Theorem isbnd3 33583

Description: A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 13-Sep-2015.)

Assertion

Ref	Expression
isbnd3	⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)))

Distinct variable groups:   𝑥,𝑀   𝑥,𝑋

Proof of Theorem isbnd3

Dummy variables 𝑟 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bndmet 33580	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
2		0re 10040	. . . . . 6 ⊢ 0 ∈ ℝ
3	2	ne0ii 3923	. . . . 5 ⊢ ℝ ≠ ∅
4		metf 22135	. . . . . . . . . 10 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
5		ffn 6045	. . . . . . . . . 10 ⊢ (𝑀:(𝑋 × 𝑋)⟶ℝ → 𝑀 Fn (𝑋 × 𝑋))
6	4, 5	syl 17	. . . . . . . . 9 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 Fn (𝑋 × 𝑋))
7	1, 6	syl 17	. . . . . . . 8 ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 Fn (𝑋 × 𝑋))
8	7	ad2antrr 762	. . . . . . 7 ⊢ (((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 = ∅) ∧ 𝑥 ∈ ℝ) → 𝑀 Fn (𝑋 × 𝑋))
9	1, 4	syl 17	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
10		fdm 6051	. . . . . . . . . . . 12 ⊢ (𝑀:(𝑋 × 𝑋)⟶ℝ → dom 𝑀 = (𝑋 × 𝑋))
11	9, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (Bnd‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
12		xpeq2 5129	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → (𝑋 × 𝑋) = (𝑋 × ∅))
13		xp0 5552	. . . . . . . . . . . 12 ⊢ (𝑋 × ∅) = ∅
14	12, 13	syl6eq 2672	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → (𝑋 × 𝑋) = ∅)
15	11, 14	sylan9eq 2676	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 = ∅) → dom 𝑀 = ∅)
16	15	adantr 481	. . . . . . . . 9 ⊢ (((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 = ∅) ∧ 𝑥 ∈ ℝ) → dom 𝑀 = ∅)
17		dm0rn0 5342	. . . . . . . . 9 ⊢ (dom 𝑀 = ∅ ↔ ran 𝑀 = ∅)
18	16, 17	sylib 208	. . . . . . . 8 ⊢ (((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 = ∅) ∧ 𝑥 ∈ ℝ) → ran 𝑀 = ∅)
19		0ss 3972	. . . . . . . 8 ⊢ ∅ ⊆ (0[,]𝑥)
20	18, 19	syl6eqss 3655	. . . . . . 7 ⊢ (((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 = ∅) ∧ 𝑥 ∈ ℝ) → ran 𝑀 ⊆ (0[,]𝑥))
21		df-f 5892	. . . . . . 7 ⊢ (𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥) ↔ (𝑀 Fn (𝑋 × 𝑋) ∧ ran 𝑀 ⊆ (0[,]𝑥)))
22	8, 20, 21	sylanbrc 698	. . . . . 6 ⊢ (((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 = ∅) ∧ 𝑥 ∈ ℝ) → 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥))
23	22	ralrimiva 2966	. . . . 5 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 = ∅) → ∀𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥))
24		r19.2z 4060	. . . . 5 ⊢ ((ℝ ≠ ∅ ∧ ∀𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) → ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥))
25	3, 23, 24	sylancr 695	. . . 4 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 = ∅) → ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥))
26		isbnd2 33582	. . . . . 6 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
27	26	simprbi 480	. . . . 5 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) → ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟))
28		2re 11090	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
29		simprlr 803	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) → 𝑟 ∈ ℝ+)
30	29	rpred 11872	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) → 𝑟 ∈ ℝ)
31		remulcl 10021	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 𝑟 ∈ ℝ) → (2 · 𝑟) ∈ ℝ)
32	28, 30, 31	sylancr 695	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) → (2 · 𝑟) ∈ ℝ)
33	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) → 𝑀 Fn (𝑋 × 𝑋))
34		simpll 790	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑀 ∈ (Met‘𝑋))
35		simprl 794	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
36		simprr 796	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
37		metcl 22137	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥𝑀𝑧) ∈ ℝ)
