Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  totbndbnd Structured version   Visualization version   GIF version

Theorem totbndbnd 36960
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 36940 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 ∈ (TotBndβ€˜π‘‹) β†’ 𝑀 ∈ (Bndβ€˜π‘‹))

Proof of Theorem totbndbnd
Dummy variables 𝑣 𝑑 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 totbndmet 36943 . 2 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ 𝑀 ∈ (Metβ€˜π‘‹))
2 1rp 12982 . . 3 1 ∈ ℝ+
3 istotbnd3 36942 . . . 4 (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
43simprbi 495 . . 3 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)
5 oveq2 7419 . . . . . . 7 (𝑑 = 1 β†’ (π‘₯(ballβ€˜π‘€)𝑑) = (π‘₯(ballβ€˜π‘€)1))
65iuneq2d 5025 . . . . . 6 (𝑑 = 1 β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1))
76eqeq1d 2732 . . . . 5 (𝑑 = 1 β†’ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋 ↔ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋))
87rexbidv 3176 . . . 4 (𝑑 = 1 β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋 ↔ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋))
98rspcv 3607 . . 3 (1 ∈ ℝ+ β†’ (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋 β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋))
102, 4, 9mpsyl 68 . 2 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)
11 simplll 771 . . . . . . . . . . 11 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 𝑀 ∈ (Metβ€˜π‘‹))
12 elfpw 9356 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 βŠ† 𝑋 ∧ 𝑣 ∈ Fin))
1312simplbi 496 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑣 βŠ† 𝑋)
1413ad2antrl 724 . . . . . . . . . . . 12 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑣 βŠ† 𝑋)
1514sselda 3981 . . . . . . . . . . 11 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 𝑧 ∈ 𝑋)
16 simpllr 772 . . . . . . . . . . 11 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 𝑦 ∈ 𝑋)
17 metcl 24058 . . . . . . . . . . 11 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑧𝑀𝑦) ∈ ℝ)
1811, 15, 16, 17syl3anc 1369 . . . . . . . . . 10 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ (𝑧𝑀𝑦) ∈ ℝ)
19 metge0 24071 . . . . . . . . . . 11 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ 0 ≀ (𝑧𝑀𝑦))
2011, 15, 16, 19syl3anc 1369 . . . . . . . . . 10 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 0 ≀ (𝑧𝑀𝑦))
2118, 20ge0p1rpd 13050 . . . . . . . . 9 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ((𝑧𝑀𝑦) + 1) ∈ ℝ+)
2221fmpttd 7115 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)):π‘£βŸΆβ„+)
2322frnd 6724 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ+)
2412simprbi 495 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑣 ∈ Fin)
25 mptfi 9353 . . . . . . . . . 10 (𝑣 ∈ Fin β†’ (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
26 rnfi 9337 . . . . . . . . . 10 ((𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
2724, 25, 263syl 18 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
2827ad2antrl 724 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
29 simplr 765 . . . . . . . . . 10 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑦 ∈ 𝑋)
30 simprr 769 . . . . . . . . . 10 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)
3129, 30eleqtrrd 2834 . . . . . . . . 9 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑦 ∈ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1))
32 ne0i 4333 . . . . . . . . 9 (𝑦 ∈ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) β‰  βˆ…)
33 dm0rn0 5923 . . . . . . . . . . 11 (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ… ↔ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ…)
34 ovex 7444 . . . . . . . . . . . . . . 15 ((𝑧𝑀𝑦) + 1) ∈ V
35 eqid 2730 . . . . . . . . . . . . . . 15 (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
3634, 35dmmpti 6693 . . . . . . . . . . . . . 14 dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
3736eqeq1i 2735 . . . . . . . . . . . . 13 (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ… ↔ 𝑣 = βˆ…)
38 iuneq1 5012 . . . . . . . . . . . . 13 (𝑣 = βˆ… β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = βˆͺ π‘₯ ∈ βˆ… (π‘₯(ballβ€˜π‘€)1))
3937, 38sylbi 216 . . . . . . . . . . . 12 (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ… β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = βˆͺ π‘₯ ∈ βˆ… (π‘₯(ballβ€˜π‘€)1))
