‘ ran 𝑀) ∈ 𝐽)
174, 16jca 512 . . . 4 (𝜑 → (𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘ ran 𝑀) ∈ 𝐽))
185mopni2 23849 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘ ran 𝑀) ∈ 𝐽𝑛 ∈ ((int‘𝐽)‘ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
19183expa 1118 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘ ran 𝑀) ∈ 𝐽) ∧ 𝑛 ∈ ((int‘𝐽)‘ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
2017, 19sylan 580 . . 3 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
215mopnuni 23794 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
224, 21syl 17 . . . . . . . . . . 11 (𝜑𝑋 = 𝐽)
2310topopn 22255 . . . . . . . . . . . 12 (𝐽 ∈ Top → 𝐽𝐽)
247, 23syl 17 . . . . . . . . . . 11 (𝜑 𝐽𝐽)
2522, 24eqeltrd 2838 . . . . . . . . . 10 (𝜑𝑋𝐽)
26 reex 11142 . . . . . . . . . . 11 ℝ ∈ V
27 rpssre 12922 . . . . . . . . . . 11 + ⊆ ℝ
2826, 27ssexi 5279 . . . . . . . . . 10 + ∈ V
29 xpexg 7684 . . . . . . . . . 10 ((𝑋𝐽 ∧ ℝ+ ∈ V) → (𝑋 × ℝ+) ∈ V)
3025, 28, 29sylancl 586 . . . . . . . . 9 (𝜑 → (𝑋 × ℝ+) ∈ V)
31303ad2ant1 1133 . . . . . . . 8 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → (𝑋 × ℝ+) ∈ V)
3210ntrss3 22411 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((int‘𝐽)‘ ran 𝑀) ⊆ 𝐽)
337, 14, 32syl2anc 584 . . . . . . . . . . . 12 (𝜑 → ((int‘𝐽)‘ ran 𝑀) ⊆ 𝐽)
3433, 22sseqtrrd 3985 . . . . . . . . . . 11 (𝜑 → ((int‘𝐽)‘ ran 𝑀) ⊆ 𝑋)
35343ad2ant1 1133 . . . . . . . . . 10 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ((int‘𝐽)‘ ran 𝑀) ⊆ 𝑋)
36 simp2 1137 . . . . . . . . . 10 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑛 ∈ ((int‘𝐽)‘ ran 𝑀))
3735, 36sseldd 3945 . . . . . . . . 9 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑛𝑋)
38 simp3 1138 . . . . . . . . 9 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ+)
3937, 38opelxpd 5671 . . . . . . . 8 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ⟨𝑛, 𝑚⟩ ∈ (𝑋 × ℝ+))
40 opabssxp 5724 . . . . . . . . . . . . 13 {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ⊆ (𝑋 × ℝ+)
41 elpw2g 5301 . . . . . . . . . . . . . . 15 ((𝑋 × ℝ+) ∈ V → ({⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ∈ 𝒫 (𝑋 × ℝ+) ↔ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ⊆ (𝑋 × ℝ+)))
4230, 41syl 17 . . . . . . . . . . . . . 14 (𝜑 → ({⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ∈ 𝒫 (𝑋 × ℝ+) ↔ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ⊆ (𝑋 × ℝ+)))
4342adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ({⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ∈ 𝒫 (𝑋 × ℝ+) ↔ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ⊆ (𝑋 × ℝ+)))
4440, 43mpbiri 257 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ∈ 𝒫 (𝑋 × ℝ+))
45 bcthlem5.7 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
46 simpl 483 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) → 𝑘 ∈ ℕ)
47 rspa 3231 . . . . . . . . . . . . . . . 16 ((∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅ ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(𝑀𝑘)) = ∅)
4845, 46, 47syl2an 596 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((int‘𝐽)‘(𝑀𝑘)) = ∅)
49 ssdif0 4323 . . . . . . . . . . . . . . . . 17 (((ball‘𝐷)‘𝑧) ⊆ (𝑀𝑘) ↔ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) = ∅)
50 1st2nd2 7960 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑋 × ℝ+) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
5150ad2antll 727 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
5251fveq2d 6846 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘⟨(1st𝑧), (2nd𝑧)⟩))
53 df-ov 7360 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑧)(ball‘𝐷)(2nd𝑧)) = ((ball‘𝐷)‘⟨(1st𝑧), (2nd𝑧)⟩)
5452, 53eqtr4di 2794 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) = ((1st𝑧)(ball‘𝐷)(2nd𝑧)))
554adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝐷 ∈ (∞Met‘𝑋))
56 xp1st 7953 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ (𝑋 × ℝ+) → (1st𝑧) ∈ 𝑋)
5756ad2antll 727 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (1st𝑧) ∈ 𝑋)
