MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caublcls Structured version   Visualization version   GIF version

Theorem caublcls 25357
Description: The convergent point of a sequence of nested balls is in the closures of any of the balls (i.e. it is in the intersection of the closures). Indeed, it is the only point in the intersection because a metric space is Hausdorff, but we don't prove this here. (Contributed by Mario Carneiro, 21-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
caubl.2 (𝜑𝐷 ∈ (∞Met‘𝑋))
caubl.3 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
caubl.4 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))
caublcls.6 𝐽 = (MetOpen‘𝐷)
Assertion
Ref Expression
caublcls ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹𝐴))))
Distinct variable groups:   𝐷,𝑛   𝑛,𝐹   𝑛,𝑋
Allowed substitution hints:   𝜑(𝑛)   𝐴(𝑛)   𝑃(𝑛)   𝐽(𝑛)

Proof of Theorem caublcls
Dummy variables 𝑘 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . 2 (ℤ𝐴) = (ℤ𝐴)
2 caubl.2 . . . 4 (𝜑𝐷 ∈ (∞Met‘𝑋))
323ad2ant1 1132 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
4 caublcls.6 . . . 4 𝐽 = (MetOpen‘𝐷)
54mopntopon 24465 . . 3 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
63, 5syl 17 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝑋))
7 simp3 1137 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
87nnzd 12638 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐴 ∈ ℤ)
9 simp2 1136 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (1st𝐹)(⇝𝑡𝐽)𝑃)
10 2fveq3 6912 . . . . . . . 8 (𝑟 = 𝐴 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝐴)))
1110sseq1d 4027 . . . . . . 7 (𝑟 = 𝐴 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1211imbi2d 340 . . . . . 6 (𝑟 = 𝐴 → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
13 2fveq3 6912 . . . . . . . 8 (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑘)))
1413sseq1d 4027 . . . . . . 7 (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1514imbi2d 340 . . . . . 6 (𝑟 = 𝑘 → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
16 2fveq3 6912 . . . . . . . 8 (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
1716sseq1d 4027 . . . . . . 7 (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1817imbi2d 340 . . . . . 6 (𝑟 = (𝑘 + 1) → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
19 ssid 4018 . . . . . . 7 ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))
20192a1i 12 . . . . . 6 (𝐴 ∈ ℤ → ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
21 caubl.4 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))
22 eluznn 12958 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝐴)) → 𝑘 ∈ ℕ)
23 fvoveq1 7454 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
2423fveq2d 6911 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
25 2fveq3 6912 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘(𝐹𝑘)))
2624, 25sseq12d 4029 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘))))
2726rspccva 3621 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2821, 22, 27syl2an 596 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝐴))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2928anassrs 467 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
30 sstr2 4002 . . . . . . . . 9 (((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
3129, 30syl 17 . . . . . . . 8 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
3231expcom 413 . . . . . . 7 (𝑘 ∈ (ℤ𝐴) → ((𝜑𝐴 ∈ ℕ) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
3332a2d 29 . . . . . 6 (𝑘 ∈ (ℤ𝐴) → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) → ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
3412, 15, 18, 15, 20, 33uzind4 12946 . . . . 5 (𝑘 ∈ (ℤ𝐴) → ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
3534impcom 407 . . . 4 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))
36353adantl2 1166 . . 3 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))
373adantr 480 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
38 simpl1 1190 . . . . . . . 8 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝜑)
39 caubl.3 . . . . . . . 8 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
4038, 39syl 17 . . . . . . 7 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
41223ad2antl3 1186 . . . . . . 7 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝑘 ∈ ℕ)
4240, 41ffvelcdmd 7105 . . . . . 6 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (𝐹𝑘) ∈ (𝑋 × ℝ+))
43 xp1st 8045 . . . . . 6 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
4442, 43syl 17 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
45 xp2nd 8046 . . . . . 6 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
4642, 45syl 17 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
47 blcntr 24439 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ+) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
4837, 44, 46, 47syl3anc 1370 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
49 fvco3 7008 . . . . 5 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
5040, 41, 49syl2anc 584 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
51 1st2nd2 8052 . . . . . . 7 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
5242, 51syl 17 . . . . . 6 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
5352fveq2d 6911 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
54 df-ov 7434 . . . . 5 ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
5553, 54eqtr4di 2793 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
5648, 50, 553eltr4d 2854 . . 3 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹𝑘)))
5736, 56sseldd 3996 . 2 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹𝐴)))
5839ffvelcdmda 7104 . . . . . . 7 ((𝜑𝐴 ∈ ℕ) → (𝐹𝐴) ∈ (𝑋 × ℝ+))
59583adant2 1130 . . . . . 6 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (𝐹𝐴) ∈ (𝑋 × ℝ+))
60 1st2nd2 8052 . . . . . 6 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6159, 60syl 17 . . . . 5 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6261fveq2d 6911 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
63 df-ov 7434 . . . 4 ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6462, 63eqtr4di 2793 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) = ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))))
65 xp1st 8045 . . . . 5 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝐴)) ∈ 𝑋)
6659, 65syl 17 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (1st ‘(𝐹𝐴)) ∈ 𝑋)
67 xp2nd 8046 . . . . . 6 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝐴)) ∈ ℝ+)
6859, 67syl 17 . . . . 5 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (2nd ‘(𝐹𝐴)) ∈ ℝ+)
6968rpxrd 13076 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (2nd ‘(𝐹𝐴)) ∈ ℝ*)
70 blssm 24444 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝐴)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝐴)) ∈ ℝ*) → ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) ⊆ 𝑋)
713, 66, 69, 70syl3anc 1370 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) ⊆ 𝑋)
7264, 71eqsstrd 4034 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ 𝑋)
731, 6, 8, 9, 57, 72lmcls 23326 1 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963  cop 4637   class class class wbr 5148   × cxp 5687  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  1c1 11154   + caddc 11156  *cxr 11292  cn 12264  cz 12611  cuz 12876  +crp 13032  ∞Metcxmet 21367  ballcbl 21369  MetOpencmopn 21372  TopOnctopon 22932  clsccl 23042  𝑡clm 23250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-bl 21377  df-mopn 21378  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-lm 23253
This theorem is referenced by:  bcthlem3  25374  heiborlem8  37805
  Copyright terms: Public domain W3C validator