40 0iun 5065 . . . . . . . . . . . 12 βˆͺ π‘₯ ∈ βˆ… (π‘₯(ballβ€˜π‘€)1) = βˆ…
4139, 40eqtrdi 2786 . . . . . . . . . . 11 (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ… β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = βˆ…)
4233, 41sylbir 234 . . . . . . . . . 10 (ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ… β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = βˆ…)
4342necon3i 2971 . . . . . . . . 9 (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) β‰  βˆ… β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ…)
4431, 32, 433syl 18 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ…)
45 rpssre 12985 . . . . . . . . 9 ℝ+ βŠ† ℝ
4623, 45sstrdi 3993 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ)
47 ltso 11298 . . . . . . . . 9 < Or ℝ
48 fisupcl 9466 . . . . . . . . 9 (( < Or ℝ ∧ (ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ… ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ)) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
4947, 48mpan 686 . . . . . . . 8 ((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ… ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
5028, 44, 46, 49syl3anc 1369 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
5123, 50sseldd 3982 . . . . . 6 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+)
52 metxmet 24060 . . . . . . . . . . . . . 14 (𝑀 ∈ (Metβ€˜π‘‹) β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
5352ad2antrr 722 . . . . . . . . . . . . 13 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
5453adantr 479 . . . . . . . . . . . 12 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
55 1red 11219 . . . . . . . . . . . 12 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 1 ∈ ℝ)
5646, 50sseldd 3982 . . . . . . . . . . . . 13 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5756adantr 479 . . . . . . . . . . . 12 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5846adantr 479 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ)
5944adantr 479 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ…)
6028adantr 479 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
61 fimaxre2 12163 . . . . . . . . . . . . . . 15 ((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin) β†’ βˆƒπ‘‘ ∈ ℝ βˆ€π‘€ ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑀 ≀ 𝑑)
6258, 60, 61syl2anc 582 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ βˆƒπ‘‘ ∈ ℝ βˆ€π‘€ ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑀 ≀ 𝑑)
6335elrnmpt1 5956 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ 𝑣 ∧ ((𝑧𝑀𝑦) + 1) ∈ V) β†’ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6434, 63mpan2 687 . . . . . . . . . . . . . . 15 (𝑧 ∈ 𝑣 β†’ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6564adantl 480 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
66 suprub 12179 . . . . . . . . . . . . . 14 (((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ… ∧ βˆƒπ‘‘ ∈ ℝ βˆ€π‘€ ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑀 ≀ 𝑑) ∧ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))) β†’ ((𝑧𝑀𝑦) + 1) ≀ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
6758, 59, 62, 65, 66syl31anc 1371 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ((𝑧𝑀𝑦) + 1) ≀ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
68 leaddsub 11694 . . . . . . . . . . . . . 14 (((𝑧𝑀𝑦) ∈ ℝ ∧ 1 ∈ ℝ ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ) β†’ (((𝑧𝑀𝑦) + 1) ≀ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ↔ (𝑧𝑀𝑦) ≀ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) βˆ’ 1)))
6918, 55, 57, 68syl3anc 1369 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ (((𝑧𝑀𝑦) + 1) ≀ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ↔ (𝑧𝑀𝑦) ≀ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) βˆ’ 1)))
7067, 69mpbid 231 . . . . . . . . . . . 12 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ (𝑧𝑀𝑦) ≀ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) βˆ’ 1))
71 blss2 24130 . . . . . . . . . . . 12 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (1 ∈ ℝ ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ ∧ (𝑧𝑀𝑦) ≀ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) βˆ’ 1))) β†’ (𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
7254, 15, 16, 55, 57, 70, 71syl33anc 1383 . . . . . . . . . . 11 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ (𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
