rrxtopnfi - Mathbox for Glauco Siliprandi

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Theorem rrxtopnfi 46984

Description: The topology of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypothesis**
Ref	Expression
rrxtopnfi.1	⊢ (𝜑 → 𝐼 ∈ Fin)

**Assertion**
Ref	Expression
rrxtopnfi	⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))))

Distinct variable groups: 𝑓,𝐼,𝑔,𝑘 𝜑,𝑓,𝑔,𝑘

Proof of Theorem rrxtopnfi Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxtopnfi.1	. . 3 ⊢ (𝜑 → 𝐼 ∈ Fin)
2	1	rrxtopn 46981	. 2 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))))
3		eqid 2763	. . . . 5 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
4		eqid 2763	. . . . 5 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
5	1, 3, 4	rrxbasefi 25550	. . . 4 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
6	5	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(ℝ^‘𝐼))) → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
7		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝜑)
8		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑓 ∈ (Base‘(ℝ^‘𝐼)))
9		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(ℝ^‘𝐼))) → 𝑓 ∈ (Base‘(ℝ^‘𝐼)))
10	9, 6	eleqtrd 2865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(ℝ^‘𝐼))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
11	8, 10	syldan 602	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
12		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑔 ∈ (Base‘(ℝ^‘𝐼)))
13		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼))) → 𝑔 ∈ (Base‘(ℝ^‘𝐼)))
14	5	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼))) → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
15	13, 14	eleqtrd 2865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
16	12, 15	syldan 602	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
17		elmapi 8847	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℝ)
18	17	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℝ)
19	18	ffvelcdmda 7081	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
20		elmapi 8847	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (ℝ ↑_m 𝐼) → 𝑔:𝐼⟶ℝ)
21	20	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑔:𝐼⟶ℝ)
22	21	ffvelcdmda 7081	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℝ)
23	19, 22	resubcld 11643	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℝ)
24	23	resqcld 14163	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) ∈ ℝ)
25		eqid 2763	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))
26	24, 25	fmptd 7111	. . . . . . . . 9 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℝ)
27	26	3adant1 1148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℝ)
28	1	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝐼 ∈ Fin)
29		0red 11212	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 0 ∈ ℝ)
30	27, 28, 29	fidmfisupp 9333	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) finSupp 0)
31		regsumsupp 21753	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℝ ∧ (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) finSupp 0 ∧ 𝐼 ∈ Fin) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))) = Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) supp 0)((𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))‘𝑘))
32	27, 30, 28, 31	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))) = Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) supp 0)((𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))‘𝑘))
33		ax-resscn 11158	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
34	33	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → ℝ ⊆ ℂ)
35	17, 34	fssd 6725	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℂ)
36	35	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℂ)
37	36	ffvelcdmda 7081	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
38	33	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (ℝ ↑_m 𝐼) → ℝ ⊆ ℂ)
39	20, 38	fssd 6725	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (ℝ ↑_m 𝐼) → 𝑔:𝐼⟶ℂ)
40	39	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑔:𝐼⟶ℂ)
41	40	ffvelcdmda 7081	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℂ)
42	37, 41	subcld 11570	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℂ)
43	42	sqcld 14182	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) ∈ ℂ)
44	43, 25	fmptd 7111	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℂ)
45	28, 44	fsumsupp0 46277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) supp 0)((𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))‘𝑘) = Σ𝑘 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))‘𝑘))
46		eqidd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))
47		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → (𝑓‘𝑥) = (𝑓‘𝑘))
48		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → (𝑔‘𝑥) = (𝑔‘𝑘))
49	47, 48	oveq12d 7430	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑘) − (𝑔‘𝑘)))
50	49	oveq1d 7427	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) = (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))
51	50	adantl 486	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) = (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))
52		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
53		ovexd 7447	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑘 ∈ 𝐼) → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) ∈ V)
54	46, 51, 52, 53	fvmptd 6999	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑘 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))‘𝑘) = (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))
55	54	sumeq2dv 15755	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → Σ𝑘 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))‘𝑘) = Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))
56	32, 45, 55	3eqtrd 2802	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))) = Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))
57	56	fveq2d 6887	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))) = (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))
58	7, 11, 16, 57	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))) = (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))
59	5, 6, 58	mpoeq123dva 7486	. . 3 ⊢ (𝜑 → (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
60	59	fveq2d 6887	. 2 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))) = (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))))
61	2, 60	eqtrd 2798	1 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ⊆ wss 3906 class class class wbr 5110 ↦ cmpt 5193 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 ∈ cmpo 7414 supp csupp 8157 ↑_m cmap 8825 Fincfn 8944 finSupp cfsupp 9322 ℂcc 11099 ℝcr 11100 0cc0 11101 − cmin 11442 2c2 12296 ↑cexp 14099 √csqrt 15286 Σcsu 15739 Basecbs 17270 TopOpenctopn 17475 Σ_g cgsu 17494 MetOpencmopn 21493 ℝ_fldcrefld 21735 ℝ^crrx 25523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180 ax-mulf 11181

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 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-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-tpos 8223 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-prds 17501 df-pws 17503 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-mhm 18842 df-grp 19004 df-minusg 19005 df-sbg 19006 df-subg 19190 df-ghm 19285 df-cntz 19388 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-ring 20318 df-cring 20319 df-oppr 20420 df-dvdsr 20440 df-unit 20441 df-invr 20471 df-dvr 20484 df-rhm 20555 df-subrng 20632 df-subrg 20656 df-drng 20816 df-field 20817 df-staf 20923 df-srng 20924 df-lmod 20964 df-lss 21034 df-sra 21275 df-rgmod 21276 df-psmet 21495 df-xmet 21496 df-bl 21498 df-mopn 21499 df-cnfld 21504 df-refld 21736 df-dsmm 21863 df-frlm 21878 df-top 23032 df-topon 23049 df-bases 23084 df-nm 24720 df-tng 24722 df-tcph 25309 df-rrx 25525

This theorem is referenced by: qndenserrnopnlem 46994 ioorrnopnlem 47001

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