isbnd2 - Mathbox for Jeff Madsen

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Theorem isbnd2 35076

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 35075	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
2	1	anbi1i 625	. 2 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) ↔ ((𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) ∧ 𝑋 ≠ ∅))
3		anass 471	. 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 4440	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
5	4	ancoms 461	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ∧ 𝑋 ≠ ∅) → ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
6		oveq1 7163	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥(ball‘𝑀)𝑟) = (𝑦(ball‘𝑀)𝑟))
7	6	eqeq2d 2832	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
8		oveq2 7164	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑦(ball‘𝑀)𝑟) = (𝑦(ball‘𝑀)𝑠))
9	8	eqeq2d 2832	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑦(ball‘𝑀)𝑠)))
10	7, 9	cbvrex2vw 3462	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ ∃𝑦 ∈ 𝑋 ∃𝑠 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑠))
11		2rp 12395	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
12		rpmulcl 12413	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) → (2 · 𝑠) ∈ ℝ⁺)
13	11, 12	mpan 688	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ⁺)
14	13	ad2antll 727	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (2 · 𝑠) ∈ ℝ⁺)
15	14	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → (2 · 𝑠) ∈ ℝ⁺)
16		rpcn 12400	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℂ)
17		2cnd 11716	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 2 ∈ ℂ)
18		2ne0 11742	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
19	18	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 2 ≠ 0)
20		divcan3 11324	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑠) / 2) = 𝑠)
21	20	eqcomd 2827	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → 𝑠 = ((2 · 𝑠) / 2))
22	16, 17, 19, 21	syl3anc 1367	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 = ((2 · 𝑠) / 2))
23	22	oveq2d 7172	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℝ⁺ → (𝑦(ball‘𝑀)𝑠) = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
24	23	eqeq2d 2832	. . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑋 = (𝑦(ball‘𝑀)𝑠) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
27	26	adantr 483	. . . . . . . . . . . . 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 409	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
29		simpr 487	. . . . . . . . . . . . 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 2901	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
31	30	biimpac 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))) → 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
32		2re 11712	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ
33		rpre 12398	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℝ)
34		remulcl 10622	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (2 · 𝑠) ∈ ℝ)
35	32, 33, 34	sylancr 589	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ)
36		blhalf 23015	. . . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . 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 469	. . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . 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 470	. . . . . . . . . . . . 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 4005	. . . . . . . . . . . 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 593	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 ⊆ (𝑥(ball‘𝑀)(2 · 𝑠)))
45	13	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺) → (2 · 𝑠) ∈ ℝ⁺)
46		rpxr 12399	. . . . . . . . . . . . . . . 16 ⊢ ((2 · 𝑠) ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ^*)
47		blssm 23028	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (2 · 𝑠) ∈ ℝ^*) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
48	46, 47	syl3an3 1161	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (2 · 𝑠) ∈ ℝ⁺) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
49	48	3expa 1114	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (2 · 𝑠) ∈ ℝ⁺) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
50	45, 49	sylan2 594	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
51	50	an32s 650	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
52	51	adantr 483	. . . . . . . . . . 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 3984	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 = (𝑥(ball‘𝑀)(2 · 𝑠)))
54		oveq2 7164	. . . . . . . . . . 11 ⊢ (𝑟 = (2 · 𝑠) → (𝑥(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)(2 · 𝑠)))
55	54	rspceeqv 3638	. . . . . . . . . 10 ⊢ (((2 · 𝑠) ∈ ℝ⁺ ∧ 𝑋 = (𝑥(ball‘𝑀)(2 · 𝑠))) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
56	15, 53, 55	syl2anc 586	. . . . . . . . 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 415	. . . . . . . 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 3189	. . . . . . 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 3294	. . . . . 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	syl5bi 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 4454	. . . . . 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 515	. . . 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 577	. 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 ∅c0 4291 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 · cmul 10542 ℝ^*cxr 10674 / cdiv 11297 2c2 11693 ℝ⁺crp 12390 ∞Metcxmet 20530 ballcbl 20532 Bndcbnd 35060

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-ec 8291 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-bnd 35072

This theorem is referenced by: isbnd3 35077 blbnd 35080 ssbnd 35081

W3C validator