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Theorem totbndbnd 37776
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 37756 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 37759 . 2 (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
2 1rp 13036 . . 3 1 ∈ ℝ+
3 istotbnd3 37758 . . . 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 7439 . . . . . . 7 (𝑑 = 1 → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)1))
65iuneq2d 5027 . . . . . 6 (𝑑 = 1 → 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑥𝑣 (𝑥(ball‘𝑀)1))
76eqeq1d 2737 . . . . 5 (𝑑 = 1 → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
87rexbidv 3177 . . . 4 (𝑑 = 1 → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
98rspcv 3618 . . 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 775 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑀 ∈ (Met‘𝑋))
12 elfpw 9392 . . . . . . . . . . . . . 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 3995 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑧𝑋)
16 simpllr 776 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑦𝑋)
17 metcl 24358 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) → (𝑧𝑀𝑦) ∈ ℝ)
1811, 15, 16, 17syl3anc 1370 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → (𝑧𝑀𝑦) ∈ ℝ)
19 metge0 24371 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) → 0 ≤ (𝑧𝑀𝑦))
2011, 15, 16, 19syl3anc 1370 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 0 ≤ (𝑧𝑀𝑦))
2118, 20ge0p1rpd 13105 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ℝ+)
2221fmpttd 7135 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)):𝑣⟶ℝ+)
2322frnd 6745 . . . . . . 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 9389 . . . . . . . . . 10 (𝑣 ∈ Fin → (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
26 rnfi 9378 . . . . . . . . . 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 769 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦𝑋)
30 simprr 773 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
3129, 30eleqtrrd 2842 . . . . . . . . 9 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 𝑥𝑣 (𝑥(ball‘𝑀)1))
32 ne0i 4347 . . . . . . . . 9 (𝑦 𝑥𝑣 (𝑥(ball‘𝑀)1) → 𝑥𝑣 (𝑥(ball‘𝑀)1) ≠ ∅)
33 dm0rn0 5938 . . . . . . . . . . 11 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅)
34 ovex 7464 . . . . . . . . . . . . . . 15 ((𝑧𝑀𝑦) + 1) ∈ V
35 eqid 2735 . . . . . . . . . . . . . . 15 (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
3634, 35dmmpti 6713 . . . . . . . . . . . . . 14 dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
3736eqeq1i 2740 . . . . . . . . . . . . 13 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ 𝑣 = ∅)
38 iuneq1 5013 . . . . . . . . . . . . 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 5068 . . . . . . . . . . . 12 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1) = ∅
4139, 40eqtrdi 2791 . . . . . . . . . . 11 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = ∅)
4233, 41sylbir 235 . . . . . . . . . 10 (ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = ∅)
4342necon3i 2971 . . . . . . . . 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 13040 . . . . . . . . 9 + ⊆ ℝ
4623, 45sstrdi 4008 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
47 ltso 11339 . . . . . . . . 9 < Or ℝ
48 fisupcl 9507 . . . . . . . . 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 1370 . . . . . . 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 3996 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+)
52 metxmet 24360 . . . . . . . . . . . . . 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 11260 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 1 ∈ ℝ)
5646, 50sseldd 3996 . . . . . . . . . . . . 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 12211 . . . . . . . . . . . . . . 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 5974 . . . . . . . . . . . . . . . 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 12227 . . . . . . . . . . . . . 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 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, < ))
68 leaddsub 11737 . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . 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 24430 . . . . . . . . . . . 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 1384 . . . . . . . . . . 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 3144 . . . . . . . . . 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 2903 . . . . . . . . . . . 12 𝑧(𝑥(ball‘𝑀)1)
75 nfcv 2903 . . . . . . . . . . . . 13 𝑧𝑦
76 nfcv 2903 . . . . . . . . . . . . 13 𝑧(ball‘𝑀)
77 nfmpt1 5256 . . . . . . . . . . . . . . 15 𝑧(𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
7877nfrn 5966 . . . . . . . . . . . . . 14 𝑧ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
79 nfcv 2903 . . . . . . . . . . . . . 14 𝑧
80 nfcv 2903 . . . . . . . . . . . . . 14 𝑧 <
8178, 79, 80nfsup 9489 . . . . . . . . . . . . 13 𝑧sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
8275, 76, 81nfov 7461 . . . . . . . . . . . 12 𝑧(𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
8374, 82nfss 3988 . . . . . . . . . . 11 𝑧(𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
84 nfv 1912 . . . . . . . . . . 11 𝑥(𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85 oveq1 7438 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥(ball‘𝑀)1) = (𝑧(ball‘𝑀)1))
8685sseq1d 4027 . . . . . . . . . . 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 3304 . . . . . . . . . 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 5050 . . . . . . . . 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 4035 . . . . . . 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 13076 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*)
93 blssm 24444 . . . . . . . 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 1370 . . . . . . 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 4013 . . . . . 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 7439 . . . . . . 7 (𝑑 = sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) → (𝑦(ball‘𝑀)𝑑) = (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9796rspceeqv 3645 . . . . . 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 3154 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
10099ralrimdva 3152 . . 3 (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
101 isbnd 37767 . . . 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 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   ciun 4996   class class class wbr 5148  cmpt 5231   Or wor 5596  dom cdm 5689  ran crn 5690  cfv 6563  (class class class)co 7431  Fincfn 8984  supcsup 9478  cr 11152  0cc0 11153  1c1 11154   + caddc 11156  *cxr 11292   < clt 11293  cle 11294  cmin 11490  +crp 13032  ∞Metcxmet 21367  Metcmet 21368  ballcbl 21369  TotBndctotbnd 37753  Bndcbnd 37754
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-2 12327  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-totbnd 37755  df-bnd 37766
This theorem is referenced by:  equivbnd2  37779  prdsbnd2  37782  cntotbnd  37783  cnpwstotbnd  37784
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