38	34, 35, 36, 37	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝑀𝑧) ∈ ℝ)
39		metge0 22150	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → 0 ≤ (𝑥𝑀𝑧))
40	34, 35, 36, 39	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 0 ≤ (𝑥𝑀𝑧))
41	32	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (2 · 𝑟) ∈ ℝ)
42		simprll 802	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) → 𝑦 ∈ 𝑋)
43	42	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
44		metcl 22137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑦𝑀𝑥) ∈ ℝ)
45	34, 43, 35, 44	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦𝑀𝑥) ∈ ℝ)
46		metcl 22137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦𝑀𝑧) ∈ ℝ)
47	34, 43, 36, 46	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦𝑀𝑧) ∈ ℝ)
48	45, 47	readdcld 10069	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦𝑀𝑥) + (𝑦𝑀𝑧)) ∈ ℝ)
49		mettri2 22146	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝑀𝑧) ≤ ((𝑦𝑀𝑥) + (𝑦𝑀𝑧)))
50	34, 43, 35, 36, 49	syl13anc 1328	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝑀𝑧) ≤ ((𝑦𝑀𝑥) + (𝑦𝑀𝑧)))
51	30	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑟 ∈ ℝ)
52		simplrr 801	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑋 = (𝑦(ball‘𝑀)𝑟))
53	35, 52	eleqtrd 2703	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ (𝑦(ball‘𝑀)𝑟))
54		metxmet 22139	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
55	34, 54	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑀 ∈ (∞Met‘𝑋))
56		rpxr 11840	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
57	56	ad2antlr 763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → 𝑟 ∈ ℝ*)
58	57	ad2antlr 763	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑟 ∈ ℝ*)
59		elbl2 22195	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑟 ∈ ℝ*) ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝑀)𝑟) ↔ (𝑦𝑀𝑥) < 𝑟))
60	55, 58, 43, 35, 59	syl22anc 1327	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 ∈ (𝑦(ball‘𝑀)𝑟) ↔ (𝑦𝑀𝑥) < 𝑟))
61	53, 60	mpbid 222	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦𝑀𝑥) < 𝑟)
62	36, 52	eleqtrd 2703	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ (𝑦(ball‘𝑀)𝑟))
63		elbl2 22195	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑟 ∈ ℝ*) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧 ∈ (𝑦(ball‘𝑀)𝑟) ↔ (𝑦𝑀𝑧) < 𝑟))
64	55, 58, 43, 36, 63	syl22anc 1327	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧 ∈ (𝑦(ball‘𝑀)𝑟) ↔ (𝑦𝑀𝑧) < 𝑟))
65	62, 64	mpbid 222	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦𝑀𝑧) < 𝑟)
66	45, 47, 51, 51, 61, 65	lt2addd 10650	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦𝑀𝑥) + (𝑦𝑀𝑧)) < (𝑟 + 𝑟))
67	51	recnd 10068	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑟 ∈ ℂ)
68	67	2timesd 11275	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (2 · 𝑟) = (𝑟 + 𝑟))
69	66, 68	breqtrrd 4681	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦𝑀𝑥) + (𝑦𝑀𝑧)) < (2 · 𝑟))
70	38, 48, 41, 50, 69	lelttrd 10195	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝑀𝑧) < (2 · 𝑟))
71	38, 41, 70	ltled 10185	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝑀𝑧) ≤ (2 · 𝑟))
72		elicc2 12238	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ (2 · 𝑟) ∈ ℝ) → ((𝑥𝑀𝑧) ∈ (0[,](2 · 𝑟)) ↔ ((𝑥𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑥𝑀𝑧) ∧ (𝑥𝑀𝑧) ≤ (2 · 𝑟))))
73	2, 41, 72	sylancr 695	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝑀𝑧) ∈ (0[,](2 · 𝑟)) ↔ ((𝑥𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑥𝑀𝑧) ∧ (𝑥𝑀𝑧) ≤ (2 · 𝑟))))
74	38, 40, 71, 73	mpbir3and 1245	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝑀𝑧) ∈ (0[,](2 · 𝑟)))
75	74	ralrimivva 2971	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) → ∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑀𝑧) ∈ (0[,](2 · 𝑟)))
