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Mathbox for Alexander van der Vekens |
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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 45144 | . . 3 ⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟})) | |
4 | fveq2 6645 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
5 | spheres.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑊) | |
6 | 5 | eqcomi 2807 | . . . . . 6 ⊢ (Base‘𝑊) = 𝐵 |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵) |
8 | 4, 7 | eqtrd 2833 | . . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
9 | eqidd 2799 | . . . 4 ⊢ (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞)) | |
10 | fveq2 6645 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊)) | |
11 | spheres.d | . . . . . . . . . 10 ⊢ 𝐷 = (dist‘𝑊) | |
12 | 11 | eqcomi 2807 | . . . . . . . . 9 ⊢ (dist‘𝑊) = 𝐷 |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷) |
14 | 10, 13 | eqtrd 2833 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷) |
15 | 14 | oveqd 7152 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥)) |
16 | 15 | eqeq1d 2800 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟)) |
17 | 8, 16 | rabeqbidv 3433 | . . . 4 ⊢ (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) |
18 | 8, 9, 17 | mpoeq123dv 7208 | . . 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 3459 | . . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
20 | fvex 6658 | . . . . . 6 ⊢ (Base‘𝑊) ∈ V | |
21 | 5, 20 | eqeltri 2886 | . . . . 5 ⊢ 𝐵 ∈ V |
22 | ovex 7168 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
23 | 21, 22 | mpoex 7760 | . . . 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 6768 | . 2 ⊢ (𝑊 ∈ 𝑉 → (Sphere‘𝑊) = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})) |
26 | 2, 25 | eqtrd 2833 | 1 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 0cc0 10526 +∞cpnf 10661 [,]cicc 12729 Basecbs 16475 distcds 16566 Spherecsph 45142 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-sph 45144 |
This theorem is referenced by: sphere 45161 rrxsphere 45162 |
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