rrxtopnfi - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > rrxtopnfi		Structured version Visualization version GIF version

Theorem rrxtopnfi 45462

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 45459	. 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 25258	. . . 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 8849	. . . . . . . . . . . . . 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 8849	. . . . . . . . . . . . . 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 11649	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℝ)
24	23	resqcld 14097	. . . . . . . . . 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 11224	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → 0 ∈ ℝ)
30	27, 28, 29	fidmfisupp 9377	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) finSupp 0)
31		regsumsupp 21485	. . . . . . . 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 11173	. . . . . . . . . . . . . . 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 11578	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼) ∧ 𝑔 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) − (𝑔‘𝑥)) ∈ ℂ)
43	42	sqcld 14116	. . . . . . . . 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 44753	. . . . . . 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 15656	. . . . . . 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 8151 ↑_m cmap 8826 Fincfn 8945 finSupp cfsupp 9367 ℂcc 11114 ℝcr 11115 0cc0 11116 − cmin 11451 2c2 12274 ↑cexp 14034 √csqrt 15187 Σcsu 15639 Basecbs 17151 TopOpenctopn 17374 Σ_g cgsu 17393 MetOpencmopn 21223 ℝ_fldcrefld 21467 ℝ^crrx 25231

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 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196

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 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-prds 17400 df-pws 17402 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-drng 20585 df-field 20586 df-staf 20684 df-srng 20685 df-lmod 20704 df-lss 20775 df-sra 21019 df-rgmod 21020 df-psmet 21225 df-xmet 21226 df-bl 21228 df-mopn 21229 df-cnfld 21234 df-refld 21468 df-dsmm 21597 df-frlm 21612 df-top 22716 df-topon 22733 df-bases 22769 df-nm 24411 df-tng 24413 df-tcph 25017 df-rrx 25233

This theorem is referenced by: qndenserrnopnlem 45472 ioorrnopnlem 45479

W3C validator