rrxtopnfi - Mathbox for Glauco Siliprandi

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Theorem rrxtopnfi 46324

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 46321	. 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 25335	. . . 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 770	. . . . . 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 2833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(ℝ^‘𝐼))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
11	8, 10	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
12		simprr 772	. . . . . 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 2833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
16	12, 15	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
17		elmapi 8773	. . . . . . . . . . . . . 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 7017	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
20		elmapi 8773	. . . . . . . . . . . . . 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 7017	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℝ)
23	19, 22	resubcld 11542	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℝ)
24	23	resqcld 14029	. . . . . . . . . 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 7047	. . . . . . . . 9 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℝ)
27	26	3adant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℝ)
28	1	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝐼 ∈ Fin)
29		0red 11112	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 0 ∈ ℝ)
30	27, 28, 29	fidmfisupp 9256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) finSupp 0)
31		regsumsupp 21557	. . . . . . . 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 1373	. . . . . . 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 11060	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
34	33	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → ℝ ⊆ ℂ)
35	17, 34	fssd 6668	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℂ)
36	35	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℂ)
37	36	ffvelcdmda 7017	. . . . . . . . . . 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 6668	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (ℝ ↑_m 𝐼) → 𝑔:𝐼⟶ℂ)
40	39	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑔:𝐼⟶ℂ)
41	40	ffvelcdmda 7017	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℂ)
42	37, 41	subcld 11469	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℂ)
43	42	sqcld 14048	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) ∈ ℂ)
44	43, 25	fmptd 7047	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℂ)
45	28, 44	fsumsupp0 45617	. . . . . . 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 6822	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → (𝑓‘𝑥) = (𝑓‘𝑘))
48		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → (𝑔‘𝑥) = (𝑔‘𝑘))
49	47, 48	oveq12d 7364	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑘) − (𝑔‘𝑘)))
50	49	oveq1d 7361	. . . . . . . . . 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 7381	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑘 ∈ 𝐼) → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) ∈ V)
54	46, 51, 52, 53	fvmptd 6936	. . . . . . . 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 15606	. . . . . . 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 2770	. . . . . 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 6826	. . . . 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 1373	. . . 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 7420	. . 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 6826	. 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 2766	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 1541 ∈ wcel 2111 Vcvv 3436 ⊆ wss 3902 class class class wbr 5091 ↦ cmpt 5172 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 supp csupp 8090 ↑_m cmap 8750 Fincfn 8869 finSupp cfsupp 9245 ℂcc 11001 ℝcr 11002 0cc0 11003 − cmin 11341 2c2 12177 ↑cexp 13965 √csqrt 15137 Σcsu 15590 Basecbs 17117 TopOpenctopn 17322 Σ_g cgsu 17341 MetOpencmopn 21279 ℝ_fldcrefld 21539 ℝ^crrx 25308

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081 ax-addf 11082 ax-mulf 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-fz 13405 df-fzo 13552 df-seq 13906 df-exp 13966 df-hash 14235 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-clim 15392 df-sum 15591 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-rest 17323 df-topn 17324 df-0g 17342 df-gsum 17343 df-topgen 17344 df-prds 17348 df-pws 17350 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-mhm 18688 df-grp 18846 df-minusg 18847 df-sbg 18848 df-subg 19033 df-ghm 19123 df-cntz 19227 df-cmn 19692 df-abl 19693 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-rhm 20388 df-subrng 20459 df-subrg 20483 df-drng 20644 df-field 20645 df-staf 20752 df-srng 20753 df-lmod 20793 df-lss 20863 df-sra 21105 df-rgmod 21106 df-psmet 21281 df-xmet 21282 df-bl 21284 df-mopn 21285 df-cnfld 21290 df-refld 21540 df-dsmm 21667 df-frlm 21682 df-top 22807 df-topon 22824 df-bases 22859 df-nm 24495 df-tng 24497 df-tcph 25094 df-rrx 25310

This theorem is referenced by: qndenserrnopnlem 46334 ioorrnopnlem 46341

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