Theorem spheres 49103

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 49087	. . 3 ⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
4		fveq2 6842	. . . . 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 6842	. . . . . . . 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 7385	. . . . . 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 3419	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})
18	8, 9, 17	mpoeq123dv 7443	. . 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 3463	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
20		fvex 6855	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
21	5, 20	eqeltri 2833	. . . . 5 ⊢ 𝐵 ∈ V
22		ovex 7401	. . . . 5 ⊢ (0[,]+∞) ∈ V
23	21, 22	mpoex 8033	. . . 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 6973	. 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 3401  Vcvv 3442  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  0cc0 11038  +∞cpnf 11175  [,]cicc 13276  Basecbs 17148  distcds 17198  Spherecsph 49085

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-sph 49087

This theorem is referenced by:  sphere  49104  rrxsphere  49105