Theorem totbndbnd 38418

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 38398 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.

Step	Hyp	Ref	Expression
1		totbndmet 38401	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
2		1rp 13021	. . 3 ⊢ 1 ∈ ℝ+
3		istotbnd3 38400	. . . 4 ⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
4	3	simprbi 502	. . 3 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5		oveq2 7420	. . . . . . 7 ⊢ (𝑑 = 1 → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)1))
6	5	iuneq2d 4988	. . . . . 6 ⊢ (𝑑 = 1 → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1))
7	6	eqeq1d 2765	. . . . 5 ⊢ (𝑑 = 1 → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
8	7	rexbidv 3189	. . . 4 ⊢ (𝑑 = 1 → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
9	8	rspcv 3578	. . 3 ⊢ (1 ∈ ℝ+ → (∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
10	2, 4, 9	mpsyl 69	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
11		simplll 786	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑀 ∈ (Met‘𝑋))
12		elfpw 9312	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ⊆ 𝑋 ∧ 𝑣 ∈ Fin))
13	12	simplbi 501	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ⊆ 𝑋)
14	13	ad2antrl 740	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑣 ⊆ 𝑋)
15	14	sselda 3938	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝑋)
16		simpllr 787	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑦 ∈ 𝑋)
17		metcl 24470	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝑀𝑦) ∈ ℝ)
18	11, 15, 16, 17	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → (𝑧𝑀𝑦) ∈ ℝ)
19		metge0 24483	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑧𝑀𝑦))
20	11, 15, 16, 19	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 0 ≤ (𝑧𝑀𝑦))
21	18, 20	ge0p1rpd 13091	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ℝ+)
22	21	fmpttd 7112	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)):𝑣⟶ℝ+)
23	22	frnd 6716	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ+)
24	12	simprbi 502	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
25		mptfi 9309	. . . . . . . . . 10 ⊢ (𝑣 ∈ Fin → (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
26		rnfi 9298	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
27	24, 25, 26	3syl 19	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
28	27	ad2antrl 740	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
29		simplr 780	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 ∈ 𝑋)
30		simprr 784	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
31	29, 30	eleqtrrd 2866	. . . . . . . . 9 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 ∈ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1))
32		ne0i 4295	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ≠ ∅)
33		dm0rn0 5916	. . . . . . . . . . 11 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅)
34		ovex 7445	. . . . . . . . . . . . . . 15 ⊢ ((𝑧𝑀𝑦) + 1) ∈ V
35		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
36	34, 35	dmmpti 6681	. . . . . . . . . . . . . 14 ⊢ dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
37	36	eqeq1i 2768	. . . . . . . . . . . . 13 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ 𝑣 = ∅)
38		iuneq1 4974	. . . . . . . . . . . . 13 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
39	37, 38	sylbi 220	. . . . . . . . . . . 12 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
40		0iun 5028	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1) = ∅
41	39, 40	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∅)
42	33, 41	sylbir 238	. . . . . . . . . 10 ⊢ (ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∅)
43	42	necon3i 2990	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ≠ ∅ → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
44	31, 32, 43	3syl 19	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
45		rpssre 13025	. . . . . . . . 9 ⊢ ℝ+ ⊆ ℝ
46	23, 45	sstrdi 3950	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
47		ltso 11291	. . . . . . . . 9 ⊢ < Or ℝ
48		fisupcl 9431	. . . . . . . . 9 ⊢ (( < Or ℝ ∧ (ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
49	47, 48	mpan 702	. . . . . . . 8 ⊢ ((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
50	28, 44, 46, 49	syl3anc 1398	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
51	23, 50	sseldd 3939	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+)
52		metxmet 24472	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
53	52	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑀 ∈ (∞Met‘𝑋))
54	53	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑀 ∈ (∞Met‘𝑋))
55		1red 11210	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 1 ∈ ℝ)
56	46, 50	sseldd 3939	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
57	56	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
58	46	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
59	44	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
60	28	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
61		fimaxre2 12161	. . . . . . . . . . . . . . 15 ⊢ ((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin) → ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤 ≤ 𝑑)
62	58, 60, 61	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤 ≤ 𝑑)
63	35	elrnmpt1 5952	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑣 ∧ ((𝑧𝑀𝑦) + 1) ∈ V) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
64	34, 63	mpan2 703	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑣 → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
