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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 48651 | . . 3 ⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟})) | |
| 4 | fveq2 6906 | . . . . 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 2777 | . . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
| 9 | eqidd 2738 | . . . 4 ⊢ (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞)) | |
| 10 | fveq2 6906 | . . . . . . . 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 2777 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷) |
| 15 | 14 | oveqd 7448 | . . . . . 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 3455 | . . . 4 ⊢ (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) |
| 18 | 8, 9, 17 | mpoeq123dv 7508 | . . 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 3501 | . . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
| 20 | fvex 6919 | . . . . . 6 ⊢ (Base‘𝑊) ∈ V | |
| 21 | 5, 20 | eqeltri 2837 | . . . . 5 ⊢ 𝐵 ∈ V |
| 22 | ovex 7464 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
| 23 | 21, 22 | mpoex 8104 | . . . 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 7039 | . 2 ⊢ (𝑊 ∈ 𝑉 → (Sphere‘𝑊) = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})) |
| 26 | 2, 25 | eqtrd 2777 | 1 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 0cc0 11155 +∞cpnf 11292 [,]cicc 13390 Basecbs 17247 distcds 17306 Spherecsph 48649 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-sph 48651 |
| This theorem is referenced by: sphere 48668 rrxsphere 48669 |
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