isbnd2 - Mathbox for Jeff Madsen

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Theorem isbnd2 38810

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 38809	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
2	1	anbi1i 636	. 2 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) ↔ ((𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) ∧ 𝑋 ≠ ∅))
3		anass 474	. 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 4454	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
5	4	ancoms 464	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ∧ 𝑋 ≠ ∅) → ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
6		oveq1 7427	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥(ball‘𝑀)𝑟) = (𝑦(ball‘𝑀)𝑟))
7	6	eqeq2d 2771	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
8		oveq2 7428	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝑦(ball‘𝑀)𝑟) = (𝑦(ball‘𝑀)𝑠))
9	8	eqeq2d 2771	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑦(ball‘𝑀)𝑠)))
10	7, 9	cbvrex2vw 3245	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ ∃𝑦 ∈ 𝑋 ∃𝑠 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑠))
11		2rp 13128	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
12		rpmulcl 13148	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) → (2 · 𝑠) ∈ ℝ⁺)
13	11, 12	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ⁺)
14	13	ad2antll 742	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (2 · 𝑠) ∈ ℝ⁺)
15	14	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → (2 · 𝑠) ∈ ℝ⁺)
16		rpcn 13134	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℂ)
17		2cnd 12424	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 2 ∈ ℂ)
18		2ne0 12452	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
19	18	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 2 ≠ 0)
20		divcan3 12003	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑠) / 2) = 𝑠)
21	20	eqcomd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → 𝑠 = ((2 · 𝑠) / 2))
22	16, 17, 19, 21	syl3anc 1398	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 = ((2 · 𝑠) / 2))
23	22	oveq2d 7436	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℝ⁺ → (𝑦(ball‘𝑀)𝑠) = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
24	23	eqeq2d 2771	. . . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑋 = (𝑦(ball‘𝑀)𝑠) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
27	26	adantr 486	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
29		simpr 490	. . . . . . . . . . . . 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 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2)) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2))))
31	30	biimpac 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑋 = (𝑦(ball‘𝑀)((2 · 𝑠) / 2))) → 𝑥 ∈ (𝑦(ball‘𝑀)((2 · 𝑠) / 2)))
32		2re 12420	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ
33		rpre 13132	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℝ)
34		remulcl 11288	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (2 · 𝑠) ∈ ℝ)
35	32, 33, 34	sylancr 599	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ)
36		blhalf 24729	. . . . . . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . 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 3942	. . . . . . . . . . . 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 603	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 ⊆ (𝑥(ball‘𝑀)(2 · 𝑠)))
45	13	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺) → (2 · 𝑠) ∈ ℝ⁺)
46		rpxr 13133	. . . . . . . . . . . . . . . 16 ⊢ ((2 · 𝑠) ∈ ℝ⁺ → (2 · 𝑠) ∈ ℝ^*)
47		blssm 24742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (2 · 𝑠) ∈ ℝ^*) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
48	46, 47	syl3an3 1183	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (2 · 𝑠) ∈ ℝ⁺) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
49	48	3expa 1136	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (2 · 𝑠) ∈ ℝ⁺) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
50	45, 49	sylan2 605	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
51	50	an32s 665	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝑀)(2 · 𝑠)) ⊆ 𝑋)
52	51	adantr 486	. . . . . . . . . . 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 3925	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝑋 ∧ 𝑠 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑋 = (𝑦(ball‘𝑀)𝑠)) → 𝑋 = (𝑥(ball‘𝑀)(2 · 𝑠)))
54		oveq2 7428	. . . . . . . . . . 11 ⊢ (𝑟 = (2 · 𝑠) → (𝑥(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)(2 · 𝑠)))
55	54	rspceeqv 3598	. . . . . . . . . 10 ⊢ (((2 · 𝑠) ∈ ℝ⁺ ∧ 𝑋 = (𝑥(ball‘𝑀)(2 · 𝑠))) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
56	15, 53, 55	syl2anc 596	. . . . . . . . 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 418	. . . . . . . 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 3162	. . . . . . 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 3219	. . . . . 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 4451	. . . . . 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 522	. . . 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 585	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ⊆ wss 3895 ∅c0 4278 ‘cfv 6538 (class class class)co 7420 ℂcc 11201 ℝcr 11202 0cc0 11203 · cmul 11208 ℝ^*cxr 11345 / cdiv 11976 2c2 12400 ℝ⁺crp 13123 ∞Metcxmet 21668 ballcbl 21670 Bndcbnd 38794

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-er 8720 df-ec 8722 df-map 8852 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-psmet 21675 df-xmet 21676 df-met 21677 df-bl 21678 df-bnd 38806

This theorem is used by: isbnd3 38811 blbnd 38814 ssbnd 38815

W3C validator