Theorem spheres 49234

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 49218	. . 3 ⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
4		fveq2 6834	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5		spheres.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
6	5	eqcomi 2746	. . . . . 6 ⊢ (Base‘𝑊) = 𝐵
7	6	a1i 11	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵)
8	4, 7	eqtrd 2772	. . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
9		eqidd 2738	. . . 4 ⊢ (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞))
10		fveq2 6834	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
11		spheres.d	. . . . . . . . . 10 ⊢ 𝐷 = (dist‘𝑊)
12	11	eqcomi 2746	. . . . . . . . 9 ⊢ (dist‘𝑊) = 𝐷
13	12	a1i 11	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷)
14	10, 13	eqtrd 2772	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
15	14	oveqd 7377	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥))
16	15	eqeq1d 2739	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟))
17	8, 16	rabeqbidv 3408	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})
18	8, 9, 17	mpoeq123dv 7435	. . 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 3451	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
20		fvex 6847	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
21	5, 20	eqeltri 2833	. . . . 5 ⊢ 𝐵 ∈ V
22		ovex 7393	. . . . 5 ⊢ (0[,]+∞) ∈ V
23	21, 22	mpoex 8025	. . . 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 6965	. 2 ⊢ (𝑊 ∈ 𝑉 → (Sphere‘𝑊) = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
26	2, 25	eqtrd 2772	1 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3390  Vcvv 3430  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  0cc0 11029  +∞cpnf 11167  [,]cicc 13292  Basecbs 17170  distcds 17220  Spherecsph 49216

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-sph 49218

This theorem is referenced by:  sphere  49235  rrxsphere  49236