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Theorem spheres 47897
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.)
Hypotheses
Ref Expression
spheres.b 𝐡 = (Baseβ€˜π‘Š)
spheres.l 𝑆 = (Sphereβ€˜π‘Š)
spheres.d 𝐷 = (distβ€˜π‘Š)
Assertion
Ref Expression
spheres (π‘Š ∈ 𝑉 β†’ 𝑆 = (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}))
Distinct variable groups:   𝐡,𝑝,π‘Ÿ,π‘₯   π‘Š,𝑝,π‘Ÿ,π‘₯
Allowed substitution hints:   𝐷(π‘₯,π‘Ÿ,𝑝)   𝑆(π‘₯,π‘Ÿ,𝑝)   𝑉(π‘₯,π‘Ÿ,𝑝)

Proof of Theorem spheres
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 spheres.l . . 3 𝑆 = (Sphereβ€˜π‘Š)
21a1i 11 . 2 (π‘Š ∈ 𝑉 β†’ 𝑆 = (Sphereβ€˜π‘Š))
3 df-sph 47881 . . 3 Sphere = (𝑀 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘€), π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ (Baseβ€˜π‘€) ∣ (𝑝(distβ€˜π‘€)π‘₯) = π‘Ÿ}))
4 fveq2 6902 . . . . 5 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
5 spheres.b . . . . . . 7 𝐡 = (Baseβ€˜π‘Š)
65eqcomi 2737 . . . . . 6 (Baseβ€˜π‘Š) = 𝐡
76a1i 11 . . . . 5 (𝑀 = π‘Š β†’ (Baseβ€˜π‘Š) = 𝐡)
84, 7eqtrd 2768 . . . 4 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝐡)
9 eqidd 2729 . . . 4 (𝑀 = π‘Š β†’ (0[,]+∞) = (0[,]+∞))
10 fveq2 6902 . . . . . . . 8 (𝑀 = π‘Š β†’ (distβ€˜π‘€) = (distβ€˜π‘Š))
11 spheres.d . . . . . . . . . 10 𝐷 = (distβ€˜π‘Š)
1211eqcomi 2737 . . . . . . . . 9 (distβ€˜π‘Š) = 𝐷
1312a1i 11 . . . . . . . 8 (𝑀 = π‘Š β†’ (distβ€˜π‘Š) = 𝐷)
1410, 13eqtrd 2768 . . . . . . 7 (𝑀 = π‘Š β†’ (distβ€˜π‘€) = 𝐷)
1514oveqd 7443 . . . . . 6 (𝑀 = π‘Š β†’ (𝑝(distβ€˜π‘€)π‘₯) = (𝑝𝐷π‘₯))
1615eqeq1d 2730 . . . . 5 (𝑀 = π‘Š β†’ ((𝑝(distβ€˜π‘€)π‘₯) = π‘Ÿ ↔ (𝑝𝐷π‘₯) = π‘Ÿ))
178, 16rabeqbidv 3448 . . . 4 (𝑀 = π‘Š β†’ {𝑝 ∈ (Baseβ€˜π‘€) ∣ (𝑝(distβ€˜π‘€)π‘₯) = π‘Ÿ} = {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ})
188, 9, 17mpoeq123dv 7501 . . 3 (𝑀 = π‘Š β†’ (π‘₯ ∈ (Baseβ€˜π‘€), π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ (Baseβ€˜π‘€) ∣ (𝑝(distβ€˜π‘€)π‘₯) = π‘Ÿ}) = (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}))
19 elex 3492 . . 3 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
20 fvex 6915 . . . . . 6 (Baseβ€˜π‘Š) ∈ V
215, 20eqeltri 2825 . . . . 5 𝐡 ∈ V
22 ovex 7459 . . . . 5 (0[,]+∞) ∈ V
2321, 22mpoex 8090 . . . 4 (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}) ∈ V
2423a1i 11 . . 3 (π‘Š ∈ 𝑉 β†’ (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}) ∈ V)
253, 18, 19, 24fvmptd3 7033 . 2 (π‘Š ∈ 𝑉 β†’ (Sphereβ€˜π‘Š) = (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}))
262, 25eqtrd 2768 1 (π‘Š ∈ 𝑉 β†’ 𝑆 = (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  {crab 3430  Vcvv 3473  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  0cc0 11146  +∞cpnf 11283  [,]cicc 13367  Basecbs 17187  distcds 17249  Spherecsph 47879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-sph 47881
This theorem is referenced by:  sphere  47898  rrxsphere  47899
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