rrxsphere - Mathbox for Alexander van der Vekens

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Theorem rrxsphere 46994

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 6883	. . . . 5 ⊢ 𝐸 ∈ V
3		rrxspheres.p	. . . . . . . . . 10 ⊢ 𝑃 = (ℝ ↑_m 𝐼)
4		id 22	. . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin)
5		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝐸) = (Base‘𝐸)
6	4, 1, 5	rrxbasefi 24833	. . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = (ℝ ↑_m 𝐼))
7	3, 6	eqtr4id 2790	. . . . . . . . 9 ⊢ (𝐼 ∈ Fin → 𝑃 = (Base‘𝐸))
8	7	eleq2d 2818	. . . . . . . 8 ⊢ (𝐼 ∈ Fin → (𝑀 ∈ 𝑃 ↔ 𝑀 ∈ (Base‘𝐸)))
9	8	biimpa 477	. . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃) → 𝑀 ∈ (Base‘𝐸))
10	9	3adant3 1132	. . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → 𝑀 ∈ (Base‘𝐸))
11	10	adantl 482	. . . . 5 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → 𝑀 ∈ (Base‘𝐸))
12		rexr 11232	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → 𝑅 ∈ ℝ^*)
13	12	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → 𝑅 ∈ ℝ^*)
14	13	anim2i 617	. . . . . . 7 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → (0 ≤ 𝑅 ∧ 𝑅 ∈ ℝ^*))
15	14	ancomd 462	. . . . . 6 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → (𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅))
16		elxrge0 13406	. . . . . 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 46993	. . . . 5 ⊢ ((𝐸 ∈ V ∧ 𝑀 ∈ (Base‘𝐸) ∧ 𝑅 ∈ (0[,]+∞)) → (𝑀𝑆𝑅) = {𝑝 ∈ (Base‘𝐸) ∣ (𝑝𝐷𝑀) = 𝑅})
21	2, 11, 17, 20	mp3an2i 1466	. . . 4 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → (𝑀𝑆𝑅) = {𝑝 ∈ (Base‘𝐸) ∣ (𝑝𝐷𝑀) = 𝑅})
22		simp1 1136	. . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → 𝐼 ∈ Fin)
23	22, 1, 5	rrxbasefi 24833	. . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (Base‘𝐸) = (ℝ ↑_m 𝐼))
24	23, 3	eqtr4di 2789	. . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (Base‘𝐸) = 𝑃)
25	24	adantl 482	. . . . 5 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → (Base‘𝐸) = 𝑃)
26	25	rabeqdv 3440	. . . 4 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → {𝑝 ∈ (Base‘𝐸) ∣ (𝑝𝐷𝑀) = 𝑅} = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})
27	21, 26	eqtrd 2771	. . 3 ⊢ ((0 ≤ 𝑅 ∧ (𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ)) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})
28	27	ex 413	. 2 ⊢ (0 ≤ 𝑅 → ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅}))
29	5, 18, 19	spheres 46992	. . . . . . 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 6882	. . . . . . 7 ⊢ (Base‘𝐸) ∈ V
32	31	rabex 5316	. . . . . 6 ⊢ {𝑝 ∈ (Base‘𝐸) ∣ (𝑝𝐷𝑥) = 𝑟} ∈ V
33	30, 32	dmmpo 8030	. . . . 5 ⊢ dom 𝑆 = ((Base‘𝐸) × (0[,]+∞))
34		0xr 11233	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
35		pnfxr 11240	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
36	34, 35	pm3.2i 471	. . . . . . . . . 10 ⊢ (0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*)
37		elicc1 13340	. . . . . . . . . 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 1137	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞) → 0 ≤ 𝑅)
40	38, 39	syl6bi 252	. . . . . . . 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 407	. . . . . 6 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → ¬ 𝑅 ∈ (0[,]+∞))
43	42	intnand 489	. . . . 5 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → ¬ (𝑀 ∈ (Base‘𝐸) ∧ 𝑅 ∈ (0[,]+∞)))
44		ndmovg 7564	. . . . 5 ⊢ ((dom 𝑆 = ((Base‘𝐸) × (0[,]+∞)) ∧ ¬ (𝑀 ∈ (Base‘𝐸) ∧ 𝑅 ∈ (0[,]+∞))) → (𝑀𝑆𝑅) = ∅)
45	33, 43, 44	sylancr 587	. . . 4 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → (𝑀𝑆𝑅) = ∅)
46	1	fveq2i 6872	. . . . . . . . . . . . . . . 16 ⊢ (dist‘𝐸) = (dist‘(ℝ^‘𝐼))
47	19, 46	eqtri 2759	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
48	47	rrxmetfi 24835	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
49	48	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
50	49	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
51	3	fveq2i 6872	. . . . . . . . . . . 12 ⊢ (Met‘𝑃) = (Met‘(ℝ ↑_m 𝐼))
52	50, 51	eleqtrrdi 2843	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → 𝐷 ∈ (Met‘𝑃))
53		simpr 485	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ 𝑃)
54		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → 𝑀 ∈ 𝑃)
55	54	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → 𝑀 ∈ 𝑃)
56		metge0 23757	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (Met‘𝑃) ∧ 𝑝 ∈ 𝑃 ∧ 𝑀 ∈ 𝑃) → 0 ≤ (𝑝𝐷𝑀))
57	52, 53, 55, 56	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ 𝑝 ∈ 𝑃) → 0 ≤ (𝑝𝐷𝑀))
58		breq2 5136	. . . . . . . . . 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 452	. . . . . . 7 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → (𝑝 ∈ 𝑃 → ¬ (𝑝𝐷𝑀) = 𝑅))
62	61	imp 407	. . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) ∧ 𝑝 ∈ 𝑃) → ¬ (𝑝𝐷𝑀) = 𝑅)
63	62	ralrimiva 3145	. . . . 5 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → ∀𝑝 ∈ 𝑃 ¬ (𝑝𝐷𝑀) = 𝑅)
64		eqcom 2738	. . . . . 6 ⊢ (∅ = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅} ↔ {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅} = ∅)
65		rabeq0 4371	. . . . . 6 ⊢ ({𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅} = ∅ ↔ ∀𝑝 ∈ 𝑃 ¬ (𝑝𝐷𝑀) = 𝑅)
66	64, 65	bitri 274	. . . . 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 2771	. . 3 ⊢ (((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) ∧ ¬ 0 ≤ 𝑅) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})
69	68	expcom 414	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3060 {crab 3425 Vcvv 3466 ∅c0 4309 class class class wbr 5132 × cxp 5658 dom cdm 5660 ‘cfv 6523 (class class class)co 7384 ∈ cmpo 7386 ↑_m cmap 8794 Fincfn 8912 ℝcr 11081 0cc0 11082 +∞cpnf 11217 ℝ^*cxr 11219 ≤ cle 11221 [,]cicc 13299 Basecbs 17116 distcds 17178 Metcmet 20841 ℝ^crrx 24806 Spherecsph 46974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5269 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-inf2 9608 ax-cnex 11138 ax-resscn 11139 ax-1cn 11140 ax-icn 11141 ax-addcl 11142 ax-addrcl 11143 ax-mulcl 11144 ax-mulrcl 11145 ax-mulcom 11146 ax-addass 11147 ax-mulass 11148 ax-distr 11149 ax-i2m1 11150 ax-1ne0 11151 ax-1rid 11152 ax-rnegex 11153 ax-rrecex 11154 ax-cnre 11155 ax-pre-lttri 11156 ax-pre-lttrn 11157 ax-pre-ltadd 11158 ax-pre-mulgt0 11159 ax-pre-sup 11160 ax-addf 11161 ax-mulf 11162

