hoiqssbl - Mathbox for Glauco Siliprandi

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Theorem hoiqssbl 46987

Description: A n-dimensional ball contains a nonempty half-open interval with vertices with rational components. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
hoiqssbl.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hoiqssbl.y	⊢ (𝜑 → 𝑌 ∈ (ℝ ↑_m 𝑋))
hoiqssbl.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)

**Assertion**
Ref	Expression
hoiqssbl	⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))

Distinct variable groups: 𝐸,𝑐,𝑑,𝑖 𝑋,𝑐,𝑑,𝑖 𝑌,𝑐,𝑑,𝑖 𝜑,𝑐,𝑑,𝑖

**Proof of Theorem hoiqssbl**
Step	Hyp	Ref	Expression
1		0ex 5254	. . . . . . 7 ⊢ ∅ ∈ V
2	1	snid 4621	. . . . . 6 ⊢ ∅ ∈ {∅}
3	2	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ {∅})
4		hoiqssbl.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑_m 𝑋))
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑌 ∈ (ℝ ↑_m 𝑋))
6		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
7		reex 11129	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
8		mapdm0 8791	. . . . . . . . . . . 12 ⊢ (ℝ ∈ V → (ℝ ↑_m ∅) = {∅})
9	7, 8	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℝ ↑_m ∅) = {∅}
10	9	a1i 11	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (ℝ ↑_m ∅) = {∅})
11	6, 10	eqtrd 2772	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = {∅})
12	11	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑_m 𝑋) = {∅})
13	5, 12	eleqtrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑌 ∈ {∅})
14		0fi 8991	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
15		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (dist‘(ℝ^‘∅)) = (dist‘(ℝ^‘∅))
16	15	rrxmetfi 25383	. . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → (dist‘(ℝ^‘∅)) ∈ (Met‘(ℝ ↑_m ∅)))
17	14, 16	ax-mp 5	. . . . . . . . . . . 12 ⊢ (dist‘(ℝ^‘∅)) ∈ (Met‘(ℝ ↑_m ∅))
18		metxmet 24293	. . . . . . . . . . . 12 ⊢ ((dist‘(ℝ^‘∅)) ∈ (Met‘(ℝ ↑_m ∅)) → (dist‘(ℝ^‘∅)) ∈ (∞Met‘(ℝ ↑_m ∅)))
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ (dist‘(ℝ^‘∅)) ∈ (∞Met‘(ℝ ↑_m ∅))
20	19	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (dist‘(ℝ^‘∅)) ∈ (∞Met‘(ℝ ↑_m ∅)))
21	3, 9	eleqtrrdi 2848	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ (ℝ ↑_m ∅))
22		hoiqssbl.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐸 ∈ ℝ⁺)
24		blcntr 24372	. . . . . . . . . 10 ⊢ (((dist‘(ℝ^‘∅)) ∈ (∞Met‘(ℝ ↑_m ∅)) ∧ ∅ ∈ (ℝ ↑_m ∅) ∧ 𝐸 ∈ ℝ⁺) → ∅ ∈ (∅(ball‘(dist‘(ℝ^‘∅)))𝐸))
25	20, 21, 23, 24	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ (∅(ball‘(dist‘(ℝ^‘∅)))𝐸))
26		elsni 4599	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ {∅} → 𝑌 = ∅)
27	13, 26	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑌 = ∅)
28	27	eqcomd 2743	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ = 𝑌)
29	28	oveq1d 7383	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (∅(ball‘(dist‘(ℝ^‘∅)))𝐸) = (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))
30	25, 29	eleqtrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))
31	30	snssd 4767	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))
32	13, 31	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
33		biidd 262	. . . . . . 7 ⊢ (𝑑 = ∅ → ((𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)) ↔ (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))))
34	33	rspcev 3578	. . . . . 6 ⊢ ((∅ ∈ {∅} ∧ (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))) → ∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
35	3, 32, 34	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
36		biidd 262	. . . . . 6 ⊢ (𝑐 = ∅ → (∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)) ↔ ∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))))
37	36	rspcev 3578	. . . . 5 ⊢ ((∅ ∈ {∅} ∧ ∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))) → ∃𝑐 ∈ {∅}∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
38	3, 35, 37	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∃𝑐 ∈ {∅}∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
39		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (ℚ ↑_m 𝑋) = (ℚ ↑_m ∅))
40		qex 12886	. . . . . . . . . . . 12 ⊢ ℚ ∈ V
41		mapdm0 8791	. . . . . . . . . . . 12 ⊢ (ℚ ∈ V → (ℚ ↑_m ∅) = {∅})
42	40, 41	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℚ ↑_m ∅) = {∅}
43	42	a1i 11	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (ℚ ↑_m ∅) = {∅})
44	39, 43	eqtr2d 2773	. . . . . . . . 9 ⊢ (𝑋 = ∅ → {∅} = (ℚ ↑_m 𝑋))
