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Theorem bcthlem5 23171
Description: Lemma for bcth 23172. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 12823, 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 23168) 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 23170). 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 23169) 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 23130 . . . . . 6 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3 metxmet 22186 . . . . . 6 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
41, 2, 33syl 18 . . . . 5 (𝜑𝐷 ∈ (∞Met‘𝑋))
5 bcth.2 . . . . . . . 8 𝐽 = (MetOpen‘𝐷)
65mopntop 22292 . . . . . . 7 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
74, 6syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
8 bcthlem.6 . . . . . . . . 9 (𝜑𝑀:ℕ⟶(Clsd‘𝐽))
9 frn 6091 . . . . . . . . 9 (𝑀:ℕ⟶(Clsd‘𝐽) → ran 𝑀 ⊆ (Clsd‘𝐽))
108, 9syl 17 . . . . . . . 8 (𝜑 → ran 𝑀 ⊆ (Clsd‘𝐽))
11 eqid 2651 . . . . . . . . 9 𝐽 = 𝐽
1211cldss2 20882 . . . . . . . 8 (Clsd‘𝐽) ⊆ 𝒫 𝐽
1310, 12syl6ss 3648 . . . . . . 7 (𝜑 → ran 𝑀 ⊆ 𝒫 𝐽)
14 sspwuni 4643 . . . . . . 7 (ran 𝑀 ⊆ 𝒫 𝐽 ran 𝑀 𝐽)
1513, 14sylib 208 . . . . . 6 (𝜑 ran 𝑀 𝐽)
1611ntropn 20901 . . . . . 6 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((int‘𝐽)‘ ran 𝑀) ∈ 𝐽)
177, 15, 16syl2anc 694 . . . . 5 (𝜑 → ((int‘𝐽)‘ ran 𝑀) ∈ 𝐽)
184, 17jca 553 . . . 4 (𝜑 → (𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘ ran 𝑀) ∈ 𝐽))
195mopni2 22345 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘ ran 𝑀) ∈ 𝐽𝑛 ∈ ((int‘𝐽)‘ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
20193expa 1284 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘ ran 𝑀) ∈ 𝐽) ∧ 𝑛 ∈ ((int‘𝐽)‘ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
2118, 20sylan 487 . . 3 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
225mopnuni 22293 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
234, 22syl 17 . . . . . . . . . . 11 (𝜑𝑋 = 𝐽)
2411topopn 20759 . . . . . . . . . . . 12 (𝐽 ∈ Top → 𝐽𝐽)
257, 24syl 17 . . . . . . . . . . 11 (𝜑 𝐽𝐽)
2623, 25eqeltrd 2730 . . . . . . . . . 10 (𝜑𝑋𝐽)
27 reex 10065 . . . . . . . . . . 11 ℝ ∈ V
28 rpssre 11881 . . . . . . . . . . 11 + ⊆ ℝ
2927, 28ssexi 4836 . . . . . . . . . 10 + ∈ V
30 xpexg 7002 . . . . . . . . . 10 ((𝑋𝐽 ∧ ℝ+ ∈ V) → (𝑋 × ℝ+) ∈ V)
3126, 29, 30sylancl 695 . . . . . . . . 9 (𝜑 → (𝑋 × ℝ+) ∈ V)
32313ad2ant1 1102 . . . . . . . 8 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → (𝑋 × ℝ+) ∈ V)
3311ntrss3 20912 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((int‘𝐽)‘ ran 𝑀) ⊆ 𝐽)
347, 15, 33syl2anc 694 . . . . . . . . . . . 12 (𝜑 → ((int‘𝐽)‘ ran 𝑀) ⊆ 𝐽)
3534, 23sseqtr4d 3675 . . . . . . . . . . 11 (𝜑 → ((int‘𝐽)‘ ran 𝑀) ⊆ 𝑋)
36353ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ((int‘𝐽)‘ ran 𝑀) ⊆ 𝑋)
37 simp2 1082 . . . . . . . . . 10 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑛 ∈ ((int‘𝐽)‘ ran 𝑀))
3836, 37sseldd 3637 . . . . . . . . 9 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑛𝑋)
39 simp3 1083 . . . . . . . . 9 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ+)
40 opelxpi 5182 . . . . . . . . 9 ((𝑛𝑋𝑚 ∈ ℝ+) → ⟨𝑛, 𝑚⟩ ∈ (𝑋 × ℝ+))
4138, 39, 40syl2anc 694 . . . . . . . 8 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ⟨𝑛, 𝑚⟩ ∈ (𝑋 × ℝ+))
42 opabssxp 5227 . . . . . . . . . . . . 13 {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ⊆ (𝑋 × ℝ+)
43 elpw2g 4857 . . . . . . . . . . . . . . 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+)))
4431, 43syl 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+)))
4544adantr 480 . . . . . . . . . . . . 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+)))
4642, 45mpbiri 248 . . . . . . . . . . . 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+))
47 bcthlem5.7 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)
48 simpl 472 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) → 𝑘 ∈ ℕ)
49 rspa 2959 . . . . . . . . . . . . . . . 16 ((∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅ ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(𝑀𝑘)) = ∅)
