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Theorem totbndbnd 33920
Description: A totally bounded metric space is bounded. This theorem fails for extended metrics - a bounded extended metric is a metric, but there are totally bounded extended metrics that are not metrics (if we were to weaken istotbnd 33900 to only require that 𝑀 be an extended metric). A counterexample is the discrete extended metric (assigning distinct points distance +∞) on a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
totbndbnd (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Bnd‘𝑋))

Proof of Theorem totbndbnd
Dummy variables 𝑣 𝑑 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 totbndmet 33903 . 2 (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
2 1rp 12039 . . 3 1 ∈ ℝ+
3 istotbnd3 33902 . . . 4 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
43simprbi 484 . . 3 (𝑀 ∈ (TotBnd‘𝑋) → ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5 oveq2 6801 . . . . . . 7 (𝑑 = 1 → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)1))
65iuneq2d 4681 . . . . . 6 (𝑑 = 1 → 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑥𝑣 (𝑥(ball‘𝑀)1))
76eqeq1d 2773 . . . . 5 (𝑑 = 1 → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
87rexbidv 3200 . . . 4 (𝑑 = 1 → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
98rspcv 3456 . . 3 (1 ∈ ℝ+ → (∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
102, 4, 9mpsyl 68 . 2 (𝑀 ∈ (TotBnd‘𝑋) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
11 simplll 758 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑀 ∈ (Met‘𝑋))
12 elfpw 8424 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
1312simplbi 485 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
1413ad2antrl 707 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑣𝑋)
1514sselda 3752 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑧𝑋)
16 simpllr 760 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑦𝑋)
17 metcl 22357 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) → (𝑧𝑀𝑦) ∈ ℝ)
1811, 15, 16, 17syl3anc 1476 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → (𝑧𝑀𝑦) ∈ ℝ)
19 metge0 22370 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) → 0 ≤ (𝑧𝑀𝑦))
2011, 15, 16, 19syl3anc 1476 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 0 ≤ (𝑧𝑀𝑦))
2118, 20ge0p1rpd 12105 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ℝ+)
22 eqid 2771 . . . . . . . . 9 (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
2321, 22fmptd 6527 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)):𝑣⟶ℝ+)
24 frn 6193 . . . . . . . 8 ((𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)):𝑣⟶ℝ+ → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ+)
2523, 24syl 17 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ+)
2612simprbi 484 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
27 mptfi 8421 . . . . . . . . . 10 (𝑣 ∈ Fin → (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
28 rnfi 8405 . . . . . . . . . 10 ((𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
2926, 27, 283syl 18 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
3029ad2antrl 707 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
31 simplr 752 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦𝑋)
32 simprr 756 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
3331, 32eleqtrrd 2853 . . . . . . . . 9 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 𝑥𝑣 (𝑥(ball‘𝑀)1))
34 ne0i 4069 . . . . . . . . 9 (𝑦 𝑥𝑣 (𝑥(ball‘𝑀)1) → 𝑥𝑣 (𝑥(ball‘𝑀)1) ≠ ∅)
35 dm0rn0 5480 . . . . . . . . . . 11 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅)
36 ovex 6823 . . . . . . . . . . . . . . 15 ((𝑧𝑀𝑦) + 1) ∈ V
3736, 22dmmpti 6163 . . . . . . . . . . . . . 14 dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
3837eqeq1i 2776 . . . . . . . . . . . . 13 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ 𝑣 = ∅)
39 iuneq1 4668 . . . . . . . . . . . . 13 (𝑣 = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
4038, 39sylbi 207 . . . . . . . . . . . 12 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
41 0iun 4711 . . . . . . . . . . . 12 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1) = ∅
4240, 41syl6eq 2821 . . . . . . . . . . 11 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = ∅)
4335, 42sylbir 225 . . . . . . . . . 10 (ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = ∅)
4443necon3i 2975 . . . . . . . . 9 ( 𝑥𝑣 (𝑥(ball‘𝑀)1) ≠ ∅ → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
4533, 34, 443syl 18 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
46 rpssre 12046 . . . . . . . . 9 + ⊆ ℝ
4725, 46syl6ss 3764 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
48 ltso 10320 . . . . . . . . 9 < Or ℝ
49 fisupcl 8531 . . . . . . . . 9 (( < Or ℝ ∧ (ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin ∧ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅ ∧ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
5048, 49mpan 670 . . . . . . . 8 ((ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin ∧ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅ ∧ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
5130, 45, 47, 50syl3anc 1476 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
5225, 51sseldd 3753 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+)
53 metxmet 22359 . . . . . . . . . . . . . 14 (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
5453ad2antrr 705 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑀 ∈ (∞Met‘𝑋))
5554adantr 466 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑀 ∈ (∞Met‘𝑋))
56 1red 10257 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 1 ∈ ℝ)
5747, 51sseldd 3753 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5857adantr 466 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5947adantr 466 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
6045adantr 466 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
6130adantr 466 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
