rrxtopnfi - Mathbox for Glauco Siliprandi

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Theorem rrxtopnfi 43457

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 43454	. 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 2734	. . . . 5 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
4		eqid 2734	. . . . 5 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
5	1, 3, 4	rrxbasefi 24279	. . . 4 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
6	5	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(ℝ^‘𝐼))) → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
7		simpl 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝜑)
8		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑓 ∈ (Base‘(ℝ^‘𝐼)))
9		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(ℝ^‘𝐼))) → 𝑓 ∈ (Base‘(ℝ^‘𝐼)))
10	9, 6	eleqtrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Base‘(ℝ^‘𝐼))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
11	8, 10	syldan 594	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑓 ∈ (ℝ ↑_m 𝐼))
12		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑔 ∈ (Base‘(ℝ^‘𝐼)))
13		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼))) → 𝑔 ∈ (Base‘(ℝ^‘𝐼)))
14	5	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼))) → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
15	13, 14	eleqtrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
16	12, 15	syldan 594	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘(ℝ^‘𝐼)) ∧ 𝑔 ∈ (Base‘(ℝ^‘𝐼)))) → 𝑔 ∈ (ℝ ↑_m 𝐼))
17		elmapi 8519	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℝ)
18	17	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℝ)
19	18	ffvelrnda 6893	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
20		elmapi 8519	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (ℝ ↑_m 𝐼) → 𝑔:𝐼⟶ℝ)
21	20	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑔:𝐼⟶ℝ)
22	21	ffvelrnda 6893	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℝ)
23	19, 22	resubcld 11243	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℝ)
24	23	resqcld 13800	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) ∈ ℝ)
25		eqid 2734	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))
26	24, 25	fmptd 6920	. . . . . . . . 9 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℝ)
27	26	3adant1 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℝ)
28	1	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝐼 ∈ Fin)
29		0red 10819	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 0 ∈ ℝ)
30	27, 28, 29	fidmfisupp 42364	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) finSupp 0)
31		regsumsupp 20556	. . . . . . . 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 10769	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
34	33	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → ℝ ⊆ ℂ)
35	17, 34	fssd 6552	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℂ)
36	35	3ad2ant2 1136	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℂ)
37	36	ffvelrnda 6893	. . . . . . . . . . 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 6552	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (ℝ ↑_m 𝐼) → 𝑔:𝐼⟶ℂ)
40	39	3ad2ant3 1137	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 𝑔:𝐼⟶ℂ)
41	40	ffvelrnda 6893	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ ℂ)
42	37, 41	subcld 11172	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℂ)
43	42	sqcld 13697	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) ∈ ℂ)
44	43, 25	fmptd 6920	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)):𝐼⟶ℂ)
45	28, 44	fsumsupp0 42748	. . . . . . 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 2735	. . . . . . . . 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 6706	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → (𝑓‘𝑥) = (𝑓‘𝑘))
48		fveq2 6706	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → (𝑔‘𝑥) = (𝑔‘𝑘))
49	47, 48	oveq12d 7220	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝑓‘𝑘) − (𝑔‘𝑘)))
50	49	oveq1d 7217	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) = (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))
51	50	adantl 485	. . . . . . . . 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 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
53		ovexd 7237	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑘 ∈ 𝐼) → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) ∈ V)
54	46, 51, 52, 53	fvmptd 6814	. . . . . . . 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 15250	. . . . . . 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 2778	. . . . . 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 6710	. . . . 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 7274	. . 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 6710	. 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 2774	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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ⊆ wss 3857 class class class wbr 5043 ↦ cmpt 5124 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ∈ cmpo 7204 supp csupp 7892 ↑_m cmap 8497 Fincfn 8615 finSupp cfsupp 8974 ℂcc 10710 ℝcr 10711 0cc0 10712 − cmin 11045 2c2 11868 ↑cexp 13618 √csqrt 14779 Σcsu 15232 Basecbs 16684 TopOpenctopn 16898 Σ_g cgsu 16917 MetOpencmopn 20325 ℝ_fldcrefld 20538 ℝ^crrx 24252

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-sum 15233 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-prds 16924 df-pws 16926 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-ghm 18592 df-cntz 18683 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-cring 19537 df-oppr 19613 df-dvdsr 19631 df-unit 19632 df-invr 19662 df-dvr 19673 df-rnghom 19707 df-drng 19741 df-field 19742 df-subrg 19770 df-staf 19853 df-srng 19854 df-lmod 19873 df-lss 19941 df-sra 20181 df-rgmod 20182 df-psmet 20327 df-xmet 20328 df-bl 20330 df-mopn 20331 df-cnfld 20336 df-refld 20539 df-dsmm 20666 df-frlm 20681 df-top 21763 df-topon 21780 df-bases 21815 df-nm 23452 df-tng 23454 df-tcph 24038 df-rrx 24254

This theorem is referenced by: qndenserrnopnlem 43467 ioorrnopnlem 43474

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