5047, 48, 49syl2an 493 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((int‘𝐽)‘(𝑀𝑘)) = ∅)
51 ssdif0 3975 . . . . . . . . . . . . . . . . 17 (((ball‘𝐷)‘𝑧) ⊆ (𝑀𝑘) ↔ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) = ∅)
52 1st2nd2 7249 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑋 × ℝ+) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
5352ad2antll 765 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
5453fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘⟨(1st𝑧), (2nd𝑧)⟩))
55 df-ov 6693 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑧)(ball‘𝐷)(2nd𝑧)) = ((ball‘𝐷)‘⟨(1st𝑧), (2nd𝑧)⟩)
5654, 55syl6eqr 2703 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) = ((1st𝑧)(ball‘𝐷)(2nd𝑧)))
574adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝐷 ∈ (∞Met‘𝑋))
58 xp1st 7242 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ (𝑋 × ℝ+) → (1st𝑧) ∈ 𝑋)
5958ad2antll 765 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (1st𝑧) ∈ 𝑋)
60 xp2nd 7243 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ (𝑋 × ℝ+) → (2nd𝑧) ∈ ℝ+)
6160ad2antll 765 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (2nd𝑧) ∈ ℝ+)
62 bln0 22267 . . . . . . . . . . . . . . . . . . . 20 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st𝑧) ∈ 𝑋 ∧ (2nd𝑧) ∈ ℝ+) → ((1st𝑧)(ball‘𝐷)(2nd𝑧)) ≠ ∅)
6357, 59, 61, 62syl3anc 1366 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((1st𝑧)(ball‘𝐷)(2nd𝑧)) ≠ ∅)
6456, 63eqnetrd 2890 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ≠ ∅)
657adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝐽 ∈ Top)
66 ffvelrn 6397 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀:ℕ⟶(Clsd‘𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀𝑘) ∈ (Clsd‘𝐽))
678, 48, 66syl2an 493 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑀𝑘) ∈ (Clsd‘𝐽))
6811cldss 20881 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝑘) ∈ (Clsd‘𝐽) → (𝑀𝑘) ⊆ 𝐽)
6967, 68syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑀𝑘) ⊆ 𝐽)
7061rpxrd 11911 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (2nd𝑧) ∈ ℝ*)
715blopn 22352 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st𝑧) ∈ 𝑋 ∧ (2nd𝑧) ∈ ℝ*) → ((1st𝑧)(ball‘𝐷)(2nd𝑧)) ∈ 𝐽)
7257, 59, 70, 71syl3anc 1366 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((1st𝑧)(ball‘𝐷)(2nd𝑧)) ∈ 𝐽)
7356, 72eqeltrd 2730 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ∈ 𝐽)
7411ssntr 20910 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ (𝑀𝑘) ⊆ 𝐽) ∧ (((ball‘𝐷)‘𝑧) ∈ 𝐽 ∧ ((ball‘𝐷)‘𝑧) ⊆ (𝑀𝑘))) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀𝑘)))
7574expr 642 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ (𝑀𝑘) ⊆ 𝐽) ∧ ((ball‘𝐷)‘𝑧) ∈ 𝐽) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀𝑘) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀𝑘))))
7665, 69, 73, 75syl21anc 1365 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀𝑘) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀𝑘))))
77 ssn0 4009 . . . . . . . . . . . . . . . . . . 19 ((((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀𝑘)) ∧ ((ball‘𝐷)‘𝑧) ≠ ∅) → ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅)
7877expcom 450 . . . . . . . . . . . . . . . . . 18 (((ball‘𝐷)‘𝑧) ≠ ∅ → (((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀𝑘)) → ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅))
7964, 76, 78sylsyld 61 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀𝑘) → ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅))
8051, 79syl5bir 233 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) = ∅ → ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅))
8180necon2d 2846 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((int‘𝐽)‘(𝑀𝑘)) = ∅ → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ≠ ∅))
8250, 81mpd 15 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ≠ ∅)
83 n0 3964 . . . . . . . . . . . . . . 15 ((((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))
