rrxtopnfi - Mathbox for Glauco Siliprandi

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Theorem rrxtopnfi 45302

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 45299	. 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 2731	. . . . 5 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
4		eqid 2731	. . . . 5 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
5	1, 3, 4	rrxbasefi 25159	. . . 4 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(ℝ^‘𝐼))) → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
7		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝜑)
8		simprl 768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑓 ∈ (Base‘(ℝ^‘𝐼)))
9		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(ℝ^‘𝐼))) → 𝑓 ∈ (Base‘(ℝ^‘𝐼)))
10	9, 6	eleqtrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(ℝ^‘𝐼))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
11	8, 10	syldan 590	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
12		simprr 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑔 ∈ (Base‘(ℝ^‘𝐼)))
13		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼))) → 𝑔 ∈ (Base‘(ℝ^‘𝐼)))
14	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼))) → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
15	13, 14	eleqtrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
16	12, 15	syldan 590	. . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℝ)
19	18	ffvelcdmda 7086	. . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑔:𝐼⟶ℝ)
22	21	ffvelcdmda 7086	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℝ)
23	19, 22	resubcld 11647	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℝ)
24	23	resqcld 14095	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) ∈ ℝ)
25		eqid 2731	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))
26	24, 25	fmptd 7115	. . . . . . . . 9 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℝ)
27	26	3adant1 1129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℝ)
28	1	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝐼 ∈ Fin)
29		0red 11222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 0 ∈ ℝ)
30	27, 28, 29	fidmfisupp 9375	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) finSupp 0)
31		regsumsupp 21395	. . . . . . . 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 1370	. . . . . . 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 11171	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
34	33	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → ℝ ⊆ ℂ)
35	17, 34	fssd 6735	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℂ)
36	35	3ad2ant2 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℂ)
37	36	ffvelcdmda 7086	. . . . . . . . . . 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 6735	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (ℝ ↑_m 𝐼) → 𝑔:𝐼⟶ℂ)
40	39	3ad2ant3 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑔:𝐼⟶ℂ)
41	40	ffvelcdmda 7086	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℂ)
42	37, 41	subcld 11576	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℂ)
43	42	sqcld 14114	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) ∈ ℂ)
44	43, 25	fmptd 7115	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℂ)
45	28, 44	fsumsupp0 44593	. . . . . . 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 2732	. . . . . . . . 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 6891	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → (𝑓‘𝑥) = (𝑓‘𝑘))
48		fveq2 6891	. . . . . . . . . . . 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 481	. . . . . . . . 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 484	. . . . . . . . 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 7005	. . . . . . . 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 15654	. . . . . . 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 2775	. . . . . 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 6895	. . . . 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 1370	. . . 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 6895	. 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 2771	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3948 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 supp csupp 8150 ↑_m cmap 8824 Fincfn 8943 finSupp cfsupp 9365 ℂcc 11112 ℝcr 11113 0cc0 11114 − cmin 11449 2c2 12272 ↑cexp 14032 √csqrt 15185 Σcsu 15637 Basecbs 17149 TopOpenctopn 17372 Σ_g cgsu 17391 MetOpencmopn 21135 ℝ_fldcrefld 21377 ℝ^crrx 25132

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-prds 17398 df-pws 17400 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-drng 20503 df-field 20504 df-staf 20597 df-srng 20598 df-lmod 20617 df-lss 20688 df-sra 20931 df-rgmod 20932 df-psmet 21137 df-xmet 21138 df-bl 21140 df-mopn 21141 df-cnfld 21146 df-refld 21378 df-dsmm 21507 df-frlm 21522 df-top 22617 df-topon 22634 df-bases 22670 df-nm 24312 df-tng 24314 df-tcph 24918 df-rrx 25134

This theorem is referenced by: qndenserrnopnlem 45312 ioorrnopnlem 45319

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