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Theorem caublcls 23010
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 2626 . 2 (ℤ𝐴) = (ℤ𝐴)
2 caubl.2 . . . 4 (𝜑𝐷 ∈ (∞Met‘𝑋))
323ad2ant1 1080 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
4 caublcls.6 . . . 4 𝐽 = (MetOpen‘𝐷)
54mopntopon 22149 . . 3 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
63, 5syl 17 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝑋))
7 simp3 1061 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
87nnzd 11425 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐴 ∈ ℤ)
9 simp2 1060 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (1st𝐹)(⇝𝑡𝐽)𝑃)
10 fveq2 6150 . . . . . . . . 9 (𝑟 = 𝐴 → (𝐹𝑟) = (𝐹𝐴))
1110fveq2d 6154 . . . . . . . 8 (𝑟 = 𝐴 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝐴)))
1211sseq1d 3616 . . . . . . 7 (𝑟 = 𝐴 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1312imbi2d 330 . . . . . 6 (𝑟 = 𝐴 → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
14 fveq2 6150 . . . . . . . . 9 (𝑟 = 𝑘 → (𝐹𝑟) = (𝐹𝑘))
1514fveq2d 6154 . . . . . . . 8 (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑘)))
1615sseq1d 3616 . . . . . . 7 (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1716imbi2d 330 . . . . . 6 (𝑟 = 𝑘 → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
18 fveq2 6150 . . . . . . . . 9 (𝑟 = (𝑘 + 1) → (𝐹𝑟) = (𝐹‘(𝑘 + 1)))
1918fveq2d 6154 . . . . . . . 8 (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
2019sseq1d 3616 . . . . . . 7 (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
2120imbi2d 330 . . . . . 6 (𝑟 = (𝑘 + 1) → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
22 ssid 3608 . . . . . . 7 ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))
23222a1i 12 . . . . . 6 (𝐴 ∈ ℤ → ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
24 caubl.4 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))
25 eluznn 11702 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝐴)) → 𝑘 ∈ ℕ)
26 oveq1 6612 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
2726fveq2d 6154 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
2827fveq2d 6154 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
29 fveq2 6150 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
3029fveq2d 6154 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘(𝐹𝑘)))
3128, 30sseq12d 3618 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘))))
3231rspccva 3299 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
3324, 25, 32syl2an 494 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝐴))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
3433anassrs 679 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
35 sstr2 3595 . . . . . . . . 9 (((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
3634, 35syl 17 . . . . . . . 8 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
3736expcom 451 . . . . . . 7 (𝑘 ∈ (ℤ𝐴) → ((𝜑𝐴 ∈ ℕ) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
3837a2d 29 . . . . . 6 (𝑘 ∈ (ℤ𝐴) → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) → ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
3913, 17, 21, 17, 23, 38uzind4 11690 . . . . 5 (𝑘 ∈ (ℤ𝐴) → ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
4039impcom 446 . . . 4 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))
41403adantl2 1216 . . 3 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))
423adantr 481 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
43 simpl1 1062 . . . . . . . 8 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝜑)
44 caubl.3 . . . . . . . 8 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
4543, 44syl 17 . . . . . . 7 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
46253ad2antl3 1223 . . . . . . 7 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝑘 ∈ ℕ)
4745, 46ffvelrnd 6317 . . . . . 6 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (𝐹𝑘) ∈ (𝑋 × ℝ+))
48 xp1st 7146 . . . . . 6 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
4947, 48syl 17 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
50 xp2nd 7147 . . . . . 6 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
5147, 50syl 17 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
52 blcntr 22123 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ+) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
5342, 49, 51, 52syl3anc 1323 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
54 fvco3 6233 . . . . 5 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
5545, 46, 54syl2anc 692 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
56 1st2nd2 7153 . . . . . . 7 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
5747, 56syl 17 . . . . . 6 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
5857fveq2d 6154 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
59 df-ov 6608 . . . . 5 ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
6058, 59syl6eqr 2678 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
6153, 55, 603eltr4d 2719 . . 3 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹𝑘)))
6241, 61sseldd 3589 . 2 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹𝐴)))
6344ffvelrnda 6316 . . . . . . 7 ((𝜑𝐴 ∈ ℕ) → (𝐹𝐴) ∈ (𝑋 × ℝ+))
64633adant2 1078 . . . . . 6 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (𝐹𝐴) ∈ (𝑋 × ℝ+))
65 1st2nd2 7153 . . . . . 6 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6664, 65syl 17 . . . . 5 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6766fveq2d 6154 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
68 df-ov 6608 . . . 4 ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6967, 68syl6eqr 2678 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) = ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))))
70 xp1st 7146 . . . . 5 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝐴)) ∈ 𝑋)
7164, 70syl 17 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (1st ‘(𝐹𝐴)) ∈ 𝑋)
72 xp2nd 7147 . . . . . 6 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝐴)) ∈ ℝ+)
7364, 72syl 17 . . . . 5 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (2nd ‘(𝐹𝐴)) ∈ ℝ+)
7473rpxrd 11817 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (2nd ‘(𝐹𝐴)) ∈ ℝ*)
75 blssm 22128 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝐴)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝐴)) ∈ ℝ*) → ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) ⊆ 𝑋)
763, 71, 74, 75syl3anc 1323 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) ⊆ 𝑋)
7769, 76eqsstrd 3623 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ 𝑋)
781, 6, 8, 9, 62, 77lmcls 21011 1 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wss 3560  cop 4159   class class class wbr 4618   × cxp 5077  ccom 5083  wf 5846  cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  1c1 9882   + caddc 9884  *cxr 10018  cn 10965  cz 11322  cuz 11631  +crp 11776  ∞Metcxmt 19645  ballcbl 19647  MetOpencmopn 19650  TopOnctopon 20613  clsccl 20727  𝑡clm 20935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-sup 8293  df-inf 8294  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-topgen 16020  df-psmet 19652  df-xmet 19653  df-bl 19655  df-mopn 19656  df-top 20616  df-bases 20617  df-topon 20618  df-cld 20728  df-ntr 20729  df-cls 20730  df-lm 20938
This theorem is referenced by:  bcthlem3  23026  heiborlem8  33235
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