isbnd2 - Mathbox for Jeff Madsen

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Theorem isbnd2 38242

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 38241	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
2	1	anbi1i 633	. 2 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) ↔ ((𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) ∧ 𝑋 ≠ ∅))
3		anass 472	. 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 4450	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
5	4	ancoms 462	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ∧ 𝑋 ≠ ∅) → ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
6		oveq1 7397	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥(ball‘𝑀)𝑟) = (𝑦(ball‘𝑀)𝑟))
7	6	eqeq2d 2772	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
8		oveq2 7398	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑦(ball‘𝑀)𝑟) = (𝑦(ball‘𝑀)𝑠))
9	8	eqeq2d 2772	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑦(ball‘𝑀)𝑠)))
10	7, 9	cbvrex2vw 3244	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ ∃𝑦 ∈ 𝑋 ∃𝑠 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑠))
11		2rp 12991	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
12		rpmulcl 13011	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) → (2 · 𝑠) ∈ ℝ⁺)
13	11, 12	mpan 700	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ⁺)
14	13	ad2antll 739	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (2 · 𝑠) ∈ ℝ⁺)
15	14	ad2antrr 736	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → (2 · 𝑠) ∈ ℝ⁺)
16		rpcn 12997	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℂ)
17		2cnd 12289	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 2 ∈ ℂ)
18		2ne0 12317	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
19	18	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 2 ≠ 0)
20		divcan3 11864	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑠) / 2) = 𝑠)
21	20	eqcomd 2767	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → 𝑠 = ((2 · 𝑠) / 2))
22	16, 17, 19, 21	syl3anc 1389	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 = ((2 · 𝑠) / 2))
23	22	oveq2d 7406	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℝ⁺ → (𝑦(ball‘𝑀)𝑠) = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
24	23	eqeq2d 2772	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℝ⁺ → (𝑋 = (𝑦(ball‘𝑀)𝑠) ↔ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
25	24	biimpd 231	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ ℝ⁺ → (𝑋 = (𝑦(ball‘𝑀)𝑠) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
26	25	ad2antll 739	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑋 = (𝑦(ball‘𝑀)𝑠) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
27	26	adantr 484	. . . . . . . . . . . . 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 410	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
29		simpr 488	. . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
31	30	biimpac 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))) → 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
32		2re 12285	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ
33		rpre 12995	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℝ)
34		remulcl 11151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (2 · 𝑠) ∈ ℝ)
35	32, 33, 34	sylancr 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ)
36		blhalf 24452	. . . . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . . . 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 602	. . . . . . . . . . . . . . . . 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 470	. . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . 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 602	. . . . . . . . . . . . . 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 471	. . . . . . . . . . . . 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 3968	. . . . . . . . . . . 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 600	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 ⊆ (𝑥(ball‘𝑀)(2 · 𝑠)))
45	13	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺) → (2 · 𝑠) ∈ ℝ⁺)
46		rpxr 12996	. . . . . . . . . . . . . . . 16 ⊢ ((2 · 𝑠) ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ^*)
47		blssm 24465	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (2 · 𝑠) ∈ ℝ^*) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
48	46, 47	syl3an3 1177	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (2 · 𝑠) ∈ ℝ⁺) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
49	48	3expa 1130	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (2 · 𝑠) ∈ ℝ⁺) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
50	45, 49	sylan2 602	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
51	50	an32s 662	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
52	51	adantr 484	. . . . . . . . . . 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 3951	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 = (𝑥(ball‘𝑀)(2 · 𝑠)))
54		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑟 = (2 · 𝑠) → (𝑥(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)(2 · 𝑠)))
55	54	rspceeqv 3603	. . . . . . . . . 10 ⊢ (((2 · 𝑠) ∈ ℝ⁺ ∧ 𝑋 = (𝑥(ball‘𝑀)(2 · 𝑠))) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
56	15, 53, 55	syl2anc 593	. . . . . . . . 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 416	. . . . . . . 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 3161	. . . . . . 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 3218	. . . . . 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 244	. . . . 5 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
61		rexn0 4447	. . . . . 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 520	. . . 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 228	. . 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 582	. 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 299	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 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 ⊆ wss 3902 ∅c0 4283 ‘cfv 6515 (class class class)co 7390 ℂcc 11064 ℝcr 11065 0cc0 11066 · cmul 11071 ℝ^*cxr 11208 / cdiv 11837 2c2 12265 ℝ⁺crp 12986 ∞Metcxmet 21396 ballcbl 21398 Bndcbnd 38226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-er 8671 df-ec 8673 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-rp 12987 df-xneg 13107 df-xadd 13108 df-xmul 13109 df-psmet 21403 df-xmet 21404 df-met 21405 df-bl 21406 df-bnd 38238

This theorem is referenced by: isbnd3 38243 blbnd 38246 ssbnd 38247

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