hoiqssbl - Mathbox for Glauco Siliprandi

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Theorem hoiqssbl 46616

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 5246	. . . . . . 7 ⊢ ∅ ∈ V
2	1	snid 4614	. . . . . 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 7357	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
7		reex 11100	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
8		mapdm0 8769	. . . . . . . . . . . 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 8967	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
15		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (dist‘(ℝ^‘∅)) = (dist‘(ℝ^‘∅))
16	15	rrxmetfi 25310	. . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → (dist‘(ℝ^‘∅)) ∈ (Met‘(ℝ ↑_m ∅)))
17	14, 16	ax-mp 5	. . . . . . . . . . . 12 ⊢ (dist‘(ℝ^‘∅)) ∈ (Met‘(ℝ ↑_m ∅))
18		metxmet 24220	. . . . . . . . . . . 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 24299	. . . . . . . . . 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 4594	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ {∅} → 𝑌 = ∅)
27	13, 26	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑌 = ∅)
28	27	eqcomd 2735	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ = 𝑌)
29	28	oveq1d 7364	. . . . . . . . 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 4760	. . . . . . 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 3577	. . . . . 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 3577	. . . . 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 7357	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (ℚ ↑_m 𝑋) = (ℚ ↑_m ∅))
40		qex 12862	. . . . . . . . . . . 12 ⊢ ℚ ∈ V
41		mapdm0 8769	. . . . . . . . . . . 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 3149	. . . . . . 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 3149	. . . . 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 8835	. . . . . . . . 9 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = X𝑖 ∈ ∅ ((𝑐‘𝑖)[,)(𝑑‘𝑖)))
55		ixp0x 8853	. . . . . . . . . 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 6827	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘∅)))
60	59	fveq2d 6826	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (ball‘(dist‘(ℝ^‘𝑋))) = (ball‘(dist‘(ℝ^‘∅))))
61	60	oveqd 7366	. . . . . . . 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 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 3153	. . . . 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 3153	. . . 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 46615	. 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 3436 ⊆ wss 3903 ∅c0 4284 {csn 4577 ‘cfv 6482 (class class class)co 7349 ↑_m cmap 8753 Xcixp 8824 Fincfn 8872 ℝcr 11008 ℚcq 12849 ℝ⁺crp 12893 [,)cico 13250 distcds 17170 ∞Metcxmet 21246 Metcmet 21247 ballcbl 21248 ℝ^crrx 25281

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xadd 13015 df-ioo 13252 df-ico 13254 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-mhm 18657 df-grp 18815 df-minusg 18816 df-sbg 18817 df-subg 19002 df-ghm 19092 df-cntz 19196 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-rhm 20357 df-subrng 20431 df-subrg 20455 df-drng 20616 df-field 20617 df-staf 20724 df-srng 20725 df-lmod 20765 df-lss 20835 df-sra 21077 df-rgmod 21078 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-cnfld 21262 df-refld 21512 df-dsmm 21639 df-frlm 21654 df-nm 24468 df-tng 24470 df-tcph 25067 df-rrx 25283

This theorem is referenced by: opnvonmbllem2 46624

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