hoiqssbl - Mathbox for Glauco Siliprandi

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Theorem hoiqssbl 46581

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 5313	. . . . . . 7 ⊢ ∅ ∈ V
2	1	snid 4667	. . . . . 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 7439	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
7		reex 11244	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
8		mapdm0 8881	. . . . . . . . . . . 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 2775	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = {∅})
12	11	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑_m 𝑋) = {∅})
13	5, 12	eleqtrd 2841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑌 ∈ {∅})
14		0fi 9081	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
15		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (dist‘(ℝ^‘∅)) = (dist‘(ℝ^‘∅))
16	15	rrxmetfi 25460	. . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → (dist‘(ℝ^‘∅)) ∈ (Met‘(ℝ ↑_m ∅)))
17	14, 16	ax-mp 5	. . . . . . . . . . . 12 ⊢ (dist‘(ℝ^‘∅)) ∈ (Met‘(ℝ ↑_m ∅))
18		metxmet 24360	. . . . . . . . . . . 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 2850	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ (ℝ ↑_m ∅))
22		hoiqssbl.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐸 ∈ ℝ⁺)
24		blcntr 24439	. . . . . . . . . 10 ⊢ (((dist‘(ℝ^‘∅)) ∈ (∞Met‘(ℝ ↑_m ∅)) ∧ ∅ ∈ (ℝ ↑_m ∅) ∧ 𝐸 ∈ ℝ⁺) → ∅ ∈ (∅(ball‘(dist‘(ℝ^‘∅)))𝐸))
25	20, 21, 23, 24	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ (∅(ball‘(dist‘(ℝ^‘∅)))𝐸))
26		elsni 4648	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ {∅} → 𝑌 = ∅)
27	13, 26	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑌 = ∅)
28	27	eqcomd 2741	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ = 𝑌)
29	28	oveq1d 7446	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (∅(ball‘(dist‘(ℝ^‘∅)))𝐸) = (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))
30	25, 29	eleqtrd 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))
31	30	snssd 4814	. . . . . . 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 3622	. . . . . 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 3622	. . . . 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 7439	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (ℚ ↑_m 𝑋) = (ℚ ↑_m ∅))
40		qex 13001	. . . . . . . . . . . 12 ⊢ ℚ ∈ V
41		mapdm0 8881	. . . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ (𝑋 = ∅ → {∅} = (ℚ ↑_m 𝑋))
45	44	eqcomd 2741	. . . . . . . 8 ⊢ (𝑋 = ∅ → (ℚ ↑_m 𝑋) = {∅})
46	45	eleq2d 2825	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝑐 ∈ (ℚ ↑_m 𝑋) ↔ 𝑐 ∈ {∅}))
47	45	eleq2d 2825	. . . . . . . . 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 3173	. . . . . . 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 3173	. . . . 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 8947	. . . . . . . . 9 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = X𝑖 ∈ ∅ ((𝑐‘𝑖)[,)(𝑑‘𝑖)))
55		ixp0x 8965	. . . . . . . . . 10 ⊢ X𝑖 ∈ ∅ ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = {∅}
56	55	a1i 11	. . . . . . . . 9 ⊢ (𝑋 = ∅ → X𝑖 ∈ ∅ ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = {∅})
57	54, 56	eqtrd 2775	. . . . . . . 8 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) = {∅})
58	57	eleq2d 2825	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ↔ 𝑌 ∈ {∅}))
59		2fveq3 6912	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘∅)))
60	59	fveq2d 6911	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (ball‘(dist‘(ℝ^‘𝑋))) = (ball‘(dist‘(ℝ^‘∅))))
61	60	oveqd 7448	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) = (𝑌(ball‘(dist‘(ℝ^‘∅)))𝐸))
62	57, 61	sseq12d 4029	. . . . . . 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 3177	. . . . 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 3177	. . . 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 2946	. . . 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 46580	. 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 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 {csn 4631 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 Xcixp 8936 Fincfn 8984 ℝcr 11152 ℚcq 12988 ℝ⁺crp 13032 [,)cico 13386 distcds 17307 ∞Metcxmet 21367 Metcmet 21368 ballcbl 21369 ℝ^crrx 25431

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ioo 13388 df-ico 13390 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-cnfld 21383 df-refld 21641 df-dsmm 21770 df-frlm 21785 df-nm 24611 df-tng 24613 df-tcph 25217 df-rrx 25433

This theorem is referenced by: opnvonmbllem2 46589

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