Theorem totbndbnd 37113

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 37093 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 37096	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
2		1rp 12974	. . 3 ⊢ 1 ∈ ℝ+
3		istotbnd3 37095	. . . 4 ⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
4	3	simprbi 496	. . 3 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5		oveq2 7409	. . . . . . 7 ⊢ (𝑑 = 1 → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)1))
6	5	iuneq2d 5016	. . . . . 6 ⊢ (𝑑 = 1 → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1))
7	6	eqeq1d 2726	. . . . 5 ⊢ (𝑑 = 1 → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
8	7	rexbidv 3170	. . . 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 3600	. . 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 68	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
11		simplll 772	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑀 ∈ (Met‘𝑋))
12		elfpw 9349	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ⊆ 𝑋 ∧ 𝑣 ∈ Fin))
13	12	simplbi 497	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ⊆ 𝑋)
14	13	ad2antrl 725	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑣 ⊆ 𝑋)
15	14	sselda 3974	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝑋)
16		simpllr 773	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑦 ∈ 𝑋)
17		metcl 24148	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝑀𝑦) ∈ ℝ)
18	11, 15, 16, 17	syl3anc 1368	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → (𝑧𝑀𝑦) ∈ ℝ)
19		metge0 24161	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑧𝑀𝑦))
20	11, 15, 16, 19	syl3anc 1368	. . . . . . . . . 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 13042	. . . . . . . . 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 7106	. . . . . . . 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 6715	. . . . . . 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 496	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
25		mptfi 9346	. . . . . . . . . 10 ⊢ (𝑣 ∈ Fin → (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
26		rnfi 9330	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
27	24, 25, 26	3syl 18	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
28	27	ad2antrl 725	. . . . . . . 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 766	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 ∈ 𝑋)
30		simprr 770	. . . . . . . . . 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 2828	. . . . . . . . 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 4326	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ≠ ∅)
33		dm0rn0 5914	. . . . . . . . . . 11 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅)
34		ovex 7434	. . . . . . . . . . . . . . 15 ⊢ ((𝑧𝑀𝑦) + 1) ∈ V
35		eqid 2724	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
36	34, 35	dmmpti 6684	. . . . . . . . . . . . . 14 ⊢ dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
37	36	eqeq1i 2729	. . . . . . . . . . . . 13 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ 𝑣 = ∅)
38		iuneq1 5003	. . . . . . . . . . . . 13 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
39	37, 38	sylbi 216	. . . . . . . . . . . 12 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
40		0iun 5056	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1) = ∅
41	39, 40	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∅)
42	33, 41	sylbir 234	. . . . . . . . . 10 ⊢ (ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∅)
43	42	necon3i 2965	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ≠ ∅ → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
44	31, 32, 43	3syl 18	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
45		rpssre 12977	. . . . . . . . 9 ⊢ ℝ+ ⊆ ℝ
46	23, 45	sstrdi 3986	. . . . . . . 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 11290	. . . . . . . . 9 ⊢ < Or ℝ
48		fisupcl 9459	. . . . . . . . 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 687	. . . . . . . 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 1368	. . . . . . 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 3975	. . . . . 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 24150	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
53	52	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑀 ∈ (∞Met‘𝑋))
54	53	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑀 ∈ (∞Met‘𝑋))
55		1red 11211	. . . . . . . . . . . 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 3975	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 12155	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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 5947	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑣 ∧ ((𝑧𝑀𝑦) + 1) ∈ V) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
64	34, 63	mpan2 688	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑣 → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
65	64	adantl 481	. . . . . . . . . . . . . 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 12171	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 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 11686	. . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 24220	. . . . . . . . . . . 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 1382	. . . . . . . . . . 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 3138	. . . . . . . . . 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 2895	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑥(ball‘𝑀)1)
75		nfcv 2895	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧𝑦
76		nfcv 2895	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧(ball‘𝑀)
77		nfmpt1 5246	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
78	77	nfrn 5941	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
79		nfcv 2895	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧ℝ
80		nfcv 2895	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 <
81	78, 79, 80	nfsup 9441	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
82	75, 76, 81	nfov 7431	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
83	74, 82	nfss 3966	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
84		nfv 1909	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85		oveq1 7408	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥(ball‘𝑀)1) = (𝑧(ball‘𝑀)1))
86	85	sseq1d 4005	. . . . . . . . . . 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 3295	. . . . . . . . . 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 233	. . . . . . . . 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 5038	. . . . . . . . 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 233	. . . . . . . 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 4013	. . . . . . 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 13013	. . . . . . . 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 24234	. . . . . . . 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 1368	. . . . . . 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 3991	. . . . . 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 7409	. . . . . . 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 3625	. . . . . 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 583	. . . . 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 3148	. . . 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 3146	. . 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 37104	. . . 4 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
102	101	baib 535	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → (𝑀 ∈ (Bnd‘𝑋) ↔ ∀𝑦 ∈ 𝑋 ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
103	100, 102	sylibrd 259	. 2 ⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → 𝑀 ∈ (Bnd‘𝑋)))
104	1, 10, 103	sylc 65	1 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Bnd‘𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053  ∃wrex 3062  Vcvv 3466   ∩ cin 3939   ⊆ wss 3940  ∅c0 4314  𝒫 cpw 4594  ∪ ciun 4987   class class class wbr 5138   ↦ cmpt 5221   Or wor 5577  dom cdm 5666  ran crn 5667  ‘cfv 6533  (class class class)co 7401  Fincfn 8934  supcsup 9430  ℝcr 11104  0cc0 11105  1c1 11106   + caddc 11108  ℝ^*cxr 11243   < clt 11244   ≤ cle 11245   − cmin 11440  ℝ⁺crp 12970  ∞Metcxmet 21208  Metcmet 21209  ballcbl 21210  TotBndctotbnd 37090  Bndcbnd 37091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-1o 8461  df-er 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-sup 9432  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-2 12271  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-psmet 21215  df-xmet 21216  df-met 21217  df-bl 21218  df-totbnd 37092  df-bnd 37103

This theorem is referenced by:  equivbnd2  37116  prdsbnd2  37119  cntotbnd  37120  cnpwstotbnd  37121