8443ad2ant1 1102 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → 𝐷 ∈ (∞Met‘𝑋))
8511difopn 20886 . . . . . . . . . . . . . . . . . . . . 21 ((((ball‘𝐷)‘𝑧) ∈ 𝐽 ∧ (𝑀𝑘) ∈ (Clsd‘𝐽)) → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ∈ 𝐽)
8673, 67, 85syl2anc 694 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ∈ 𝐽)
87863adant3 1101 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ∈ 𝐽)
88 simp3 1083 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))
89 simp2l 1107 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → 𝑘 ∈ ℕ)
90 nnrp 11880 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
9190rpreccld 11920 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ+)
9289, 91syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → (1 / 𝑘) ∈ ℝ+)
935mopni3 22346 . . . . . . . . . . . . . . . . . . 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))))
9484, 87, 88, 92, 93syl31anc 1369 . . . . . . . . . . . . . . . . . 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))))
95 simp1 1081 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → 𝜑)
96 elssuni 4499 . . . . . . . . . . . . . . . . . . . . . . . 24 (((ball‘𝐷)‘𝑧) ∈ 𝐽 → ((ball‘𝐷)‘𝑧) ⊆ 𝐽)
9773, 96syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ⊆ 𝐽)
9823adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝑋 = 𝐽)
9997, 98sseqtr4d 3675 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ⊆ 𝑋)
10099ssdifssd 3781 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) ⊆ 𝑋)
101100sseld 3635 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)) → 𝑥𝑋))
1021013impia 1280 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → 𝑥𝑋)
103 simp2 1082 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))) → (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)))
104 rphalfcl 11896 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ℝ+ → (𝑛 / 2) ∈ ℝ+)
105 rphalflt 11898 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ℝ+ → (𝑛 / 2) < 𝑛)
106 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑟 = (𝑛 / 2) → (𝑟 < 𝑛 ↔ (𝑛 / 2) < 𝑛))
107106rspcev 3340 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 / 2) ∈ ℝ+ ∧ (𝑛 / 2) < 𝑛) → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
108104, 105, 107syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℝ+ → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
109108ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))) → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
110 df-rex 2947 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑟 ∈ ℝ+ 𝑟 < 𝑛 ↔ ∃𝑟(𝑟 ∈ ℝ+𝑟 < 𝑛))
111 simpr3 1089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ+)
112111rpred 11910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ)
113 simpr1 1087 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
114113rpred 11910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑛 ∈ ℝ)
115 simplrl 817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑘 ∈ ℕ)
116115nnrecred 11104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → (1 / 𝑘) ∈ ℝ)
117 simpr2 1088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑟 < 𝑛)
118 lttr 10152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑟 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → ((𝑟 < 𝑛𝑛 < (1 / 𝑘)) → 𝑟 < (1 / 𝑘)))
119118expdimp 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑟 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) ∧ 𝑟 < 𝑛) → (𝑛 < (1 / 𝑘) → 𝑟 < (1 / 𝑘)))
120112, 114, 116, 117, 119syl31anc 1369 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → (𝑛 < (1 / 𝑘) → 𝑟 < (1 / 𝑘)))
1214anim1i 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑥𝑋) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋))
122121adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋))
123 rpxr 11878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
124 rpxr 11878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑛 ∈ ℝ+𝑛 ∈ ℝ*)
125 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑟 < 𝑛𝑟 < 𝑛)
