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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 48580 | . . 3 ⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟})) | |
4 | fveq2 6907 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
5 | spheres.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑊) | |
6 | 5 | eqcomi 2744 | . . . . . 6 ⊢ (Base‘𝑊) = 𝐵 |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵) |
8 | 4, 7 | eqtrd 2775 | . . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
9 | eqidd 2736 | . . . 4 ⊢ (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞)) | |
10 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊)) | |
11 | spheres.d | . . . . . . . . . 10 ⊢ 𝐷 = (dist‘𝑊) | |
12 | 11 | eqcomi 2744 | . . . . . . . . 9 ⊢ (dist‘𝑊) = 𝐷 |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷) |
14 | 10, 13 | eqtrd 2775 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷) |
15 | 14 | oveqd 7448 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥)) |
16 | 15 | eqeq1d 2737 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟)) |
17 | 8, 16 | rabeqbidv 3452 | . . . 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 3499 | . . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
20 | fvex 6920 | . . . . . 6 ⊢ (Base‘𝑊) ∈ V | |
21 | 5, 20 | eqeltri 2835 | . . . . 5 ⊢ 𝐵 ∈ V |
22 | ovex 7464 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
23 | 21, 22 | mpoex 8103 | . . . 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 2775 | 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 2106 {crab 3433 Vcvv 3478 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 0cc0 11153 +∞cpnf 11290 [,]cicc 13387 Basecbs 17245 distcds 17307 Spherecsph 48578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-sph 48580 |
This theorem is referenced by: sphere 48597 rrxsphere 48598 |
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