Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sstotbnd3 Structured version   Visualization version   GIF version

Theorem sstotbnd3 35048
Description: Use a net that is not necessarily finite, but for which only finitely many balls meet the subset. (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypothesis
Ref Expression
sstotbnd.2 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
Assertion
Ref Expression
sstotbnd3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
Distinct variable groups:   𝑣,𝑑,𝑥,𝑀   𝑋,𝑑,𝑣,𝑥   𝑁,𝑑,𝑣,𝑥   𝑌,𝑑,𝑣,𝑥

Proof of Theorem sstotbnd3
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstotbnd.2 . . . 4 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
21sstotbnd2 35046 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)))
3 elin 4169 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ∈ 𝒫 𝑋𝑣 ∈ Fin))
4 rabfi 8737 . . . . . . . . . 10 (𝑣 ∈ Fin → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)
54anim2i 618 . . . . . . . . 9 ((𝑣 ∈ 𝒫 𝑋𝑣 ∈ Fin) → (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
63, 5sylbi 219 . . . . . . . 8 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
76anim2i 618 . . . . . . 7 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
87ancoms 461 . . . . . 6 ((𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)) → (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
9 an12 643 . . . . . 6 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) ↔ (𝑣 ∈ 𝒫 𝑋 ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
108, 9sylib 220 . . . . 5 ((𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)) → (𝑣 ∈ 𝒫 𝑋 ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
1110reximi2 3244 . . . 4 (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
1211ralimi 3160 . . 3 (∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → ∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
132, 12syl6bi 255 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) → ∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
14 ssrab2 4056 . . . . . . . 8 {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑣
15 elpwi 4551 . . . . . . . . 9 (𝑣 ∈ 𝒫 𝑋𝑣𝑋)
1615ad2antlr 725 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → 𝑣𝑋)
1714, 16sstrid 3978 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑋)
18 simprr 771 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)
19 elfpw 8820 . . . . . . 7 ({𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ (𝒫 𝑋 ∩ Fin) ↔ ({𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
2017, 18, 19sylanbrc 585 . . . . . 6 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ (𝒫 𝑋 ∩ Fin))
21 ssel2 3962 . . . . . . . . . . . 12 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → 𝑧 𝑥𝑣 (𝑥(ball‘𝑀)𝑑))
22 eliun 4916 . . . . . . . . . . . 12 (𝑧 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))
2321, 22sylib 220 . . . . . . . . . . 11 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → ∃𝑥𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))
24 inelcm 4414 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅)
2524expcom 416 . . . . . . . . . . . . . 14 (𝑧𝑌 → (𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅))
2625ancrd 554 . . . . . . . . . . . . 13 (𝑧𝑌 → (𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))))
2726reximdv 3273 . . . . . . . . . . . 12 (𝑧𝑌 → (∃𝑥𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → ∃𝑥𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))))
2827impcom 410 . . . . . . . . . . 11 ((∃𝑥𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → ∃𝑥𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
2923, 28sylancom 590 . . . . . . . . . 10 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → ∃𝑥𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
30 eliun 4916 . . . . . . . . . . 11 (𝑧 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑) ↔ ∃𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅}𝑧 ∈ (𝑦(ball‘𝑀)𝑑))
31 oveq1 7157 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)𝑑))
3231eleq2d 2898 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑧 ∈ (𝑦(ball‘𝑀)𝑑) ↔ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
3332rexrab2 3693 . . . . . . . . . . 11 (∃𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅}𝑧 ∈ (𝑦(ball‘𝑀)𝑑) ↔ ∃𝑥𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
3430, 33bitri 277 . . . . . . . . . 10 (𝑧 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑) ↔ ∃𝑥𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
3529, 34sylibr 236 . . . . . . . . 9 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → 𝑧 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
3635ex 415 . . . . . . . 8 (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → (𝑧𝑌𝑧 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑)))
3736ssrdv 3973 . . . . . . 7 (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → 𝑌 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
3837ad2antrl 726 . . . . . 6 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → 𝑌 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
39 iuneq1 4928 . . . . . . . 8 (𝑤 = {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} → 𝑦𝑤 (𝑦(ball‘𝑀)𝑑) = 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
4039sseq2d 3999 . . . . . . 7 (𝑤 = {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} → (𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑) ↔ 𝑌 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑)))
4140rspcev 3623 . . . . . 6 (({𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑)) → ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑))
4220, 38, 41syl2anc 586 . . . . 5 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑))
4342rexlimdva2 3287 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin) → ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑)))
4443ralimdv 3178 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin) → ∀𝑑 ∈ ℝ+𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑)))
451sstotbnd2 35046 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑)))
4644, 45sylibrd 261 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin) → 𝑁 ∈ (TotBnd‘𝑌)))
4713, 46impbid 214 1 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wne 3016  wral 3138  wrex 3139  {crab 3142  cin 3935  wss 3936  c0 4291  𝒫 cpw 4539   ciun 4912   × cxp 5548  cres 5552  cfv 6350  (class class class)co 7150  Fincfn 8503  +crp 12383  Metcmet 20525  ballcbl 20526  TotBndctotbnd 35038
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-2 11694  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-psmet 20531  df-xmet 20532  df-met 20533  df-bl 20534  df-totbnd 35040
This theorem is referenced by:  cntotbnd  35068
  Copyright terms: Public domain W3C validator