45	44	eqcomd 2743	. . . . . . . 8 ⊢ (𝑋 = ∅ → (ℚ ↑_m 𝑋) = {∅})
46	45	eleq2d 2823	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝑐 ∈ (ℚ ↑_m 𝑋) ↔ 𝑐 ∈ {∅}))
47	45	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (𝑑 ∈ (ℚ ↑_m 𝑋) ↔ 𝑑 ∈ {∅}))
48	47	anbi1d 632	. . . . . . . 8 ⊢ (𝑋 = ∅ → ((𝑑 ∈ (ℚ ↑_m 𝑋) ∧ (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))) ↔ (𝑑 ∈ {∅} ∧ (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))))
49	48	rexbidv2 3158	. . . . . . 7 ⊢ (𝑋 = ∅ → (∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)) ↔ ∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))))
50	46, 49	anbi12d 633	. . . . . 6 ⊢ (𝑋 = ∅ → ((𝑐 ∈ (ℚ ↑_m 𝑋) ∧ ∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))) ↔ (𝑐 ∈ {∅} ∧ ∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))))
51	50	rexbidv2 3158	. . . . 5 ⊢ (𝑋 = ∅ → (∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)) ↔ ∃𝑐 ∈ {∅}∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))))
52	51	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)) ↔ ∃𝑐 ∈ {∅}∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))))
53	38, 52	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
54		ixpeq1 8858	. . . . . . . . 9 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = X𝑖 ∈ ∅ ((𝑐‘𝑖)[,)(𝑑‘𝑖)))
55		ixp0x 8876	. . . . . . . . . 10 ⊢ X𝑖 ∈ ∅ ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = {∅}
56	55	a1i 11	. . . . . . . . 9 ⊢ (𝑋 = ∅ → X𝑖 ∈ ∅ ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = {∅})
57	54, 56	eqtrd 2772	. . . . . . . 8 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = {∅})
58	57	eleq2d 2823	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ↔ 𝑌 ∈ {∅}))
59		2fveq3 6847	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘∅)))
60	59	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (ball‘(dist‘(ℝ^‘𝑋))) = (ball‘(dist‘(ℝ^‘∅))))
61	60	oveqd 7385	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) = (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))
62	57, 61	sseq12d 3969	. . . . . . 7 ⊢ (𝑋 = ∅ → (X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) ↔ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
63	58, 62	anbi12d 633	. . . . . 6 ⊢ (𝑋 = ∅ → ((𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) ↔ (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))))
64	63	rexbidv 3162	. . . . 5 ⊢ (𝑋 = ∅ → (∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) ↔ ∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))))
65	64	rexbidv 3162	. . . 4 ⊢ (𝑋 = ∅ → (∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) ↔ ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))))
66	65	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)) ↔ ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))))
67	53, 66	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))
68		hoiqssbl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin)
69	68	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
70		neqne 2941	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
71	70	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
72	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑌 ∈ (ℝ ↑_m 𝑋))
73	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐸 ∈ ℝ⁺)
74	69, 71, 72, 73	hoiqssbllem3 46986	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))
75	67, 74	pm2.61dan 813	1 ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 Vcvv 3442 ⊆ wss 3903 ∅c0 4287 {csn 4582 ‘cfv 6500 (class class class)co 7368 ↑_m cmap 8775 Xcixp 8847 Fincfn 8895 ℝcr 11037 ℚcq 12873 ℝ⁺crp 12917 [,)cico 13275 distcds 17198 ∞Metcxmet 21309 Metcmet 21310 ballcbl 21311 ℝ^crrx 25354

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xadd 13039 df-ioo 13277 df-ico 13279 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-grp 18881 df-minusg 18882 df-sbg 18883 df-subg 19068 df-ghm 19157 df-cntz 19261 df-cmn 19726 df-abl 19727 df-mgp 20091 df-rng 20103 df-ur 20132 df-ring 20185 df-cring 20186 df-oppr 20288 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-rhm 20423 df-subrng 20494 df-subrg 20518 df-drng 20679 df-field 20680 df-staf 20787 df-srng 20788 df-lmod 20828 df-lss 20898 df-sra 21140 df-rgmod 21141 df-psmet 21316 df-xmet 21317 df-met 21318 df-bl 21319 df-cnfld 21325 df-refld 21575 df-dsmm 21702 df-frlm 21717 df-nm 24541 df-tng 24543 df-tcph 25140 df-rrx 25356

This theorem is referenced by: opnvonmbllem2 46995

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