Theorem spheres 49237

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 49221	. . 3 ⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
4		fveq2 6827	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5		spheres.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
6	5	eqcomi 2748	. . . . . 6 ⊢ (Base‘𝑊) = 𝐵
7	6	a1i 11	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵)
8	4, 7	eqtrd 2774	. . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
9		eqidd 2740	. . . 4 ⊢ (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞))
10		fveq2 6827	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
11		spheres.d	. . . . . . . . . 10 ⊢ 𝐷 = (dist‘𝑊)
12	11	eqcomi 2748	. . . . . . . . 9 ⊢ (dist‘𝑊) = 𝐷
13	12	a1i 11	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷)
14	10, 13	eqtrd 2774	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
15	14	oveqd 7373	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥))
16	15	eqeq1d 2741	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟))
17	8, 16	rabeqbidv 3409	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})
18	8, 9, 17	mpoeq123dv 7431	. . 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 3452	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
20		fvex 6840	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
21	5, 20	eqeltri 2835	. . . . 5 ⊢ 𝐵 ∈ V
22		ovex 7389	. . . . 5 ⊢ (0[,]+∞) ∈ V
23	21, 22	mpoex 8021	. . . 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 6959	. 2 ⊢ (𝑊 ∈ 𝑉 → (Sphere‘𝑊) = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
26	2, 25	eqtrd 2774	1 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  {crab 3391  Vcvv 3431  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  0cc0 11029  +∞cpnf 11167  [,]cicc 13292  Basecbs 17170  distcds 17220  Spherecsph 49219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-sph 49221

This theorem is referenced by:  sphere  49238  rrxsphere  49239