62 fimaxre2 11171 . . . . . . . . . . . . . . 15 ((ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ ∧ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin) → ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤𝑑)
6359, 61, 62syl2anc 573 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤𝑑)
6422elrnmpt1 5512 . . . . . . . . . . . . . . . 16 ((𝑧𝑣 ∧ ((𝑧𝑀𝑦) + 1) ∈ V) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6536, 64mpan2 671 . . . . . . . . . . . . . . 15 (𝑧𝑣 → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6665adantl 467 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
67 suprub 11186 . . . . . . . . . . . . . 14 (((ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ ∧ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅ ∧ ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤𝑑) ∧ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))) → ((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
6859, 60, 63, 66, 67syl31anc 1479 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
69 leaddsub 10706 . . . . . . . . . . . . . 14 (((𝑧𝑀𝑦) ∈ ℝ ∧ 1 ∈ ℝ ∧ sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ) → (((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ↔ (𝑧𝑀𝑦) ≤ (sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1)))
7018, 56, 58, 69syl3anc 1476 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → (((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ↔ (𝑧𝑀𝑦) ≤ (sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1)))
7168, 70mpbid 222 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → (𝑧𝑀𝑦) ≤ (sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1))
72 blss2 22429 . . . . . . . . . . . 12 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) ∧ (1 ∈ ℝ ∧ sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ ∧ (𝑧𝑀𝑦) ≤ (sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1))) → (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
7355, 15, 16, 56, 58, 71, 72syl33anc 1491 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
7473ralrimiva 3115 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∀𝑧𝑣 (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
75 nfcv 2913 . . . . . . . . . . . 12 𝑧(𝑥(ball‘𝑀)1)
76 nfcv 2913 . . . . . . . . . . . . 13 𝑧𝑦
77 nfcv 2913 . . . . . . . . . . . . 13 𝑧(ball‘𝑀)
78 nfmpt1 4881 . . . . . . . . . . . . . . 15 𝑧(𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
7978nfrn 5506 . . . . . . . . . . . . . 14 𝑧ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
80 nfcv 2913 . . . . . . . . . . . . . 14 𝑧
81 nfcv 2913 . . . . . . . . . . . . . 14 𝑧 <
8279, 80, 81nfsup 8513 . . . . . . . . . . . . 13 𝑧sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
8376, 77, 82nfov 6821 . . . . . . . . . . . 12 𝑧(𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
8475, 83nfss 3745 . . . . . . . . . . 11 𝑧(𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85 nfv 1995 . . . . . . . . . . 11 𝑥(𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
86 oveq1 6800 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥(ball‘𝑀)1) = (𝑧(ball‘𝑀)1))
8786sseq1d 3781 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))))
8884, 85, 87cbvral 3316 . . . . . . . . . 10 (∀𝑥𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ ∀𝑧𝑣 (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
8974, 88sylibr 224 . . . . . . . . 9 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∀𝑥𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
90 iunss 4695 . . . . . . . . 9 ( 𝑥𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ ∀𝑥𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9189, 90sylibr 224 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑥𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9232, 91eqsstr3d 3789 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑋 ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9352rpxrd 12076 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*)
94 blssm 22443 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*) → (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ⊆ 𝑋)
9554, 31, 93, 94syl3anc 1476 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ⊆ 𝑋)
9692, 95eqssd 3769 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑋 = (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
97 oveq2 6801 . . . . . . . 8 (𝑑 = sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) → (𝑦(ball‘𝑀)𝑑) = (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9897eqeq2d 2781 . . . . . . 7 (𝑑 = sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) → (𝑋 = (𝑦(ball‘𝑀)𝑑) ↔ 𝑋 = (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))))
9998rspcev 3460 . . . . . 6 ((sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+𝑋 = (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))) → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑))
10052, 96, 99syl2anc 573 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑))
101100rexlimdvaa 3180 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
102101ralrimdva 3118 . . 3 (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
103 isbnd 33911 . . . 4 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
104103baib 525 . . 3 (𝑀 ∈ (Met‘𝑋) → (𝑀 ∈ (Bnd‘𝑋) ↔ ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
105102, 104sylibrd 249 . 2 (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋𝑀 ∈ (Bnd‘𝑋)))
1061, 10, 105sylc 65 1 (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Bnd‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297   ciun 4654   class class class wbr 4786  cmpt 4863   Or wor 5169  dom cdm 5249  ran crn 5250  wf 6027  cfv 6031  (class class class)co 6793  Fincfn 8109  supcsup 8502  cr 10137  0cc0 10138  1c1 10139   + caddc 10141  *cxr 10275   < clt 10276  cle 10277  cmin 10468  +crp 12035  ∞Metcxmt 19946  Metcme 19947  ballcbl 19948  TotBndctotbnd 33897  Bndcbnd 33898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-2 11281  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-totbnd 33899  df-bnd 33910
This theorem is referenced by:  equivbnd2  33923  prdsbnd2  33926  cntotbnd  33927  cnpwstotbnd  33928
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