hoiqssbl - Mathbox for Glauco Siliprandi

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Theorem hoiqssbl 46623

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 5262	. . . . . . 7 ⊢ ∅ ∈ V
2	1	snid 4626	. . . . . 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 7395	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
7		reex 11159	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
8		mapdm0 8815	. . . . . . . . . . . 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 2764	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = {∅})
12	11	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑_m 𝑋) = {∅})
13	5, 12	eleqtrd 2830	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑌 ∈ {∅})
14		0fi 9013	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
15		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (dist‘(ℝ^‘∅)) = (dist‘(ℝ^‘∅))
16	15	rrxmetfi 25312	. . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → (dist‘(ℝ^‘∅)) ∈ (Met‘(ℝ ↑_m ∅)))
17	14, 16	ax-mp 5	. . . . . . . . . . . 12 ⊢ (dist‘(ℝ^‘∅)) ∈ (Met‘(ℝ ↑_m ∅))
18		metxmet 24222	. . . . . . . . . . . 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 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ (ℝ ↑_m ∅))
22		hoiqssbl.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐸 ∈ ℝ⁺)
24		blcntr 24301	. . . . . . . . . 10 ⊢ (((dist‘(ℝ^‘∅)) ∈ (∞Met‘(ℝ ↑_m ∅)) ∧ ∅ ∈ (ℝ ↑_m ∅) ∧ 𝐸 ∈ ℝ⁺) → ∅ ∈ (∅(ball‘(dist‘(ℝ^‘∅)))𝐸))
25	20, 21, 23, 24	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ (∅(ball‘(dist‘(ℝ^‘∅)))𝐸))
26		elsni 4606	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ {∅} → 𝑌 = ∅)
27	13, 26	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑌 = ∅)
28	27	eqcomd 2735	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ = 𝑌)
29	28	oveq1d 7402	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (∅(ball‘(dist‘(ℝ^‘∅)))𝐸) = (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))
30	25, 29	eleqtrd 2830	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))
31	30	snssd 4773	. . . . . . 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 3588	. . . . . 6 ⊢ ((∅ ∈ {∅} ∧ (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))) → ∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
35	3, 32, 34	syl2anc 584	. . . . 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 3588	. . . . 5 ⊢ ((∅ ∈ {∅} ∧ ∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))) → ∃𝑐 ∈ {∅}∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
38	3, 35, 37	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∃𝑐 ∈ {∅}∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
39		oveq2 7395	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (ℚ ↑_m 𝑋) = (ℚ ↑_m ∅))
40		qex 12920	. . . . . . . . . . . 12 ⊢ ℚ ∈ V
41		mapdm0 8815	. . . . . . . . . . . 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 2765	. . . . . . . . 9 ⊢ (𝑋 = ∅ → {∅} = (ℚ ↑_m 𝑋))
45	44	eqcomd 2735	. . . . . . . 8 ⊢ (𝑋 = ∅ → (ℚ ↑_m 𝑋) = {∅})
46	45	eleq2d 2814	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝑐 ∈ (ℚ ↑_m 𝑋) ↔ 𝑐 ∈ {∅}))
47	45	eleq2d 2814	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (𝑑 ∈ (ℚ ↑_m 𝑋) ↔ 𝑑 ∈ {∅}))
48	47	anbi1d 631	. . . . . . . 8 ⊢ (𝑋 = ∅ → ((𝑑 ∈ (ℚ ↑_m 𝑋) ∧ (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))) ↔ (𝑑 ∈ {∅} ∧ (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))))
49	48	rexbidv2 3153	. . . . . . 7 ⊢ (𝑋 = ∅ → (∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)) ↔ ∃𝑑 ∈ {∅} (𝑌 ∈ {∅} ∧ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))))
50	46, 49	anbi12d 632	. . . . . 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 3153	. . . . 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 8881	. . . . . . . . 9 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = X𝑖 ∈ ∅ ((𝑐‘𝑖)[,)(𝑑‘𝑖)))
55		ixp0x 8899	. . . . . . . . . 10 ⊢ X𝑖 ∈ ∅ ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = {∅}
56	55	a1i 11	. . . . . . . . 9 ⊢ (𝑋 = ∅ → X𝑖 ∈ ∅ ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = {∅})
57	54, 56	eqtrd 2764	. . . . . . . 8 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = {∅})
58	57	eleq2d 2814	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ↔ 𝑌 ∈ {∅}))
59		2fveq3 6863	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘∅)))
60	59	fveq2d 6862	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (ball‘(dist‘(ℝ^‘𝑋))) = (ball‘(dist‘(ℝ^‘∅))))
61	60	oveqd 7404	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) = (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))
62	57, 61	sseq12d 3980	. . . . . . 7 ⊢ (𝑋 = ∅ → (X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) ↔ {∅} ⊆ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸)))
63	58, 62	anbi12d 632	. . . . . 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 3157	. . . . 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 3157	. . . 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 2933	. . . 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 46622	. 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 812	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 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 Vcvv 3447 ⊆ wss 3914 ∅c0 4296 {csn 4589 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799 Xcixp 8870 Fincfn 8918 ℝcr 11067 ℚcq 12907 ℝ⁺crp 12951 [,)cico 13308 distcds 17229 ∞Metcxmet 21249 Metcmet 21250 ballcbl 21251 ℝ^crrx 25283

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xadd 13073 df-ioo 13310 df-ico 13312 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-ghm 19145 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-subrng 20455 df-subrg 20479 df-drng 20640 df-field 20641 df-staf 20748 df-srng 20749 df-lmod 20768 df-lss 20838 df-sra 21080 df-rgmod 21081 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-cnfld 21265 df-refld 21514 df-dsmm 21641 df-frlm 21656 df-nm 24470 df-tng 24472 df-tcph 25069 df-rrx 25285

This theorem is referenced by: opnvonmbllem2 46631

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