65	64	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
66		suprub 12177	. . . . . . . . . . . . . 14 ⊢ (((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅ ∧ ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤 ≤ 𝑑) ∧ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))) → ((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
67	58, 59, 62, 65, 66	syl31anc 1400	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
68		leaddsub 11691	. . . . . . . . . . . . . 14 ⊢ (((𝑧𝑀𝑦) ∈ ℝ ∧ 1 ∈ ℝ ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ) → (((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ↔ (𝑧𝑀𝑦) ≤ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1)))
69	18, 55, 57, 68	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → (((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ↔ (𝑧𝑀𝑦) ≤ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1)))
70	67, 69	mpbid 235	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → (𝑧𝑀𝑦) ≤ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1))
71		blss2 24542	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (1 ∈ ℝ ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ ∧ (𝑧𝑀𝑦) ≤ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1))) → (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
72	54, 15, 16, 55, 57, 70, 71	syl33anc 1412	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
73	72	ralrimiva 3157	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∀𝑧 ∈ 𝑣 (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
74		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑥(ball‘𝑀)1)
75		nfcv 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧𝑦
76		nfcv 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧(ball‘𝑀)
77		nfmpt1 5211	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
78	77	nfrn 5944	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
79		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧ℝ
80		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 <
81	78, 79, 80	nfsup 9412	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
82	75, 76, 81	nfov 7442	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
83	74, 82	nfss 3931	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
84		nfv 1944	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥(ball‘𝑀)1) = (𝑧(ball‘𝑀)1))
86	85	sseq1d 3969	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))))
87	83, 84, 86	cbvralw 3307	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ ∀𝑧 ∈ 𝑣 (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
88	73, 87	sylibr 237	. . . . . . . . 9 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∀𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
89		iunss 5010	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ ∀𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
90	88, 89	sylibr 237	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
91	30, 90	eqsstrrd 3973	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑋 ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
92	51	rpxrd 13062	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*)
93		blssm 24556	. . . . . . . 8 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*) → (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ⊆ 𝑋)
94	53, 29, 92, 93	syl3anc 1398	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ⊆ 𝑋)
95	91, 94	eqssd 3955	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑋 = (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
96		oveq2 7420	. . . . . . 7 ⊢ (𝑑 = sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) → (𝑦(ball‘𝑀)𝑑) = (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
97	96	rspceeqv 3605	. . . . . 6 ⊢ ((sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+ ∧ 𝑋 = (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))) → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑))
98	51, 95, 97	syl2anc 595	. . . . 5 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑))
99	98	rexlimdvaa 3167	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
100	99	ralrimdva 3165	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∀𝑦 ∈ 𝑋 ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
101		isbnd 38409	. . . 4 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
102	101	baib 544	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → (𝑀 ∈ (Bnd‘𝑋) ↔ ∀𝑦 ∈ 𝑋 ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
103	100, 102	sylibrd 262	. 2 ⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → 𝑀 ∈ (Bnd‘𝑋)))
104	1, 10, 103	sylc 66	1 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Bnd‘𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  ∪ ciun 4957   class class class wbr 5110   ↦ cmpt 5193   Or wor 5570  dom cdm 5663  ran crn 5664  ‘cfv 6538  (class class class)co 7412  Fincfn 8944  supcsup 9401  ℝcr 11100  0cc0 11101  1c1 11102   + caddc 11104  ℝ^*cxr 11243   < clt 11244   ≤ cle 11245   − cmin 11442  ℝ⁺crp 13017  ∞Metcxmet 21488  Metcmet 21489  ballcbl 21490  TotBndctotbnd 38395  Bndcbnd 38396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-pre-sup 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-sup 9403  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-rp 13018  df-xneg 13138  df-xadd 13139  df-xmul 13140  df-psmet 21495  df-xmet 21496  df-met 21497  df-bl 21498  df-totbnd 38397  df-bnd 38408

This theorem is referenced by:  equivbnd2  38421  prdsbnd2  38424  cntotbnd  38425  cnpwstotbnd  38426