isbnd2 - Mathbox for Jeff Madsen

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Theorem isbnd2 38289

Description: The predicate "is a bounded metric space". Uses a single point instead of an arbitrary point in the space. (Contributed by Jeff Madsen, 2-Sep-2009.)

**Assertion**
Ref	Expression
isbnd2	⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))

Distinct variable groups: 𝑥,𝑟,𝑀 𝑋,𝑟,𝑥

Proof of Theorem isbnd2 Dummy variables 𝑠 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbndx 38288	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
2	1	anbi1i 635	. 2 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) ↔ ((𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) ∧ 𝑋 ≠ ∅))
3		anass 473	. 2 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) ∧ 𝑋 ≠ ∅) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ∧ 𝑋 ≠ ∅)))
4		r19.2z 4456	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
5	4	ancoms 463	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ∧ 𝑋 ≠ ∅) → ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
6		oveq1 7407	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥(ball‘𝑀)𝑟) = (𝑦(ball‘𝑀)𝑟))
7	6	eqeq2d 2776	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
8		oveq2 7408	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑦(ball‘𝑀)𝑟) = (𝑦(ball‘𝑀)𝑠))
9	8	eqeq2d 2776	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑦(ball‘𝑀)𝑠)))
10	7, 9	cbvrex2vw 3248	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ ∃𝑦 ∈ 𝑋 ∃𝑠 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑠))
11		2rp 13009	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
12		rpmulcl 13029	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) → (2 · 𝑠) ∈ ℝ⁺)
13	11, 12	mpan 702	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ⁺)
14	13	ad2antll 741	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (2 · 𝑠) ∈ ℝ⁺)
15	14	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → (2 · 𝑠) ∈ ℝ⁺)
16		rpcn 13015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℂ)
17		2cnd 12307	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 2 ∈ ℂ)
18		2ne0 12335	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
19	18	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 2 ≠ 0)
20		divcan3 11886	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑠) / 2) = 𝑠)
21	20	eqcomd 2771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → 𝑠 = ((2 · 𝑠) / 2))
22	16, 17, 19, 21	syl3anc 1394	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 = ((2 · 𝑠) / 2))
23	22	oveq2d 7416	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℝ⁺ → (𝑦(ball‘𝑀)𝑠) = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
24	23	eqeq2d 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℝ⁺ → (𝑋 = (𝑦(ball‘𝑀)𝑠) ↔ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
25	24	biimpd 232	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ ℝ⁺ → (𝑋 = (𝑦(ball‘𝑀)𝑠) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
26	25	ad2antll 741	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑋 = (𝑦(ball‘𝑀)𝑠) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
27	26	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) → (𝑋 = (𝑦(ball‘𝑀)𝑠) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
28	27	imp 411	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
29		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
30		eleq2 2854	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
31	30	biimpac 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))) → 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
32		2re 12303	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ
33		rpre 13013	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℝ)
34		remulcl 11173	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (2 · 𝑠) ∈ ℝ)
35	32, 33, 34	sylancr 598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ)
36		blhalf 24519	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ ((2 · 𝑠) ∈ ℝ ∧ 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))) → (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) ⊆ (𝑥(ball‘𝑀)(2 · 𝑠)))
37	36	expr 461	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (2 · 𝑠) ∈ ℝ) → (𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) → (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) ⊆ (𝑥(ball‘𝑀)(2 · 𝑠))))
38	35, 37	sylan2 604	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑠 ∈ ℝ⁺) → (𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) → (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) ⊆ (𝑥(ball‘𝑀)(2 · 𝑠))))
39	38	anasss 471	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) → (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) ⊆ (𝑥(ball‘𝑀)(2 · 𝑠))))
40	39	imp 411	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2))) → (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) ⊆ (𝑥(ball‘𝑀)(2 · 𝑠)))
41	31, 40	sylan2 604	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))) → (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) ⊆ (𝑥(ball‘𝑀)(2 · 𝑠)))
42	41	anassrs 472	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))) → (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) ⊆ (𝑥(ball‘𝑀)(2 · 𝑠)))
43	29, 42	eqsstrd 3973	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))) → 𝑋 ⊆ (𝑥(ball‘𝑀)(2 · 𝑠)))
44	28, 43	syldan 602	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 ⊆ (𝑥(ball‘𝑀)(2 · 𝑠)))
45	13	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺) → (2 · 𝑠) ∈ ℝ⁺)
46		rpxr 13014	. . . . . . . . . . . . . . . 16 ⊢ ((2 · 𝑠) ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ^*)
47		blssm 24532	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (2 · 𝑠) ∈ ℝ^*) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
48	46, 47	syl3an3 1181	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (2 · 𝑠) ∈ ℝ⁺) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
49	48	3expa 1134	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (2 · 𝑠) ∈ ℝ⁺) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
50	45, 49	sylan2 604	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
51	50	an32s 664	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
52	51	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
53	44, 52	eqssd 3956	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 = (𝑥(ball‘𝑀)(2 · 𝑠)))
54		oveq2 7408	. . . . . . . . . . 11 ⊢ (𝑟 = (2 · 𝑠) → (𝑥(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)(2 · 𝑠)))
55	54	rspceeqv 3607	. . . . . . . . . 10 ⊢ (((2 · 𝑠) ∈ ℝ⁺ ∧ 𝑋 = (𝑥(ball‘𝑀)(2 · 𝑠))) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
56	15, 53, 55	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
57	56	ex 417	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) → (𝑋 = (𝑦(ball‘𝑀)𝑠) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
58	57	ralrimdva 3165	. . . . . . 7 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑋 = (𝑦(ball‘𝑀)𝑠) → ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
59	58	rexlimdvva 3222	. . . . . 6 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (∃𝑦 ∈ 𝑋 ∃𝑠 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑠) → ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
60	10, 59	biimtrid 245	. . . . 5 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
61		rexn0 4453	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → 𝑋 ≠ ∅)
62	61	a1i 11	. . . . 5 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → 𝑋 ≠ ∅))
63	60, 62	jcad 521	. . . 4 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ∧ 𝑋 ≠ ∅)))
64	5, 63	impbid2 229	. . 3 ⊢ (𝑀 ∈ (∞Met‘𝑋) → ((∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ∧ 𝑋 ≠ ∅) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
65	64	pm5.32i 584	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ∧ 𝑋 ≠ ∅)) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
66	2, 3, 65	3bitri 300	1 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∀wral 3079 ∃wrex 3089 ⊆ wss 3907 ∅c0 4288 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 ℝcr 11087 0cc0 11088 · cmul 11093 ℝ^*cxr 11230 / cdiv 11859 2c2 12283 ℝ⁺crp 13004 ∞Metcxmet 21464 ballcbl 21466 Bndcbnd 38273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-er 8682 df-ec 8684 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-2 12291 df-rp 13005 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-psmet 21471 df-xmet 21472 df-met 21473 df-bl 21474 df-bnd 38285

This theorem is referenced by: isbnd3 38290 blbnd 38293 ssbnd 38294

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