totbndbnd - Mathbox for Jeff Madsen

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Theorem totbndbnd 36645

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 36625 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 36628	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
2		1rp 12974	. . 3 ⊢ 1 ∈ ℝ⁺
3		istotbnd3 36627	. . . 4 ⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
4	3	simprbi 497	. . 3 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5		oveq2 7413	. . . . . . 7 ⊢ (𝑑 = 1 → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)1))
6	5	iuneq2d 5025	. . . . . 6 ⊢ (𝑑 = 1 → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1))
7	6	eqeq1d 2734	. . . . 5 ⊢ (𝑑 = 1 → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
8	7	rexbidv 3178	. . . 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 3608	. . 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 773	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑀 ∈ (Met‘𝑋))
12		elfpw 9350	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ⊆ 𝑋 ∧ 𝑣 ∈ Fin))
13	12	simplbi 498	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ⊆ 𝑋)
14	13	ad2antrl 726	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑣 ⊆ 𝑋)
15	14	sselda 3981	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝑋)
16		simpllr 774	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑦 ∈ 𝑋)
17		metcl 23829	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝑀𝑦) ∈ ℝ)
18	11, 15, 16, 17	syl3anc 1371	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → (𝑧𝑀𝑦) ∈ ℝ)
19		metge0 23842	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑧𝑀𝑦))
20	11, 15, 16, 19	syl3anc 1371	. . . . . . . . . 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 7111	. . . . . . . 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 6722	. . . . . . 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 497	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
25		mptfi 9347	. . . . . . . . . 10 ⊢ (𝑣 ∈ Fin → (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
26		rnfi 9331	. . . . . . . . . 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 726	. . . . . . . 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 767	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 ∈ 𝑋)
30		simprr 771	. . . . . . . . . 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 2836	. . . . . . . . 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 4333	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ≠ ∅)
33		dm0rn0 5922	. . . . . . . . . . 11 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅)
34		ovex 7438	. . . . . . . . . . . . . . 15 ⊢ ((𝑧𝑀𝑦) + 1) ∈ V
35		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
36	34, 35	dmmpti 6691	. . . . . . . . . . . . . 14 ⊢ dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
37	36	eqeq1i 2737	. . . . . . . . . . . . 13 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ 𝑣 = ∅)
38		iuneq1 5012	. . . . . . . . . . . . 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 5065	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1) = ∅
41	39, 40	eqtrdi 2788	. . . . . . . . . . 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 2973	. . . . . . . . 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 3993	. . . . . . . 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 9460	. . . . . . . . 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 688	. . . . . . . 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 1371	. . . . . . 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 3982	. . . . . 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 23831	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
53	52	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑀 ∈ (∞Met‘𝑋))
54	53	adantr 481	. . . . . . . . . . . 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 3982	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 5955	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑣 ∧ ((𝑧𝑀𝑦) + 1) ∈ V) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
64	34, 63	mpan2 689	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑣 → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
65	64	adantl 482	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 23901	. . . . . . . . . . . 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 1385	. . . . . . . . . . 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 3146	. . . . . . . . . 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 2903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑥(ball‘𝑀)1)
75		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧𝑦
76		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧(ball‘𝑀)
77		nfmpt1 5255	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
78	77	nfrn 5949	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
79		nfcv 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧ℝ
80		nfcv 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 <
81	78, 79, 80	nfsup 9442	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
82	75, 76, 81	nfov 7435	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
83	74, 82	nfss 3973	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
84		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85		oveq1 7412	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥(ball‘𝑀)1) = (𝑧(ball‘𝑀)1))
86	85	sseq1d 4012	. . . . . . . . . . 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 3303	. . . . . . . . . 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 5047	. . . . . . . . 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 4020	. . . . . . 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 23915	. . . . . . . 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 1371	. . . . . . 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 3998	. . . . . 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 7413	. . . . . . 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 3632	. . . . . 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 584	. . . . 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 3156	. . . 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 3154	. . 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 36636	. . . 4 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑑 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
102	101	baib 536	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → (𝑀 ∈ (Bnd‘𝑋) ↔ ∀𝑦 ∈ 𝑋 ∃𝑑 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
103	100, 102	sylibrd 258	. 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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 ∪ ciun 4996 class class class wbr 5147 ↦ cmpt 5230 Or wor 5586 dom cdm 5675 ran crn 5676 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 supcsup 9431 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 ℝ⁺crp 12970 ∞Metcxmet 20921 Metcmet 20922 ballcbl 20923 TotBndctotbnd 36622 Bndcbnd 36623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 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 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-totbnd 36624 df-bnd 36635

This theorem is referenced by: equivbnd2 36648 prdsbnd2 36651 cntotbnd 36652 cnpwstotbnd 36653

W3C validator