58 xp2nd 7954 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ (𝑋 × ℝ+) → (2nd𝑧) ∈ ℝ+)
5958ad2antll 727 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (2nd𝑧) ∈ ℝ+)
60 bln0 23768 . . . . . . . . . . . . . . . . . . . 20 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st𝑧) ∈ 𝑋 ∧ (2nd𝑧) ∈ ℝ+) → ((1st𝑧)(ball‘𝐷)(2nd𝑧)) ≠ ∅)
6155, 57, 59, 60syl3anc 1371 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((1st𝑧)(ball‘𝐷)(2nd𝑧)) ≠ ∅)
6254, 61eqnetrd 3011 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ≠ ∅)
637adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝐽 ∈ Top)
64 ffvelcdm 7032 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀:ℕ⟶(Clsd‘𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ (Clsd‘𝐽))
658, 46, 64syl2an 596 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑀𝑘) ∈ (Clsd‘𝐽))
6610cldss 22380 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝑘) ∈ (Clsd‘𝐽) → (𝑀𝑘) ⊆ 𝐽)
6765, 66syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑀𝑘) ⊆ 𝐽)
6859rpxrd 12958 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (2nd𝑧) ∈ ℝ*)
695blopn 23856 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st𝑧) ∈ 𝑋 ∧ (2nd𝑧) ∈ ℝ*) → ((1st𝑧)(ball‘𝐷)(2nd𝑧)) ∈ 𝐽)
7055, 57, 68, 69syl3anc 1371 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((1st𝑧)(ball‘𝐷)(2nd𝑧)) ∈ 𝐽)
7154, 70eqeltrd 2838 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ∈ 𝐽)
7210ssntr 22409 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ (𝑀𝑘) ⊆ 𝐽) ∧ (((ball‘𝐷)‘𝑧) ∈ 𝐽 ∧ ((ball‘𝐷)‘𝑧) ⊆ (𝑀𝑘))) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀𝑘)))
7372expr 457 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ (𝑀𝑘) ⊆ 𝐽) ∧ ((ball‘𝐷)‘𝑧) ∈ 𝐽) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀𝑘) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀𝑘))))
7463, 67, 71, 73syl21anc 836 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀𝑘) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀𝑘))))
75 ssn0 4360 . . . . . . . . . . . . . . . . . . 19 ((((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀𝑘)) ∧ ((ball‘𝐷)‘𝑧) ≠ ∅) → ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅)
7675expcom 414 . . . . . . . . . . . . . . . . . 18 (((ball‘𝐷)‘𝑧) ≠ ∅ → (((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀𝑘)) → ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅))
7762, 74, 76sylsyld 61 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀𝑘) → ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅))
7849, 77biimtrrid 242 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) = ∅ → ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅))
7978necon2d 2966 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((int‘𝐽)‘(𝑀𝑘)) = ∅ → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ≠ ∅))
8048, 79mpd 15 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ≠ ∅)
81 n0 4306 . . . . . . . . . . . . . . 15 ((((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))
8243ad2ant1 1133 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → 𝐷 ∈ (∞Met‘𝑋))
8310difopn 22385 . . . . . . . . . . . . . . . . . . . . 21 ((((ball‘𝐷)‘𝑧) ∈ 𝐽 ∧ (𝑀𝑘) ∈ (Clsd‘𝐽)) → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ∈ 𝐽)
8471, 65, 83syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ∈ 𝐽)
85843adant3 1132 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ∈ 𝐽)
86 simp3 1138 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))
87 simp2l 1199 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → 𝑘 ∈ ℕ)
88 nnrp 12926 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
8988rpreccld 12967 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ+)
9087, 89syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → (1 / 𝑘) ∈ ℝ+)
915mopni3 23850 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ∈ (∞Met‘𝑋) ∧ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ∈ 𝐽𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) ∧ (1 / 𝑘) ∈ ℝ+) → ∃𝑛 ∈ ℝ+ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))
