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Theorem totbndbnd 33247
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 33227 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 33230 . 2 (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
2 1rp 11787 . . 3 1 ∈ ℝ+
3 istotbnd3 33229 . . . 4 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
43simprbi 480 . . 3 (𝑀 ∈ (TotBnd‘𝑋) → ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5 oveq2 6618 . . . . . . 7 (𝑑 = 1 → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)1))
65iuneq2d 4518 . . . . . 6 (𝑑 = 1 → 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑥𝑣 (𝑥(ball‘𝑀)1))
76eqeq1d 2623 . . . . 5 (𝑑 = 1 → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
87rexbidv 3046 . . . 4 (𝑑 = 1 → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
98rspcv 3294 . . 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 797 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑀 ∈ (Met‘𝑋))
12 elfpw 8219 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
1312simplbi 476 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
1413ad2antrl 763 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑣𝑋)
1514sselda 3587 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑧𝑋)
16 simpllr 798 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑦𝑋)
17 metcl 22056 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) → (𝑧𝑀𝑦) ∈ ℝ)
1811, 15, 16, 17syl3anc 1323 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → (𝑧𝑀𝑦) ∈ ℝ)
19 metge0 22069 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧𝑋𝑦𝑋) → 0 ≤ (𝑧𝑀𝑦))
2011, 15, 16, 19syl3anc 1323 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 0 ≤ (𝑧𝑀𝑦))
2118, 20ge0p1rpd 11853 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ℝ+)
22 eqid 2621 . . . . . . . . 9 (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
2321, 22fmptd 6346 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)):𝑣⟶ℝ+)
24 frn 6015 . . . . . . . 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 480 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
27 mptfi 8216 . . . . . . . . . 10 (𝑣 ∈ Fin → (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
28 rnfi 8200 . . . . . . . . . 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 763 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
31 simplr 791 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦𝑋)
32 simprr 795 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
3331, 32eleqtrrd 2701 . . . . . . . . 9 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 𝑥𝑣 (𝑥(ball‘𝑀)1))
34 ne0i 3902 . . . . . . . . 9 (𝑦 𝑥𝑣 (𝑥(ball‘𝑀)1) → 𝑥𝑣 (𝑥(ball‘𝑀)1) ≠ ∅)
35 dm0rn0 5307 . . . . . . . . . . 11 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅)
36 ovex 6638 . . . . . . . . . . . . . . 15 ((𝑧𝑀𝑦) + 1) ∈ V
3736, 22dmmpti 5985 . . . . . . . . . . . . . 14 dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
3837eqeq1i 2626 . . . . . . . . . . . . 13 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ 𝑣 = ∅)
39 iuneq1 4505 . . . . . . . . . . . . 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 4548 . . . . . . . . . . . 12 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1) = ∅
4240, 41syl6eq 2671 . . . . . . . . . . 11 (dom (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = ∅)
4335, 42sylbir 225 . . . . . . . . . 10 (ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)1) = ∅)
4443necon3i 2822 . . . . . . . . 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 11794 . . . . . . . . 9 + ⊆ ℝ
4725, 46syl6ss 3599 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
48 ltso 10069 . . . . . . . . 9 < Or ℝ
49 fisupcl 8326 . . . . . . . . 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 705 . . . . . . . 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 1323 . . . . . . 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 3588 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+)
53 metxmet 22058 . . . . . . . . . . . . . 14 (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
5453ad2antrr 761 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑀 ∈ (∞Met‘𝑋))
5554adantr 481 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 𝑀 ∈ (∞Met‘𝑋))
56 1red 10006 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → 1 ∈ ℝ)
5747, 51sseldd 3588 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5857adantr 481 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5947adantr 481 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
6045adantr 481 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
6130adantr 481 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
62 fimaxre2 10920 . . . . . . . . . . . . . . 15 ((ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ ∧ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin) → ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤𝑑)
6359, 61, 62syl2anc 692 . . . . . . . . . . . . . 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 5339 . . . . . . . . . . . . . . . 16 ((𝑧𝑣 ∧ ((𝑧𝑀𝑦) + 1) ∈ V) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6536, 64mpan2 706 . . . . . . . . . . . . . . 15 (𝑧𝑣 → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6665adantl 482 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
67 suprub 10935 . . . . . . . . . . . . . 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 1326 . . . . . . . . . . . . 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 10455 . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . 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 22128 . . . . . . . . . . . 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 1338 . . . . . . . . . . 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 2961 . . . . . . . . . 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 2761 . . . . . . . . . . . 12 𝑧(𝑥(ball‘𝑀)1)
76 nfcv 2761 . . . . . . . . . . . . 13 𝑧𝑦
77 nfcv 2761 . . . . . . . . . . . . 13 𝑧(ball‘𝑀)
78 nfmpt1 4712 . . . . . . . . . . . . . . 15 𝑧(𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
7978nfrn 5333 . . . . . . . . . . . . . 14 𝑧ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1))
80 nfcv 2761 . . . . . . . . . . . . . 14 𝑧
81 nfcv 2761 . . . . . . . . . . . . . 14 𝑧 <
8279, 80, 81nfsup 8308 . . . . . . . . . . . . 13 𝑧sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
8376, 77, 82nfov 6636 . . . . . . . . . . . 12 𝑧(𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
8475, 83nfss 3580 . . . . . . . . . . 11 𝑧(𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85 nfv 1840 . . . . . . . . . . 11 𝑥(𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
86 oveq1 6617 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥(ball‘𝑀)1) = (𝑧(ball‘𝑀)1))
8786sseq1d 3616 . . . . . . . . . . 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 3158 . . . . . . . . . 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 4532 . . . . . . . . 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 3624 . . . . . . 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 11824 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*)
94 blssm 22142 . . . . . . . 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 1323 . . . . . . 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 3604 . . . . . 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 6618 . . . . . . . 8 (𝑑 = sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) → (𝑦(ball‘𝑀)𝑑) = (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9897eqeq2d 2631 . . . . . . 7 (𝑑 = sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) → (𝑋 = (𝑦(ball‘𝑀)𝑑) ↔ 𝑋 = (𝑦(ball‘𝑀)sup(ran (𝑧𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))))
9998rspcev 3298 . . . . . 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 692 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑))
101100rexlimdvaa 3026 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
102101ralrimdva 2964 . . 3 (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
103 isbnd 33238 . . . 4 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
104103baib 943 . . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3189  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135   ciun 4490   class class class wbr 4618  cmpt 4678   Or wor 4999  dom cdm 5079  ran crn 5080  wf 5848  cfv 5852  (class class class)co 6610  Fincfn 7906  supcsup 8297  cr 9886  0cc0 9887  1c1 9888   + caddc 9890  *cxr 10024   < clt 10025  cle 10026  cmin 10217  +crp 11783  ∞Metcxmt 19659  Metcme 19660  ballcbl 19661  TotBndctotbnd 33224  Bndcbnd 33225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-2 11030  df-rp 11784  df-xneg 11897  df-xadd 11898  df-xmul 11899  df-psmet 19666  df-xmet 19667  df-met 19668  df-bl 19669  df-totbnd 33226  df-bnd 33237
This theorem is referenced by:  equivbnd2  33250  prdsbnd2  33253  cntotbnd  33254  cnpwstotbnd  33255
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