126123, 124, 1253anim123i 1266 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑟 ∈ ℝ+𝑛 ∈ ℝ+𝑟 < 𝑛) → (𝑟 ∈ ℝ*𝑛 ∈ ℝ*𝑟 < 𝑛))
1271263coml 1292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+) → (𝑟 ∈ ℝ*𝑛 ∈ ℝ*𝑟 < 𝑛))
1285blsscls 22359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋) ∧ (𝑟 ∈ ℝ*𝑛 ∈ ℝ*𝑟 < 𝑛)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (𝑥(ball‘𝐷)𝑛))
129122, 127, 128syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (𝑥(ball‘𝐷)𝑛))
130 sstr2 3643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((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))))
131129, 130syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((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))))
132120, 131anim12d 585 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((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)))))
133 simpllr 815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → 𝑥𝑋)
134133, 111jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑥𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+𝑟 < 𝑛𝑟 ∈ ℝ+)) → (𝑥𝑋𝑟 ∈ ℝ+))
135132, 134jctild 565 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((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))))))
1361353exp2 1307 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((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)))))))))
137136com35 98 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((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)))))))))
138137imp5d 624 . . . . . . . . . . . . . . . . . . . . . . . 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))))))
139138eximdv 1886 . . . . . . . . . . . . . . . . . . . . . . 23 (((((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))))))
140110, 139syl5bi 232 . . . . . . . . . . . . . . . . . . . . . 22 (((((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))))))
141109, 140mpd 15 . . . . . . . . . . . . . . . . . . . . 21 (((((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)))))
142141ex 449 . . . . . . . . . . . . . . . . . . . 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))))))
143142rexlimdva 3060 . . . . . . . . . . . . . . . . . . 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))))))
14495, 102, 103, 143syl21anc 1365 . . . . . . . . . . . . . . . . . 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))))))
14594, 144mpd 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)))))
1461453expia 1286 . . . . . . . . . . . . . . . 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))))))
147146eximdv 1886 . . . . . . . . . . . . . . 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))))))
14883, 147syl5bi 232 . . . . . . . . . . . . . 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))))))
14982, 148mpd 15 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ∃𝑥𝑟((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘)))))
150 opabn0 5035 . . . . . . . . . . . . 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)))))
151149, 150sylibr 224 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))} ≠ ∅)
152 eldifsn 4350 . . . . . . . . . . . 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))))} ≠ ∅))
15346, 151, 152sylanbrc 699 . . . . . . . . . . 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+) ∖ {∅}))
154153ralrimivva 3000 . . . . . . . . . 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+) ∖ {∅}))
155 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))))})
156155fmpt2 7282 . . . . . . . . . 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+) ∖ {∅}))
157154, 156sylib 208 . . . . . . . . 9 (𝜑𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
1581573ad2ant1 1102 . . . . . . . 8 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
159 1z 11445 . . . . . . . . 9 1 ∈ ℤ
160 nnuz 11761 . . . . . . . . 9 ℕ = (ℤ‘1)
161159, 160axdc4uz 12823 . . . . . . . 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))))
16232, 41, 158, 161syl3anc 1366 . . . . . . 7 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ∃𝑔(𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛))))