9282, 85, 86, 90, 91syl31anc 1373 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → ∃𝑛 ∈ ℝ+ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))
93 simp1 1136 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → 𝜑)
94 elssuni 4898 . . . . . . . . . . . . . . . . . . . . . . . 24 (((ball‘𝐷)‘𝑧) ∈ 𝐽 → ((ball‘𝐷)‘𝑧) ⊆ 𝐽)
9571, 94syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ⊆ 𝐽)
9622adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝑋 = 𝐽)
9795, 96sseqtrrd 3985 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ⊆ 𝑋)
9897ssdifssd 4102 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ⊆ 𝑋)
9998sseld 3943 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) → 𝑥𝑋))
100993impia 1117 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → 𝑥𝑋)
101 simp2 1137 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)))
102 rphalfcl 12942 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℝ+ → (𝑛 / 2) ∈ ℝ+)
103 rphalflt 12944 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℝ+ → (𝑛 / 2) < 𝑛)
104 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑟 = (𝑛 / 2) → (𝑟 < 𝑛 ↔ (𝑛 / 2) < 𝑛))
105104rspcev 3581 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 / 2) ∈ ℝ+ ∧ (𝑛 / 2) < 𝑛) → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
106102, 103, 105syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ ℝ+ → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
107106ad2antlr 725 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))) → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
108 df-rex 3074 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑟 ∈ ℝ+ 𝑟 < 𝑛 ↔ ∃𝑟(𝑟 ∈ ℝ+𝑟 < 𝑛))
109 simpr3 1196 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ+)
110109rpred 12957 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ)
111 simpr1 1194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
112111rpred 12957 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑛 ∈ ℝ)
113 simplrl 775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑘 ∈ ℕ)
114113nnrecred 12204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → (1 / 𝑘) ∈ ℝ)
115 simpr2 1195 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑟 < 𝑛)
116 lttr 11231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑟 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → ((𝑟 < 𝑛𝑛 < (1 / 𝑘)) → 𝑟 < (1 / 𝑘)))
117116expdimp 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑟 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) ∧ 𝑟 < 𝑛) → (𝑛 < (1 / 𝑘) → 𝑟 < (1 / 𝑘)))
118110, 112, 114, 115, 117syl31anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → (𝑛 < (1 / 𝑘) → 𝑟 < (1 / 𝑘)))
1194anim1i 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑥𝑋) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋))
120119adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋))
121 rpxr 12924 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
122 rpxr 12924 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑛 ∈ ℝ+𝑛 ∈ ℝ*)
123 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑟 < 𝑛𝑟 < 𝑛)
124121, 122, 1233anim123i 1151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑟 ∈ ℝ+𝑛 ∈ ℝ+𝑟 < 𝑛) → (𝑟 ∈ ℝ*𝑛 ∈ ℝ*𝑟 < 𝑛))
1251243coml 1127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+) → (𝑟 ∈ ℝ*𝑛 ∈ ℝ*𝑟 < 𝑛))
1265blsscls 23863 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋) ∧ (𝑟 ∈ ℝ*𝑛 ∈ ℝ*𝑟 < 𝑛)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (𝑥(ball‘𝐷)𝑛))
127120, 125, 126syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (𝑥(ball‘𝐷)𝑛))
128 sstr2 3951 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (𝑥(ball‘𝐷)𝑛) → ((𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))
129127, 128syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → ((𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))
130118, 129anim12d 609 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → ((𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))))
131 simpllr 774 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑥𝑋)
132131, 109jca 512 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → (𝑥𝑋𝑟 ∈ ℝ+))
