Theorem spheres 49329

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 49313	. . 3 ⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
4		fveq2 6862	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5		spheres.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
6	5	eqcomi 2770	. . . . . 6 ⊢ (Base‘𝑊) = 𝐵
7	6	a1i 11	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵)
8	4, 7	eqtrd 2796	. . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
9		eqidd 2762	. . . 4 ⊢ (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞))
10		fveq2 6862	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
11		spheres.d	. . . . . . . . . 10 ⊢ 𝐷 = (dist‘𝑊)
12	11	eqcomi 2770	. . . . . . . . 9 ⊢ (dist‘𝑊) = 𝐷
13	12	a1i 11	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷)
14	10, 13	eqtrd 2796	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
15	14	oveqd 7408	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥))
16	15	eqeq1d 2763	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟))
17	8, 16	rabeqbidv 3431	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})
18	8, 9, 17	mpoeq123dv 7466	. . 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 3474	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
20		fvex 6875	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
21	5, 20	eqeltri 2857	. . . . 5 ⊢ 𝐵 ∈ V
22		ovex 7424	. . . . 5 ⊢ (0[,]+∞) ∈ V
23	21, 22	mpoex 8055	. . . 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 6994	. 2 ⊢ (𝑊 ∈ 𝑉 → (Sphere‘𝑊) = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
26	2, 25	eqtrd 2796	1 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  0cc0 11067  +∞cpnf 11207  [,]cicc 13346  Basecbs 17236  distcds 17286  Spherecsph 49311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-sph 49313

This theorem is referenced by:  sphere  49330  rrxsphere  49331