163 simpl1 1084 . . . . . . . . 9 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → 𝜑)
164163, 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))
165163, 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))
166 simpl3 1086 . . . . . . . 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+)
16738adantr 480 . . . . . . . 8 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)))) → 𝑛𝑋)
168 simpr1 1087 . . . . . . . 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+))
169 simpr2 1088 . . . . . . . 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⟩)
170 simpr3 1089 . . . . . . . . 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)))
171 oveq1 6697 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
172171fveq2d 6233 . . . . . . . . . . 11 (𝑛 = 𝑘 → (𝑔‘(𝑛 + 1)) = (𝑔‘(𝑘 + 1)))
173 id 22 . . . . . . . . . . . 12 (𝑛 = 𝑘𝑛 = 𝑘)
174 fveq2 6229 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (𝑔𝑛) = (𝑔𝑘))
175173, 174oveq12d 6708 . . . . . . . . . . 11 (𝑛 = 𝑘 → (𝑛𝐹(𝑔𝑛)) = (𝑘𝐹(𝑔𝑘)))
176172, 175eleq12d 2724 . . . . . . . . . 10 (𝑛 = 𝑘 → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)) ↔ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘))))
177176cbvralv 3201 . . . . . . . . 9 (∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔𝑛)) ↔ ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))
178170, 177sylib 208 . . . . . . . 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)))
1795, 164, 155, 165, 166, 167, 168, 169, 178bcthlem4 23170 . . . . . . 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) ≠ ∅)
180162, 179exlimddv 1903 . . . . . 6 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) ≠ ∅)
18111ntrss2 20909 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ ran 𝑀 𝐽) → ((int‘𝐽)‘ ran 𝑀) ⊆ ran 𝑀)
1827, 15, 181syl2anc 694 . . . . . . . . . 10 (𝜑 → ((int‘𝐽)‘ ran 𝑀) ⊆ ran 𝑀)
183 sstr2 3643 . . . . . . . . . 10 ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀) → (((int‘𝐽)‘ ran 𝑀) ⊆ ran 𝑀 → (𝑛(ball‘𝐷)𝑚) ⊆ ran 𝑀))
184182, 183syl5com 31 . . . . . . . . 9 (𝜑 → ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀) → (𝑛(ball‘𝐷)𝑚) ⊆ ran 𝑀))
185 ssdif0 3975 . . . . . . . . 9 ((𝑛(ball‘𝐷)𝑚) ⊆ ran 𝑀 ↔ ((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) = ∅)
186184, 185syl6ib 241 . . . . . . . 8 (𝜑 → ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀) → ((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) = ∅))
187186necon3ad 2836 . . . . . . 7 (𝜑 → (((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) ≠ ∅ → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀)))
1881873ad2ant1 1102 . . . . . 6 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → (((𝑛(ball‘𝐷)𝑚) ∖ ran 𝑀) ≠ ∅ → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀)))
189180, 188mpd 15 . . . . 5 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
1901893expa 1284 . . . 4 (((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀)) ∧ 𝑚 ∈ ℝ+) → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
191190nrexdv 3030 . . 3 ((𝜑𝑛 ∈ ((int‘𝐽)‘ ran 𝑀)) → ¬ ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘ ran 𝑀))
19221, 191pm2.65da 599 . 2 (𝜑 → ¬ 𝑛 ∈ ((int‘𝐽)‘ ran 𝑀))
193192eq0rdv 4012 1 (𝜑 → ((int‘𝐽)‘ 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  𝒫 cpw 4191  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685  {copab 4745   × cxp 5141  ran crn 5144  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  cr 9973  1c1 9975   + caddc 9977  *cxr 10111   < clt 10112   / cdiv 10722  cn 11058  2c2 11108  +crp 11870  ∞Metcxmt 19779  Metcme 19780  ballcbl 19781  MetOpencmopn 19784  Topctop 20746  Clsdccld 20868  intcnt 20869  clsccl 20870  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-inf2 8576  ax-dc 9306  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-1o 7605  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:  bcth  23172
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