rrxsphere - Mathbox for Alexander van der Vekens

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Theorem rrxsphere 47434

Description: The sphere with center 𝑀 and radius 𝑅 in a generalized real Euclidean space of finite dimension. Remark: this theorem holds also for the degenerate case 𝑅 < 0 (negative radius): in this case, (𝑀𝑆𝑅) is empty. (Contributed by AV, 5-Feb-2023.)

**Hypotheses**
Ref	Expression
rrxspheres.e	⊢ 𝐸 = (ℝ^‘𝐼)
rrxspheres.p	⊢ 𝑃 = (ℝ ↑_m 𝐼)
rrxspheres.d	⊢ 𝐷 = (dist‘𝐸)
rrxspheres.s	⊢ 𝑆 = (Sphere‘𝐸)

**Assertion**
Ref	Expression
rrxsphere	⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})

Distinct variable groups: 𝐸,𝑝 𝐼,𝑝 𝑀,𝑝 𝑃,𝑝 𝑅,𝑝 Allowed substitution hints: 𝐷(𝑝) 𝑆(𝑝)

Proof of Theorem rrxsphere Dummy variables 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxspheres.e	. . . . . 6 ⊢ 𝐸 = (ℝ^‘𝐼)
2	1	fvexi 6906	. . . . 5 ⊢ 𝐸 ∈ V
3		rrxspheres.p	. . . . . . . . . 10 ⊢ 𝑃 = (ℝ ↑_m 𝐼)
4		id 22	. . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin)
5		eqid 2733	. . . . . . . . . . 11 ⊢ (Base‘𝐸) = (Base‘𝐸)
6	4, 1, 5	rrxbasefi 24927	. . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = (ℝ ↑_m 𝐼))
7	3, 6	eqtr4id 2792	. . . . . . . . 9 ⊢ (𝐼 ∈ Fin → 𝑃 = (Base‘𝐸))
8	7	eleq2d 2820	. . . . . . . 8 ⊢ (𝐼 ∈ Fin → (𝑀 ∈ 𝑃 ↔ 𝑀 ∈ (Base‘𝐸)))
9	8	biimpa 478	. . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃) → 𝑀 ∈ (Base‘𝐸))
10	9	3adant3 1133	. . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → 𝑀 ∈ (Base‘𝐸))
11	10	adantl 483	. . . . 5 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → 𝑀 ∈ (Base‘𝐸))
12		rexr 11260	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → 𝑅 ∈ ℝ^*)
13	12	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → 𝑅 ∈ ℝ^*)
14	13	anim2i 618	. . . . . . 7 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → (0 ≤ 𝑅 ∧ 𝑅 ∈ ℝ^*))
15	14	ancomd 463	. . . . . 6 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → (𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅))
16		elxrge0 13434	. . . . . 6 ⊢ (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅))
17	15, 16	sylibr 233	. . . . 5 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → 𝑅 ∈ (0[,]+∞))
18		rrxspheres.s	. . . . . 6 ⊢ 𝑆 = (Sphere‘𝐸)
19		rrxspheres.d	. . . . . 6 ⊢ 𝐷 = (dist‘𝐸)
20	5, 18, 19	sphere 47433	. . . . 5 ⊢ ((𝐸 ∈ V ∧ 𝑀 ∈ (Base‘𝐸) ∧ 𝑅 ∈ (0[,]+∞)) → (𝑀𝑆𝑅) = {𝑝 ∈ (Base‘𝐸) ∣ (𝑝𝐷𝑀) = 𝑅})
21	2, 11, 17, 20	mp3an2i 1467	. . . 4 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → (𝑀𝑆𝑅) = {𝑝 ∈ (Base‘𝐸) ∣ (𝑝𝐷𝑀) = 𝑅})
22		simp1 1137	. . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → 𝐼 ∈ Fin)
23	22, 1, 5	rrxbasefi 24927	. . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (Base‘𝐸) = (ℝ ↑_m 𝐼))
24	23, 3	eqtr4di 2791	. . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (Base‘𝐸) = 𝑃)
25	24	adantl 483	. . . . 5 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → (Base‘𝐸) = 𝑃)
26	25	rabeqdv 3448	. . . 4 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → {𝑝 ∈ (Base‘𝐸) ∣ (𝑝𝐷𝑀) = 𝑅} = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})
27	21, 26	eqtrd 2773	. . 3 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})
28	27	ex 414	. 2 ⊢ (0 ≤ 𝑅 → ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅}))
29	5, 18, 19	spheres 47432	. . . . . . 7 ⊢ (𝐸 ∈ V → 𝑆 = (𝑥 ∈ (Base‘𝐸), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝐸) ∣ (𝑝𝐷𝑥) = 𝑟}))
30	2, 29	ax-mp 5	. . . . . 6 ⊢ 𝑆 = (𝑥 ∈ (Base‘𝐸), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝐸) ∣ (𝑝𝐷𝑥) = 𝑟})
31		fvex 6905	. . . . . . 7 ⊢ (Base‘𝐸) ∈ V
32	31	rabex 5333	. . . . . 6 ⊢ {𝑝 ∈ (Base‘𝐸) ∣ (𝑝𝐷𝑥) = 𝑟} ∈ V
33	30, 32	dmmpo 8057	. . . . 5 ⊢ dom 𝑆 = ((Base‘𝐸) × (0[,]+∞))
34		0xr 11261	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
35		pnfxr 11268	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
36	34, 35	pm3.2i 472	. . . . . . . . . 10 ⊢ (0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
37		elicc1 13368	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ^ ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞)))