133130, 132jctild 526 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → ((𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))))
1341333exp2 1354 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑛 ∈ ℝ+ → (𝑟 < 𝑛 → (𝑟 ∈ ℝ+ → ((𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))))))))
135134com35 98 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑛 ∈ ℝ+ → ((𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → (𝑟 ∈ ℝ+ → (𝑟 < 𝑛 → ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))))))))
136135imp5d 440 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))) → ((𝑟 ∈ ℝ+𝑟 < 𝑛) → ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))))
137136eximdv 1920 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))) → (∃𝑟(𝑟 ∈ ℝ+𝑟 < 𝑛) → ∃𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))))
138108, 137biimtrid 241 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))) → (∃𝑟 ∈ ℝ+ 𝑟 < 𝑛 → ∃𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))))
139107, 138mpd 15 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))) → ∃𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))))
140139rexlimdva2 3154 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (∃𝑛 ∈ ℝ+ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → ∃𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))))
14193, 100, 101, 140syl21anc 836 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → (∃𝑛 ∈ ℝ+ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → ∃𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))))
14292, 141mpd 15 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → ∃𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))))
1431423expia 1121 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) → ∃𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))))
144143eximdv 1920 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (∃𝑥 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) → ∃𝑥𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))))
14581, 144biimtrid 241 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ≠ ∅ → ∃𝑥𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))))
14680, 145mpd 15 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ∃𝑥𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))))
147 opabn0 5510 . . . . . . . . . . . . 13 ({⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ≠ ∅ ↔ ∃𝑥𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))))
148146, 147sylibr 233 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ≠ ∅)
149 eldifsn 4747 . . . . . . . . . . . 12 ({⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ∈ (𝒫 (𝑋 × ℝ+) ∖ {∅}) ↔ ({⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ∈ 𝒫 (𝑋 × ℝ+) ∧ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ≠ ∅))
15044, 148, 149sylanbrc 583 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ∈ (𝒫 (𝑋 × ℝ+) ∖ {∅}))
151150ralrimivva 3197 . . . . . . . . . 10 (𝜑 → ∀𝑘 ∈ ℕ ∀𝑧 ∈ (𝑋 × ℝ+){⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ∈ (𝒫 (𝑋 × ℝ+) ∖ {∅}))
152 bcthlem.5 . . . . . . . . . . 11 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})
153152fmpo 8000 . . . . . . . . . 10 (∀𝑘 ∈ ℕ ∀𝑧 ∈ (𝑋 × ℝ+){⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ∈ (𝒫 (𝑋 × ℝ+) ∖ {∅}) ↔ 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
154151, 153sylib 217 . . . . . . . . 9 (𝜑𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
1551543ad2ant1 1133 . . . . . . . 8 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
156 1z 12533 . . . . . . . . 9 1 ∈ ℤ
157 nnuz 12806 . . . . . . . . 9 ℕ = (ℤ‘1)
158156, 157axdc4uz 13889 . . . . . . . 8 (((𝑋 × ℝ+) ∈ V ∧ ⟨𝑛, 𝑚⟩ ∈ (𝑋 × ℝ+) ∧ 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅})) → ∃𝑔(𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛))))
15931, 39, 155, 158syl3anc 1371 . . . . . . 7 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ∃𝑔(𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛))))
160 simpl1 1191 . . . . . . . . 9 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → 𝜑)
