Theorem spheres 48693

Description: The spheres for given centers and radii in a metric space (or any extensible structure having a base set and a distance function). (Contributed by AV, 22-Jan-2023.)

Hypotheses

Ref	Expression
spheres.b	⊢ 𝐵 = (Base‘𝑊)
spheres.l	⊢ 𝑆 = (Sphere‘𝑊)
spheres.d	⊢ 𝐷 = (dist‘𝑊)

Assertion

Ref	Expression
spheres	⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))

Distinct variable groups:   𝐵,𝑝,𝑟,𝑥   𝑊,𝑝,𝑟,𝑥

Allowed substitution hints:   𝐷(𝑥,𝑟,𝑝)   𝑆(𝑥,𝑟,𝑝)   𝑉(𝑥,𝑟,𝑝)

Proof of Theorem spheres

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		spheres.l	. . 3 ⊢ 𝑆 = (Sphere‘𝑊)
2	1	a1i 11	. 2 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (Sphere‘𝑊))
3		df-sph 48677	. . 3 ⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
4		fveq2 6881	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5		spheres.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
6	5	eqcomi 2745	. . . . . 6 ⊢ (Base‘𝑊) = 𝐵
7	6	a1i 11	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵)
8	4, 7	eqtrd 2771	. . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
9		eqidd 2737	. . . 4 ⊢ (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞))
10		fveq2 6881	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
11		spheres.d	. . . . . . . . . 10 ⊢ 𝐷 = (dist‘𝑊)
12	11	eqcomi 2745	. . . . . . . . 9 ⊢ (dist‘𝑊) = 𝐷
13	12	a1i 11	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷)
14	10, 13	eqtrd 2771	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
15	14	oveqd 7427	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥))
16	15	eqeq1d 2738	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟))
17	8, 16	rabeqbidv 3439	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})
18	8, 9, 17	mpoeq123dv 7487	. . 3 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}) = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
19		elex 3485	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
20		fvex 6894	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
21	5, 20	eqeltri 2831	. . . . 5 ⊢ 𝐵 ∈ V
22		ovex 7443	. . . . 5 ⊢ (0[,]+∞) ∈ V
23	21, 22	mpoex 8083	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) ∈ V
24	23	a1i 11	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) ∈ V)
25	3, 18, 19, 24	fvmptd3 7014	. 2 ⊢ (𝑊 ∈ 𝑉 → (Sphere‘𝑊) = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
26	2, 25	eqtrd 2771	1 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3420  Vcvv 3464  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  0cc0 11134  +∞cpnf 11271  [,]cicc 13370  Basecbs 17233  distcds 17285  Spherecsph 48675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-sph 48677

This theorem is referenced by:  sphere  48694  rrxsphere  48695