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Theorem totbndbnd 34935
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 34915 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 34918 . 2 (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
2 1rp 12383 . . 3 1 ∈ ℝ+
3 istotbnd3 34917 . . . 4 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
43simprbi 497 . . 3 (𝑀 ∈ (TotBnd‘𝑋) → ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5 oveq2 7156 . . . . . . 7 (𝑑 = 1 → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)1))
65iuneq2d 4945 . . . . . 6 (𝑑 = 1 → 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑥𝑣 (𝑥(ball‘𝑀)1))
76eqeq1d 2828 . . . . 5 (𝑑 = 1 → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
87rexbidv 3302 . . . 4 (𝑑 = 1 → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
98rspcv 3622 . . 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 771 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑀 ∈ (Met‘𝑋))
12 elfpw 8815 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
1312simplbi 498 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
1413ad2antrl 724 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑣𝑋)
1514sselda 3971 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑧𝑋)
16 simpllr 772 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑦𝑋)
17 metcl 22857 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) → (𝑧𝑀𝑦) ∈ ℝ)
1811, 15, 16, 17syl3anc 1365 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → (𝑧𝑀𝑦) ∈ ℝ)
19 metge0 22870 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) → 0 ≤ (𝑧𝑀𝑦))
2011, 15, 16, 19syl3anc 1365 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 0 ≤ (𝑧𝑀𝑦))
2118, 20ge0p1rpd 12451 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ℝ+)
2221fmpttd 6875 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)):𝑣⟶ℝ+)
2322frnd 6518 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ+)
2412simprbi 497 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
25 mptfi 8812 . . . . . . . . . 10 (𝑣 ∈ Fin → (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
26 rnfi 8796 . . . . . . . . . 10 ((𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
2724, 25, 263syl 18 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
2827ad2antrl 724 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
29 simplr 765 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦𝑋)
30 simprr 769 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
3129, 30eleqtrrd 2921 . . . . . . . . 9 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 𝑥𝑣 (𝑥(ball‘𝑀)1))
32 ne0i 4304 . . . . . . . . 9 (𝑦 𝑥𝑣 (𝑥(ball‘𝑀)1) → 𝑥𝑣 (𝑥(ball‘𝑀)1) ≠ ∅)
33 dm0rn0 5794 . . . . . . . . . . 11 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅)
34 ovex 7181 . . . . . . . . . . . . . . 15 ((𝑧𝑀𝑦) + 1) ∈ V
35 eqid 2826 . . . . . . . . . . . . . . 15 (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
3634, 35dmmpti 6489 . . . . . . . . . . . . . 14 dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
3736eqeq1i 2831 . . . . . . . . . . . . 13 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ 𝑣 = ∅)
38 iuneq1 4932 . . . . . . . . . . . . 13 (𝑣 = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
3937, 38sylbi 218 . . . . . . . . . . . 12 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
40 0iun 4983 . . . . . . . . . . . 12 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1) = ∅
4139, 40syl6eq 2877 . . . . . . . . . . 11 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = ∅)
4233, 41sylbir 236 . . . . . . . . . 10 (ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = ∅)
4342necon3i 3053 . . . . . . . . 9 ( 𝑥𝑣 (𝑥(ball‘𝑀)1) ≠ ∅ → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
4431, 32, 433syl 18 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
45 rpssre 12386 . . . . . . . . 9 + ⊆ ℝ
4623, 45syl6ss 3983 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
47 ltso 10710 . . . . . . . . 9 < Or ℝ
48 fisupcl 8922 . . . . . . . . 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)))
4947, 48mpan 686 . . . . . . . 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)))
5028, 44, 46, 49syl3anc 1365 . . . . . . 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)))
5123, 50sseldd 3972 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+)
52 metxmet 22859 . . . . . . . . . . . . . 14 (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
5352ad2antrr 722 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑀 ∈ (∞Met‘𝑋))
5453adantr 481 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑀 ∈ (∞Met‘𝑋))
55 1red 10631 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 1 ∈ ℝ)
5646, 50sseldd 3972 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5756adantr 481 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5846adantr 481 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
5944adantr 481 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
6028adantr 481 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
