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Theorem totbndbnd 37797
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 37777 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 37780 . 2 (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
2 1rp 13039 . . 3 1 ∈ ℝ+
3 istotbnd3 37779 . . . 4 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
43simprbi 496 . . 3 (𝑀 ∈ (TotBnd‘𝑋) → ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5 oveq2 7440 . . . . . . 7 (𝑑 = 1 → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)1))
65iuneq2d 5021 . . . . . 6 (𝑑 = 1 → 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑥𝑣 (𝑥(ball‘𝑀)1))
76eqeq1d 2738 . . . . 5 (𝑑 = 1 → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
87rexbidv 3178 . . . 4 (𝑑 = 1 → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
98rspcv 3617 . . 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 774 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑀 ∈ (Met‘𝑋))
12 elfpw 9395 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
1312simplbi 497 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
1413ad2antrl 728 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑣𝑋)
1514sselda 3982 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑧𝑋)
16 simpllr 775 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑦𝑋)
17 metcl 24343 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) → (𝑧𝑀𝑦) ∈ ℝ)
1811, 15, 16, 17syl3anc 1372 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → (𝑧𝑀𝑦) ∈ ℝ)
19 metge0 24356 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) → 0 ≤ (𝑧𝑀𝑦))
2011, 15, 16, 19syl3anc 1372 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 0 ≤ (𝑧𝑀𝑦))
2118, 20ge0p1rpd 13108 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ℝ+)
2221fmpttd 7134 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)):𝑣⟶ℝ+)
2322frnd 6743 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ+)
2412simprbi 496 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
25 mptfi 9392 . . . . . . . . . 10 (𝑣 ∈ Fin → (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
26 rnfi 9381 . . . . . . . . . 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 728 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
29 simplr 768 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦𝑋)
30 simprr 772 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
3129, 30eleqtrrd 2843 . . . . . . . . 9 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 𝑥𝑣 (𝑥(ball‘𝑀)1))
32 ne0i 4340 . . . . . . . . 9 (𝑦 𝑥𝑣 (𝑥(ball‘𝑀)1) → 𝑥𝑣 (𝑥(ball‘𝑀)1) ≠ ∅)
33 dm0rn0 5934 . . . . . . . . . . 11 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅)
34 ovex 7465 . . . . . . . . . . . . . . 15 ((𝑧𝑀𝑦) + 1) ∈ V
35 eqid 2736 . . . . . . . . . . . . . . 15 (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
3634, 35dmmpti 6711 . . . . . . . . . . . . . 14 dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
3736eqeq1i 2741 . . . . . . . . . . . . 13 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ 𝑣 = ∅)
38 iuneq1 5007 . . . . . . . . . . . . 13 (𝑣 = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
3937, 38sylbi 217 . . . . . . . . . . . 12 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
40 0iun 5062 . . . . . . . . . . . 12 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1) = ∅
4139, 40eqtrdi 2792 . . . . . . . . . . 11 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = ∅)
4233, 41sylbir 235 . . . . . . . . . 10 (ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = ∅)
4342necon3i 2972 . . . . . . . . 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 13043 . . . . . . . . 9 + ⊆ ℝ
4623, 45sstrdi 3995 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
47 ltso 11342 . . . . . . . . 9 < Or ℝ
48 fisupcl 9510 . . . . . . . . 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 690 . . . . . . . 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 1372 . . . . . . 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 3983 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+)
52 metxmet 24345 . . . . . . . . . . . . . 14 (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
5352ad2antrr 726 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑀 ∈ (∞Met‘𝑋))
5453adantr 480 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑀 ∈ (∞Met‘𝑋))
55 1red 11263 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 1 ∈ ℝ)
5646, 50sseldd 3983 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5756adantr 480 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5846adantr 480 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
5944adantr 480 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
6028adantr 480 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
61 fimaxre2 12214 . . . . . . . . . . . . . . 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 5970 . . . . . . . . . . . . . . . 16 ((𝑧𝑣 ∧ ((𝑧𝑀𝑦) + 1) ∈ V) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6434, 63mpan2 691 . . . . . . . . . . . . . . 15 (𝑧𝑣 → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6564adantl 481 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
66 suprub 12230 . . . . . . . . . . . . . 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 1374 . . . . . . . . . . . . 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 11740 . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 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 24415 . . . . . . . . . . . 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 1386 . . . . . . . . . . 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 3145 . . . . . . . . . 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 2904 . . . . . . . . . . . 12 𝑧(𝑥(ball‘𝑀)1)
75 nfcv 2904 . . . . . . . . . . . . 13 𝑧𝑦
76 nfcv 2904 . . . . . . . . . . . . 13 𝑧(ball‘𝑀)
77 nfmpt1 5249 . . . . . . . . . . . . . . 15 𝑧(𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
7877nfrn 5962 . . . . . . . . . . . . . 14 𝑧ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
79 nfcv 2904 . . . . . . . . . . . . . 14 𝑧
80 nfcv 2904 . . . . . . . . . . . . . 14 𝑧 <
8178, 79, 80nfsup 9492 . . . . . . . . . . . . 13 𝑧sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
8275, 76, 81nfov 7462 . . . . . . . . . . . 12 𝑧(𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
8374, 82nfss 3975 . . . . . . . . . . 11 𝑧(𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
84 nfv 1913 . . . . . . . . . . 11 𝑥(𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85 oveq1 7439 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥(ball‘𝑀)1) = (𝑧(ball‘𝑀)1))
8685sseq1d 4014 . . . . . . . . . . 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, 86cbvralw 3305 . . . . . . . . . 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 234 . . . . . . . . 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 5044 . . . . . . . . 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 234 . . . . . . . 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 4018 . . . . . . 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 13079 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*)
93 blssm 24429 . . . . . . . 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 1372 . . . . . . 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 4000 . . . . . 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 7440 . . . . . . 7 (𝑑 = sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) → (𝑦(ball‘𝑀)𝑑) = (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9796rspceeqv 3644 . . . . . 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 3155 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
10099ralrimdva 3153 . . 3 (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
101 isbnd 37788 . . . 4 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
102101baib 535 . . 3 (𝑀 ∈ (Met‘𝑋) → (𝑀 ∈ (Bnd‘𝑋) ↔ ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
103100, 102sylibrd 259 . 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 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  Vcvv 3479  cin 3949  wss 3950  c0 4332  𝒫 cpw 4599   ciun 4990   class class class wbr 5142  cmpt 5224   Or wor 5590  dom cdm 5684  ran crn 5685  cfv 6560  (class class class)co 7432  Fincfn 8986  supcsup 9481  cr 11155  0cc0 11156  1c1 11157   + caddc 11159  *cxr 11295   < clt 11296  cle 11297  cmin 11493  +crp 13035  ∞Metcxmet 21350  Metcmet 21351  ballcbl 21352  TotBndctotbnd 37774  Bndcbnd 37775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-2 12330  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-totbnd 37776  df-bnd 37787
This theorem is referenced by:  equivbnd2  37800  prdsbnd2  37803  cntotbnd  37804  cnpwstotbnd  37805
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