76		ffnov 6764	. . . . . . . . . . 11 ⊢ (𝑀:(𝑋 × 𝑋)⟶(0[,](2 · 𝑟)) ↔ (𝑀 Fn (𝑋 × 𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝑀𝑧) ∈ (0[,](2 · 𝑟))))
77	33, 75, 76	sylanbrc 698	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) → 𝑀:(𝑋 × 𝑋)⟶(0[,](2 · 𝑟)))
78		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑥 = (2 · 𝑟) → (0[,]𝑥) = (0[,](2 · 𝑟)))
79	78	feq3d 6032	. . . . . . . . . . 11 ⊢ (𝑥 = (2 · 𝑟) → (𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥) ↔ 𝑀:(𝑋 × 𝑋)⟶(0[,](2 · 𝑟))))
80	79	rspcev 3309	. . . . . . . . . 10 ⊢ (((2 · 𝑟) ∈ ℝ ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,](2 · 𝑟))) → ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥))
81	32, 77, 80	syl2anc 693	. . . . . . . . 9 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑟))) → ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥))
82	81	expr 643	. . . . . . . 8 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+)) → (𝑋 = (𝑦(ball‘𝑀)𝑟) → ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)))
83	82	rexlimdvva 3038	. . . . . . 7 ⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟) → ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)))
84	1, 83	syl 17	. . . . . 6 ⊢ (𝑀 ∈ (Bnd‘𝑋) → (∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟) → ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)))
85	84	adantr 481	. . . . 5 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) → (∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟) → ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)))
86	27, 85	mpd 15	. . . 4 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) → ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥))
87	25, 86	pm2.61dane 2881	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) → ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥))
88	1, 87	jca 554	. 2 ⊢ (𝑀 ∈ (Bnd‘𝑋) → (𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)))
89		simpll 790	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) → 𝑀 ∈ (Met‘𝑋))
90		simpllr 799	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ ℝ)
91	89	adantr 481	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → 𝑀 ∈ (Met‘𝑋))
92		simpr 477	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
93		met0 22148	. . . . . . . . 9 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑦𝑀𝑦) = 0)
94	91, 92, 93	syl2anc 693	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → (𝑦𝑀𝑦) = 0)
95		simplr 792	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥))
96	95, 92, 92	fovrnd 6806	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → (𝑦𝑀𝑦) ∈ (0[,]𝑥))
97		elicc2 12238	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑦𝑀𝑦) ∈ (0[,]𝑥) ↔ ((𝑦𝑀𝑦) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑦) ∧ (𝑦𝑀𝑦) ≤ 𝑥)))
98	2, 90, 97	sylancr 695	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝑀𝑦) ∈ (0[,]𝑥) ↔ ((𝑦𝑀𝑦) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑦) ∧ (𝑦𝑀𝑦) ≤ 𝑥)))
99	96, 98	mpbid 222	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝑀𝑦) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑦) ∧ (𝑦𝑀𝑦) ≤ 𝑥))
100	99	simp3d 1075	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → (𝑦𝑀𝑦) ≤ 𝑥)
101	94, 100	eqbrtrrd 4677	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → 0 ≤ 𝑥)
102	90, 101	ge0p1rpd 11902	. . . . . 6 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → (𝑥 + 1) ∈ ℝ+)
103		fovrn 6804	. . . . . . . . . . . . . 14 ⊢ ((𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦𝑀𝑧) ∈ (0[,]𝑥))
104	103	3expa 1265	. . . . . . . . . . . . 13 ⊢ (((𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (𝑦𝑀𝑧) ∈ (0[,]𝑥))
105	104	adantlll 754	. . . . . . . . . . . 12 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (𝑦𝑀𝑧) ∈ (0[,]𝑥))