61 fimaxre2 11575 . . . . . . . . . . . . . . 15 ((ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ ∧ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin) → ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤𝑑)
6258, 60, 61syl2anc 584 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤𝑑)
6335elrnmpt1 5829 . . . . . . . . . . . . . . . 16 ((𝑧𝑣 ∧ ((𝑧𝑀𝑦) + 1) ∈ V) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6434, 63mpan2 687 . . . . . . . . . . . . . . 15 (𝑧𝑣 → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6564adantl 482 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
66 suprub 11591 . . . . . . . . . . . . . 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, < ))
6758, 59, 62, 65, 66syl31anc 1367 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
68 leaddsub 11105 . . . . . . . . . . . . . 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)))
6918, 55, 57, 68syl3anc 1365 . . . . . . . . . . . . 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)))
7067, 69mpbid 233 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → (𝑧𝑀𝑦) ≤ (sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1))
71 blss2 22929 . . . . . . . . . . . 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, < )))
7254, 15, 16, 55, 57, 70, 71syl33anc 1379 . . . . . . . . . . 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, < )))
7372ralrimiva 3187 . . . . . . . . . 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, < )))
74 nfcv 2982 . . . . . . . . . . . 12 𝑧(𝑥(ball‘𝑀)1)
75 nfcv 2982 . . . . . . . . . . . . 13 𝑧𝑦
76 nfcv 2982 . . . . . . . . . . . . 13 𝑧(ball‘𝑀)
77 nfmpt1 5161 . . . . . . . . . . . . . . 15 𝑧(𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
7877nfrn 5823 . . . . . . . . . . . . . 14 𝑧ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
79 nfcv 2982 . . . . . . . . . . . . . 14 𝑧
80 nfcv 2982 . . . . . . . . . . . . . 14 𝑧 <
8178, 79, 80nfsup 8904 . . . . . . . . . . . . 13 𝑧sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
8275, 76, 81nfov 7178 . . . . . . . . . . . 12 𝑧(𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
8374, 82nfss 3964 . . . . . . . . . . 11 𝑧(𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
84 nfv 1908 . . . . . . . . . . 11 𝑥(𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85 oveq1 7155 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥(ball‘𝑀)1) = (𝑧(ball‘𝑀)1))
8685sseq1d 4002 . . . . . . . . . . 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, < ))))
8783, 84, 86cbvral 3451 . . . . . . . . . 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, < )))
8873, 87sylibr 235 . . . . . . . . 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, < )))
89 iunss 4966 . . . . . . . . 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, < )))
9088, 89sylibr 235 . . . . . . . 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, < )))
9130, 90eqsstrrd 4010 . . . . . . 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, < )))
9251rpxrd 12422 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*)
93 blssm 22943 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*) → (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ⊆ 𝑋)
9453, 29, 92, 93syl3anc 1365 . . . . . . 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)
9591, 94eqssd 3988 . . . . . 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, < )))
96 oveq2 7156 . . . . . . 7 (𝑑 = sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) → (𝑦(ball‘𝑀)𝑑) = (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9796rspceeqv 3642 . . . . . 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))
9851, 95, 97syl2anc 584 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑))
9998rexlimdvaa 3290 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
10099ralrimdva 3194 . . 3 (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
101 isbnd 34926 . . . 4 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
102101baib 536 . . 3 (𝑀 ∈ (Met‘𝑋) → (𝑀 ∈ (Bnd‘𝑋) ↔ ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
103100, 102sylibrd 260 . 2 (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋𝑀 ∈ (Bnd‘𝑋)))
1041, 10, 103sylc 65 1 (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Bnd‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  Vcvv 3500  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542   ciun 4917   class class class wbr 5063  cmpt 5143   Or wor 5472  dom cdm 5554  ran crn 5555  cfv 6352  (class class class)co 7148  Fincfn 8498  supcsup 8893  cr 10525  0cc0 10526  1c1 10527   + caddc 10529  *cxr 10663   < clt 10664  cle 10665  cmin 10859  +crp 12379  ∞Metcxmet 20446  Metcmet 20447  ballcbl 20448  TotBndctotbnd 34912  Bndcbnd 34913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-2 11689  df-rp 12380  df-xneg 12497  df-xadd 12498  df-xmul 12499  df-psmet 20453  df-xmet 20454  df-met 20455  df-bl 20456  df-totbnd 34914  df-bnd 34925
This theorem is referenced by:  equivbnd2  34938  prdsbnd2  34941  cntotbnd  34942  cnpwstotbnd  34943
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