38	36, 37	mp1i 13	. . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞)))
39		simp2 1138	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞) → 0 ≤ 𝑅)
40	38, 39	syl6bi 253	. . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑅 ∈ (0[,]+∞) → 0 ≤ 𝑅))
41	40	con3d 152	. . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (¬ 0 ≤ 𝑅 → ¬ 𝑅 ∈ (0[,]+∞)))
42	41	imp 408	. . . . . 6 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → ¬ 𝑅 ∈ (0[,]+∞))
43	42	intnand 490	. . . . 5 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → ¬ (𝑀 ∈ (Base‘𝐸) ∧ 𝑅 ∈ (0[,]+∞)))
44		ndmovg 7590	. . . . 5 ⊢ ((dom 𝑆 = ((Base‘𝐸) × (0[,]+∞)) ∧ ¬ (𝑀 ∈ (Base‘𝐸) ∧ 𝑅 ∈ (0[,]+∞))) → (𝑀𝑆𝑅) = ∅)
45	33, 43, 44	sylancr 588	. . . 4 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → (𝑀𝑆𝑅) = ∅)
46	1	fveq2i 6895	. . . . . . . . . . . . . . . 16 ⊢ (dist‘𝐸) = (dist‘(ℝ^‘𝐼))
47	19, 46	eqtri 2761	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
48	47	rrxmetfi 24929	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
49	48	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
50	49	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
51	3	fveq2i 6895	. . . . . . . . . . . 12 ⊢ (Met‘𝑃) = (Met‘(ℝ ↑_m 𝐼))
52	50, 51	eleqtrrdi 2845	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → 𝐷 ∈ (Met‘𝑃))
53		simpr 486	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ 𝑃)
54		simp2 1138	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → 𝑀 ∈ 𝑃)
55	54	adantr 482	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → 𝑀 ∈ 𝑃)
56		metge0 23851	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (Met‘𝑃) ∧ 𝑝 ∈ 𝑃 ∧ 𝑀 ∈ 𝑃) → 0 ≤ (𝑝𝐷𝑀))
57	52, 53, 55, 56	syl3anc 1372	. . . . . . . . . 10 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → 0 ≤ (𝑝𝐷𝑀))
58		breq2 5153	. . . . . . . . . 10 ⊢ ((𝑝𝐷𝑀) = 𝑅 → (0 ≤ (𝑝𝐷𝑀) ↔ 0 ≤ 𝑅))
59	57, 58	syl5ibcom 244	. . . . . . . . 9 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → ((𝑝𝐷𝑀) = 𝑅 → 0 ≤ 𝑅))
60	59	con3d 152	. . . . . . . 8 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → (¬ 0 ≤ 𝑅 → ¬ (𝑝𝐷𝑀) = 𝑅))
61	60	impancom 453	. . . . . . 7 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → (𝑝 ∈ 𝑃 → ¬ (𝑝𝐷𝑀) = 𝑅))
62	61	imp 408	. . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) ∧ 𝑝 ∈ 𝑃) → ¬ (𝑝𝐷𝑀) = 𝑅)
63	62	ralrimiva 3147	. . . . 5 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → ∀𝑝 ∈ 𝑃 ¬ (𝑝𝐷𝑀) = 𝑅)
64		eqcom 2740	. . . . . 6 ⊢ (∅ = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅} ↔ {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅} = ∅)
65		rabeq0 4385	. . . . . 6 ⊢ ({𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅} = ∅ ↔ ∀𝑝 ∈ 𝑃 ¬ (𝑝𝐷𝑀) = 𝑅)
66	64, 65	bitri 275	. . . . 5 ⊢ (∅ = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅} ↔ ∀𝑝 ∈ 𝑃 ¬ (𝑝𝐷𝑀) = 𝑅)
67	63, 66	sylibr 233	. . . 4 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → ∅ = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})
68	45, 67	eqtrd 2773	. . 3 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})
69	68	expcom 415	. 2 ⊢ (¬ 0 ≤ 𝑅 → ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅}))
70	28, 69	pm2.61i 182	1 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 Vcvv 3475 ∅c0 4323 class class class wbr 5149 × cxp 5675 dom cdm 5677 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 ↑_m cmap 8820 Fincfn 8939 ℝcr 11109 0cc0 11110 +∞cpnf 11245 ℝ^*cxr 11247 ≤ cle 11249 [,]cicc 13327 Basecbs 17144 distcds 17206 Metcmet 20930 ℝ^crrx 24900 Spherecsph 47414

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-ghm 19090 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-rnghom 20251 df-subrg 20317 df-drng 20359 df-field 20360 df-staf 20453 df-srng 20454 df-lmod 20473 df-lss 20543 df-sra 20785 df-rgmod 20786 df-xmet 20937 df-met 20938 df-cnfld 20945 df-refld 21158 df-dsmm 21287 df-frlm 21302 df-nm 24091 df-tng 24093 df-tcph 24686 df-rrx 24902 df-sph 47416

This theorem is referenced by: 2sphere 47435

W3C validator