106		elicc2 12238	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑦𝑀𝑧) ∈ (0[,]𝑥) ↔ ((𝑦𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑧) ∧ (𝑦𝑀𝑧) ≤ 𝑥)))
107	2, 90, 106	sylancr 695	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝑀𝑧) ∈ (0[,]𝑥) ↔ ((𝑦𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑧) ∧ (𝑦𝑀𝑧) ≤ 𝑥)))
108	107	adantr 481	. . . . . . . . . . . 12 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑦𝑀𝑧) ∈ (0[,]𝑥) ↔ ((𝑦𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑧) ∧ (𝑦𝑀𝑧) ≤ 𝑥)))
109	105, 108	mpbid 222	. . . . . . . . . . 11 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑦𝑀𝑧) ∈ ℝ ∧ 0 ≤ (𝑦𝑀𝑧) ∧ (𝑦𝑀𝑧) ≤ 𝑥))
110	109	simp1d 1073	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (𝑦𝑀𝑧) ∈ ℝ)
111	90	adantr 481	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝑥 ∈ ℝ)
112		peano2re 10209	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
113	90, 112	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → (𝑥 + 1) ∈ ℝ)
114	113	adantr 481	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (𝑥 + 1) ∈ ℝ)
115	109	simp3d 1075	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (𝑦𝑀𝑧) ≤ 𝑥)
116	111	ltp1d 10954	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝑥 < (𝑥 + 1))
117	110, 111, 114, 115, 116	lelttrd 10195	. . . . . . . . 9 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (𝑦𝑀𝑧) < (𝑥 + 1))
118	117	ralrimiva 2966	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) < (𝑥 + 1))
119		rabid2 3118	. . . . . . . 8 ⊢ (𝑋 = {𝑧 ∈ 𝑋 ∣ (𝑦𝑀𝑧) < (𝑥 + 1)} ↔ ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) < (𝑥 + 1))
120	118, 119	sylibr 224	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → 𝑋 = {𝑧 ∈ 𝑋 ∣ (𝑦𝑀𝑧) < (𝑥 + 1)})
121	91, 54	syl 17	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → 𝑀 ∈ (∞Met‘𝑋))
122	113	rexrd 10089	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → (𝑥 + 1) ∈ ℝ*)
123		blval 22191	. . . . . . . 8 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ (𝑥 + 1) ∈ ℝ*) → (𝑦(ball‘𝑀)(𝑥 + 1)) = {𝑧 ∈ 𝑋 ∣ (𝑦𝑀𝑧) < (𝑥 + 1)})
124	121, 92, 122, 123	syl3anc 1326	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → (𝑦(ball‘𝑀)(𝑥 + 1)) = {𝑧 ∈ 𝑋 ∣ (𝑦𝑀𝑧) < (𝑥 + 1)})
125	120, 124	eqtr4d 2659	. . . . . 6 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → 𝑋 = (𝑦(ball‘𝑀)(𝑥 + 1)))
126		oveq2 6658	. . . . . . . 8 ⊢ (𝑟 = (𝑥 + 1) → (𝑦(ball‘𝑀)𝑟) = (𝑦(ball‘𝑀)(𝑥 + 1)))
127	126	eqeq2d 2632	. . . . . . 7 ⊢ (𝑟 = (𝑥 + 1) → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑦(ball‘𝑀)(𝑥 + 1))))
128	127	rspcev 3309	. . . . . 6 ⊢ (((𝑥 + 1) ∈ ℝ+ ∧ 𝑋 = (𝑦(ball‘𝑀)(𝑥 + 1))) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟))
129	102, 125, 128	syl2anc 693	. . . . 5 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) ∧ 𝑦 ∈ 𝑋) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟))
130	129	ralrimiva 2966	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) → ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟))
131		isbnd 33579	. . . 4 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
132	89, 130, 131	sylanbrc 698	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ ℝ) ∧ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) → 𝑀 ∈ (Bnd‘𝑋))
133	132	r19.29an 3077	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)) → 𝑀 ∈ (Bnd‘𝑋))
134	88, 133	impbii 199	1 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653   × cxp 5112  dom cdm 5114  ran crn 5115   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  2c2 11070  ℝ⁺crp 11832  [,]cicc 12178  ∞Metcxmt 19731  Metcme 19732  ballcbl 19733  Bndcbnd 33566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-ec 7744  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-bnd 33578

This theorem is referenced by:  isbnd3b  33584  prdsbnd  33592