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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 44766 | . . 3 ⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟})) | |
4 | fveq2 6670 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
5 | spheres.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑊) | |
6 | 5 | eqcomi 2830 | . . . . . 6 ⊢ (Base‘𝑊) = 𝐵 |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵) |
8 | 4, 7 | eqtrd 2856 | . . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
9 | eqidd 2822 | . . . 4 ⊢ (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞)) | |
10 | fveq2 6670 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊)) | |
11 | spheres.d | . . . . . . . . . 10 ⊢ 𝐷 = (dist‘𝑊) | |
12 | 11 | eqcomi 2830 | . . . . . . . . 9 ⊢ (dist‘𝑊) = 𝐷 |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷) |
14 | 10, 13 | eqtrd 2856 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷) |
15 | 14 | oveqd 7173 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥)) |
16 | 15 | eqeq1d 2823 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟)) |
17 | 8, 16 | rabeqbidv 3485 | . . . 4 ⊢ (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) |
18 | 8, 9, 17 | mpoeq123dv 7229 | . . 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 3512 | . . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
20 | fvex 6683 | . . . . . 6 ⊢ (Base‘𝑊) ∈ V | |
21 | 5, 20 | eqeltri 2909 | . . . . 5 ⊢ 𝐵 ∈ V |
22 | ovex 7189 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
23 | 21, 22 | mpoex 7777 | . . . 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 6791 | . 2 ⊢ (𝑊 ∈ 𝑉 → (Sphere‘𝑊) = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})) |
26 | 2, 25 | eqtrd 2856 | 1 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 0cc0 10537 +∞cpnf 10672 [,]cicc 12742 Basecbs 16483 distcds 16574 Spherecsph 44764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-sph 44766 |
This theorem is referenced by: sphere 44783 rrxsphere 44784 |
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