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

Theorem caublcls 25328
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 2726 . 2 (ℤ𝐴) = (ℤ𝐴)
2 caubl.2 . . . 4 (𝜑𝐷 ∈ (∞Met‘𝑋))
323ad2ant1 1130 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
4 caublcls.6 . . . 4 𝐽 = (MetOpen‘𝐷)
54mopntopon 24436 . . 3 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
63, 5syl 17 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝑋))
7 simp3 1135 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
87nnzd 12637 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐴 ∈ ℤ)
9 simp2 1134 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (1st𝐹)(⇝𝑡𝐽)𝑃)
10 2fveq3 6906 . . . . . . . 8 (𝑟 = 𝐴 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝐴)))
1110sseq1d 4011 . . . . . . 7 (𝑟 = 𝐴 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1211imbi2d 339 . . . . . 6 (𝑟 = 𝐴 → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
13 2fveq3 6906 . . . . . . . 8 (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑘)))
1413sseq1d 4011 . . . . . . 7 (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1514imbi2d 339 . . . . . 6 (𝑟 = 𝑘 → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
16 2fveq3 6906 . . . . . . . 8 (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
1716sseq1d 4011 . . . . . . 7 (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1817imbi2d 339 . . . . . 6 (𝑟 = (𝑘 + 1) → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
19 ssid 4002 . . . . . . 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 12954 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝐴)) → 𝑘 ∈ ℕ)
23 fvoveq1 7447 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
2423fveq2d 6905 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
25 2fveq3 6906 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘(𝐹𝑘)))
2624, 25sseq12d 4013 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘))))
2726rspccva 3607 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2821, 22, 27syl2an 594 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝐴))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2928anassrs 466 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
30 sstr2 3986 . . . . . . . . 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 412 . . . . . . 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 12942 . . . . 5 (𝑘 ∈ (ℤ𝐴) → ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
3534impcom 406 . . . 4 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))
36353adantl2 1164 . . 3 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))
373adantr 479 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
38 simpl1 1188 . . . . . . . 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 1184 . . . . . . 7 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝑘 ∈ ℕ)
4240, 41ffvelcdmd 7099 . . . . . 6 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (𝐹𝑘) ∈ (𝑋 × ℝ+))
43 xp1st 8035 . . . . . 6 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
4442, 43syl 17 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
45 xp2nd 8036 . . . . . 6 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
4642, 45syl 17 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
47 blcntr 24410 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ+) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
4837, 44, 46, 47syl3anc 1368 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
49 fvco3 7001 . . . . 5 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
5040, 41, 49syl2anc 582 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
51 1st2nd2 8042 . . . . . . 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 6905 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
54 df-ov 7427 . . . . 5 ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
5553, 54eqtr4di 2784 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
5648, 50, 553eltr4d 2841 . . 3 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹𝑘)))
5736, 56sseldd 3980 . 2 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹𝐴)))
5839ffvelcdmda 7098 . . . . . . 7 ((𝜑𝐴 ∈ ℕ) → (𝐹𝐴) ∈ (𝑋 × ℝ+))
59583adant2 1128 . . . . . 6 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (𝐹𝐴) ∈ (𝑋 × ℝ+))
60 1st2nd2 8042 . . . . . 6 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6159, 60syl 17 . . . . 5 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6261fveq2d 6905 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
63 df-ov 7427 . . . 4 ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6462, 63eqtr4di 2784 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) = ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))))
65 xp1st 8035 . . . . 5 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝐴)) ∈ 𝑋)
6659, 65syl 17 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (1st ‘(𝐹𝐴)) ∈ 𝑋)
67 xp2nd 8036 . . . . . 6 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝐴)) ∈ ℝ+)
6859, 67syl 17 . . . . 5 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (2nd ‘(𝐹𝐴)) ∈ ℝ+)
6968rpxrd 13071 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (2nd ‘(𝐹𝐴)) ∈ ℝ*)
70 blssm 24415 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝐴)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝐴)) ∈ ℝ*) → ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) ⊆ 𝑋)
713, 66, 69, 70syl3anc 1368 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) ⊆ 𝑋)
7264, 71eqsstrd 4018 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ 𝑋)
731, 6, 8, 9, 57, 72lmcls 23297 1 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wss 3947  cop 4639   class class class wbr 5153   × cxp 5680  ccom 5686  wf 6550  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  1c1 11159   + caddc 11161  *cxr 11297  cn 12264  cz 12610  cuz 12874  +crp 13028  ∞Metcxmet 21328  ballcbl 21330  MetOpencmopn 21333  TopOnctopon 22903  clsccl 23013  𝑡clm 23221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-map 8857  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-sup 9485  df-inf 9486  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-n0 12525  df-z 12611  df-uz 12875  df-q 12985  df-rp 13029  df-xneg 13146  df-xadd 13147  df-xmul 13148  df-topgen 17458  df-psmet 21335  df-xmet 21336  df-bl 21338  df-mopn 21339  df-top 22887  df-topon 22904  df-bases 22940  df-cld 23014  df-ntr 23015  df-cls 23016  df-lm 23224
This theorem is referenced by:  bcthlem3  25345  heiborlem8  37519
  Copyright terms: Public domain W3C validator