161160, 1syl 17 . . . . . . . 8 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → 𝐷 ∈ (CMet‘𝑋))
162160, 8syl 17 . . . . . . . 8 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → 𝑀:ℕ⟶(Clsd‘𝐽))
163 simpl3 1193 . . . . . . . 8 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → 𝑚 ∈ ℝ+)
16437adantr 481 . . . . . . . 8 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → 𝑛𝑋)
165 simpr1 1194 . . . . . . . 8 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → 𝑔:ℕ⟶(𝑋 × ℝ+))
166 simpr2 1195 . . . . . . . 8 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → (𝑔‘1) = ⟨𝑛, 𝑚⟩)
167 simpr3 1196 . . . . . . . . 9 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))
168 fvoveq1 7380 . . . . . . . . . . 11 (𝑛 = 𝑘 → (𝑔‘(𝑛 + 1)) = (𝑔‘(𝑘 + 1)))
169 id 22 . . . . . . . . . . . 12 (𝑛 = 𝑘𝑛 = 𝑘)
170 fveq2 6842 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (𝑔𝑛) = (𝑔𝑘))
171169, 170oveq12d 7375 . . . . . . . . . . 11 (𝑛 = 𝑘 → (𝑛𝐹(𝑔𝑛)) = (𝑘𝐹(𝑔𝑘)))
172168, 171eleq12d 2832 . . . . . . . . . 10 (𝑛 = 𝑘 → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)) ↔ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘))))
173172cbvralvw 3225 . . . . . . . . 9 (∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)) ↔ ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))
174167, 173sylib 217 . . . . . . . 8 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))
1755, 161, 152, 162, 163, 164, 165, 166, 174bcthlem4 24691 . . . . . . 7 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → ((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) ≠ ∅)
176159, 175exlimddv 1938 . . . . . 6 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) ≠ ∅)
17710ntrss2 22408 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((int‘𝐽)‘ ran 𝑀) ⊆ ran 𝑀)
1787, 14, 177syl2anc 584 . . . . . . . . . 10 (𝜑 → ((int‘𝐽)‘ ran 𝑀) ⊆ ran 𝑀)
179 sstr2 3951 . . . . . . . . . 10 ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀) → (((int‘𝐽)‘ ran 𝑀) ⊆ ran 𝑀 → (𝑛(ball‘𝐷)𝑚) ⊆ ran 𝑀))
180178, 179syl5com 31 . . . . . . . . 9 (𝜑 → ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀) → (𝑛(ball‘𝐷)𝑚) ⊆ ran 𝑀))
181 ssdif0 4323 . . . . . . . . 9 ((𝑛(ball‘𝐷)𝑚) ⊆ ran 𝑀 ↔ ((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) = ∅)
182180, 181syl6ib 250 . . . . . . . 8 (𝜑 → ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀) → ((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) = ∅))
183182necon3ad 2956 . . . . . . 7 (𝜑 → (((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) ≠ ∅ → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀)))
1841833ad2ant1 1133 . . . . . 6 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → (((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) ≠ ∅ → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀)))
185176, 184mpd 15 . . . . 5 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
1861853expa 1118 . . . 4 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀)) ∧ 𝑚 ∈ ℝ+) → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
187186nrexdv 3146 . . 3 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀)) → ¬ ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
18820, 187pm2.65da 815 . 2 (𝜑 → ¬ 𝑛 ∈ ((int‘𝐽)‘ ran 𝑀))
189188eq0rdv 4364 1 (𝜑 → ((int‘𝐽)‘ 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  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   class class class wbr 5105  {copab 5167   × cxp 5631  ran crn 5634  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  cr 11050  1c1 11052   + caddc 11054  *cxr 11188   < clt 11189   / cdiv 11812  cn 12153  2c2 12208  +crp 12915  ∞Metcxmet 20781  Metcmet 20782  ballcbl 20783  MetOpencmopn 20786  Topctop 22242  Clsdccld 22367  intcnt 22368  clsccl 22369  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-inf2 9577  ax-dc 10382  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-1o 8412  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:  bcth  24693
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