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Mirrors > Home > MPE Home > Th. List > Mathboxes > spheres | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
spheres.b | ⊢ 𝐵 = (Base‘𝑊) |
spheres.l | ⊢ 𝑆 = (Sphere‘𝑊) |
spheres.d | ⊢ 𝐷 = (dist‘𝑊) |
Ref | Expression |
---|---|
spheres | ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spheres.l | . . 3 ⊢ 𝑆 = (Sphere‘𝑊) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (Sphere‘𝑊)) |
3 | df-sph 43476 | . . 3 ⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟})) | |
4 | fveq2 6448 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
5 | spheres.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑊) | |
6 | 5 | eqcomi 2787 | . . . . . 6 ⊢ (Base‘𝑊) = 𝐵 |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵) |
8 | 4, 7 | eqtrd 2814 | . . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
9 | eqidd 2779 | . . . 4 ⊢ (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞)) | |
10 | fveq2 6448 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊)) | |
11 | spheres.d | . . . . . . . . . 10 ⊢ 𝐷 = (dist‘𝑊) | |
12 | 11 | eqcomi 2787 | . . . . . . . . 9 ⊢ (dist‘𝑊) = 𝐷 |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷) |
14 | 10, 13 | eqtrd 2814 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷) |
15 | 14 | oveqd 6941 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥)) |
16 | 15 | eqeq1d 2780 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟)) |
17 | 8, 16 | rabeqbidv 3392 | . . . 4 ⊢ (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) |
18 | 8, 9, 17 | mpt2eq123dv 6996 | . . 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 3414 | . . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
20 | fvex 6461 | . . . . . 6 ⊢ (Base‘𝑊) ∈ V | |
21 | 5, 20 | eqeltri 2855 | . . . . 5 ⊢ 𝐵 ∈ V |
22 | ovex 6956 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
23 | 21, 22 | mpt2ex 7529 | . . . 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 6566 | . 2 ⊢ (𝑊 ∈ 𝑉 → (Sphere‘𝑊) = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})) |
26 | 2, 25 | eqtrd 2814 | 1 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 {crab 3094 Vcvv 3398 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 0cc0 10274 +∞cpnf 10410 [,]cicc 12494 Basecbs 16259 distcds 16351 Spherecsph 43474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-sph 43476 |
This theorem is referenced by: sphere 43493 rrxsphere 43494 |
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