isbnd2 - Mathbox for Jeff Madsen

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Theorem isbnd2 37784

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 37783	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
2	1	anbi1i 624	. 2 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) ↔ ((𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) ∧ 𝑋 ≠ ∅))
3		anass 468	. 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 4461	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
5	4	ancoms 458	. . . 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 2741	. . . . . . 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 2741	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑦(ball‘𝑀)𝑠)))
10	7, 9	cbvrex2vw 3221	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ ∃𝑦 ∈ 𝑋 ∃𝑠 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑠))
11		2rp 12963	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
12		rpmulcl 12983	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) → (2 · 𝑠) ∈ ℝ⁺)
13	11, 12	mpan 690	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ⁺)
14	13	ad2antll 729	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (2 · 𝑠) ∈ ℝ⁺)
15	14	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → (2 · 𝑠) ∈ ℝ⁺)
16		rpcn 12969	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℂ)
17		2cnd 12271	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 2 ∈ ℂ)
18		2ne0 12297	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
19	18	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 2 ≠ 0)
20		divcan3 11870	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑠) / 2) = 𝑠)
21	20	eqcomd 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → 𝑠 = ((2 · 𝑠) / 2))
22	16, 17, 19, 21	syl3anc 1373	. . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℝ⁺ → (𝑋 = (𝑦(ball‘𝑀)𝑠) ↔ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
25	24	biimpd 229	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ ℝ⁺ → (𝑋 = (𝑦(ball‘𝑀)𝑠) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
26	25	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑋 = (𝑦(ball‘𝑀)𝑠) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
27	26	adantr 480	. . . . . . . . . . . . 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 406	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
29		simpr 484	. . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
31	30	biimpac 478	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))) → 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
32		2re 12267	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ
33		rpre 12967	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℝ)
34		remulcl 11160	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (2 · 𝑠) ∈ ℝ)
35	32, 33, 34	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ)
36		blhalf 24300	. . . . . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . 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 467	. . . . . . . . . . . . 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 3984	. . . . . . . . . . . 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 591	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 ⊆ (𝑥(ball‘𝑀)(2 · 𝑠)))
45	13	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺) → (2 · 𝑠) ∈ ℝ⁺)
46		rpxr 12968	. . . . . . . . . . . . . . . 16 ⊢ ((2 · 𝑠) ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ^*)
47		blssm 24313	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (2 · 𝑠) ∈ ℝ^*) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
48	46, 47	syl3an3 1165	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (2 · 𝑠) ∈ ℝ⁺) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
49	48	3expa 1118	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (2 · 𝑠) ∈ ℝ⁺) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
50	45, 49	sylan2 593	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
51	50	an32s 652	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
52	51	adantr 480	. . . . . . . . . . 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 3967	. . . . . . . . . 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 3614	. . . . . . . . . 10 ⊢ (((2 · 𝑠) ∈ ℝ⁺ ∧ 𝑋 = (𝑥(ball‘𝑀)(2 · 𝑠))) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
56	15, 53, 55	syl2anc 584	. . . . . . . . 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 412	. . . . . . . 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 3134	. . . . . . 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 3195	. . . . . 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 242	. . . . 5 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
61		rexn0 4477	. . . . . 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 512	. . . 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 226	. . 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 574	. 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 297	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 ⊆ wss 3917 ∅c0 4299 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 · cmul 11080 ℝ^*cxr 11214 / cdiv 11842 2c2 12248 ℝ⁺crp 12958 ∞Metcxmet 21256 ballcbl 21258 Bndcbnd 37768

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-ec 8676 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-bnd 37780

This theorem is referenced by: isbnd3 37785 blbnd 37788 ssbnd 37789

W3C validator