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3371 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-pss 3954 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-tp 4618 df-op 4620 df-uni 4893 df-int 4935 df-iun 4983 df-br 5133 df-opab 5195 df-mpt 5216 df-tr 5250 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5615 df-se 5616 df-we 5617 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6280 df-ord 6347 df-on 6348 df-lim 6349 df-suc 6350 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7340 df-ov 7387 df-oprab 7388 df-mpo 7389 df-of 7644 df-om 7830 df-1st 7948 df-2nd 7949 df-supp 8120 df-tpos 8184 df-frecs 8239 df-wrecs 8270 df-recs 8344 df-rdg 8383 df-1o 8439 df-er 8677 df-map 8796 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9335 df-sup 9409 df-oi 9477 df-card 9906 df-pnf 11222 df-mnf 11223 df-xr 11224 df-ltxr 11225 df-le 11226 df-sub 11418 df-neg 11419 df-div 11844 df-nn 12185 df-2 12247 df-3 12248 df-4 12249 df-5 12250 df-6 12251 df-7 12252 df-8 12253 df-9 12254 df-n0 12445 df-z 12531 df-dec 12650 df-uz 12795 df-rp 12947 df-xneg 13064 df-xadd 13065 df-xmul 13066 df-ico 13302 df-icc 13303 df-fz 13457 df-fzo 13600 df-seq 13939 df-exp 14000 df-hash 14263 df-cj 15018 df-re 15019 df-im 15020 df-sqrt 15154 df-abs 15155 df-clim 15404 df-sum 15605 df-struct 17052 df-sets 17069 df-slot 17087 df-ndx 17099 df-base 17117 df-ress 17146 df-plusg 17182 df-mulr 17183 df-starv 17184 df-sca 17185 df-vsca 17186 df-ip 17187 df-tset 17188 df-ple 17189 df-ds 17191 df-unif 17192 df-hom 17193 df-cco 17194 df-0g 17359 df-gsum 17360 df-prds 17365 df-pws 17367 df-mgm 18533 df-sgrp 18582 df-mnd 18593 df-mhm 18637 df-grp 18787 df-minusg 18788 df-sbg 18789 df-subg 18961 df-ghm 19042 df-cntz 19133 df-cmn 19600 df-abl 19601 df-mgp 19933 df-ur 19950 df-ring 20002 df-cring 20003 df-oppr 20085 df-dvdsr 20106 df-unit 20107 df-invr 20137 df-dvr 20148 df-rnghom 20184 df-drng 20249 df-field 20250 df-subrg 20290 df-staf 20382 df-srng 20383 df-lmod 20402 df-lss 20472 df-sra 20714 df-rgmod 20715 df-xmet 20848 df-met 20849 df-cnfld 20856 df-refld 21068 df-dsmm 21197 df-frlm 21212 df-nm 23997 df-tng 23999 df-tcph 24592 df-rrx 24808 df-sph 46976

This theorem is referenced by: 2sphere 46995

W3C validator