7372ralrimiva 3144 . . . . . . . . . 10 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ βˆ€π‘§ ∈ 𝑣 (𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
74 nfcv 2901 . . . . . . . . . . . 12 Ⅎ𝑧(π‘₯(ballβ€˜π‘€)1)
75 nfcv 2901 . . . . . . . . . . . . 13 Ⅎ𝑧𝑦
76 nfcv 2901 . . . . . . . . . . . . 13 Ⅎ𝑧(ballβ€˜π‘€)
77 nfmpt1 5255 . . . . . . . . . . . . . . 15 Ⅎ𝑧(𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
7877nfrn 5950 . . . . . . . . . . . . . 14 Ⅎ𝑧ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
79 nfcv 2901 . . . . . . . . . . . . . 14 Ⅎ𝑧ℝ
80 nfcv 2901 . . . . . . . . . . . . . 14 Ⅎ𝑧 <
8178, 79, 80nfsup 9448 . . . . . . . . . . . . 13 Ⅎ𝑧sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
8275, 76, 81nfov 7441 . . . . . . . . . . . 12 Ⅎ𝑧(𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
8374, 82nfss 3973 . . . . . . . . . . 11 Ⅎ𝑧(π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
84 nfv 1915 . . . . . . . . . . 11 β„²π‘₯(𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85 oveq1 7418 . . . . . . . . . . . 12 (π‘₯ = 𝑧 β†’ (π‘₯(ballβ€˜π‘€)1) = (𝑧(ballβ€˜π‘€)1))
8685sseq1d 4012 . . . . . . . . . . 11 (π‘₯ = 𝑧 β†’ ((π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ (𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))))
8783, 84, 86cbvralw 3301 . . . . . . . . . 10 (βˆ€π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ βˆ€π‘§ ∈ 𝑣 (𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
8873, 87sylibr 233 . . . . . . . . 9 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ βˆ€π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
89 iunss 5047 . . . . . . . . 9 (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ βˆ€π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9088, 89sylibr 233 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9130, 90eqsstrrd 4020 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑋 βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9251rpxrd 13021 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*)
93 blssm 24144 . . . . . . . 8 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*) β†’ (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) βŠ† 𝑋)
9453, 29, 92, 93syl3anc 1369 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) βŠ† 𝑋)
9591, 94eqssd 3998 . . . . . 6 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑋 = (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
96 oveq2 7419 . . . . . . 7 (𝑑 = sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) β†’ (𝑦(ballβ€˜π‘€)𝑑) = (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9796rspceeqv 3632 . . . . . 6 ((sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+ ∧ 𝑋 = (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))) β†’ βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑))
9851, 95, 97syl2anc 582 . . . . 5 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑))
9998rexlimdvaa 3154 . . . 4 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋 β†’ βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑)))
10099ralrimdva 3152 . . 3 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋 β†’ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑)))
101 isbnd 36951 . . . 4 (𝑀 ∈ (Bndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑)))
102101baib 534 . . 3 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (𝑀 ∈ (Bndβ€˜π‘‹) ↔ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑)))
103100, 102sylibrd 258 . 2 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋 β†’ 𝑀 ∈ (Bndβ€˜π‘‹)))
1041, 10, 103sylc 65 1 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ 𝑀 ∈ (Bndβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  βˆͺ ciun 4996   class class class wbr 5147   ↦ cmpt 5230   Or wor 5586  dom cdm 5675  ran crn 5676  β€˜cfv 6542  (class class class)co 7411  Fincfn 8941  supcsup 9437  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115  β„*cxr 11251   < clt 11252   ≀ cle 11253   βˆ’ cmin 11448  β„+crp 12978  βˆžMetcxmet 21129  Metcmet 21130  ballcbl 21131  TotBndctotbnd 36937  Bndcbnd 36938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-2 12279  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-totbnd 36939  df-bnd 36950
This theorem is referenced by:  equivbnd2  36963  prdsbnd2  36966  cntotbnd  36967  cnpwstotbnd  36968
  Copyright terms: Public domain W3C validator