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Definition df-mir 25461
Description: Define the point inversion ("mirror") function. Definition 7.5 of [Schwabhauser] p. 49. See mirval 25463 and ismir 25467. (Contributed by Thierry Arnoux, 30-May-2019.)
Assertion
Ref Expression
df-mir pInvG = (𝑔 ∈ V ↦ (𝑚 ∈ (Base‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎))))))
Distinct variable group:   𝑎,𝑏,𝑔,𝑚

Detailed syntax breakdown of Definition df-mir
StepHypRef Expression
1 cmir 25460 . 2 class pInvG
2 vg . . 3 setvar 𝑔
3 cvv 3189 . . 3 class V
4 vm . . . 4 setvar 𝑚
52cv 1479 . . . . 5 class 𝑔
6 cbs 15788 . . . . 5 class Base
75, 6cfv 5852 . . . 4 class (Base‘𝑔)
8 va . . . . 5 setvar 𝑎
94cv 1479 . . . . . . . . 9 class 𝑚
10 vb . . . . . . . . . 10 setvar 𝑏
1110cv 1479 . . . . . . . . 9 class 𝑏
12 cds 15878 . . . . . . . . . 10 class dist
135, 12cfv 5852 . . . . . . . . 9 class (dist‘𝑔)
149, 11, 13co 6610 . . . . . . . 8 class (𝑚(dist‘𝑔)𝑏)
158cv 1479 . . . . . . . . 9 class 𝑎
169, 15, 13co 6610 . . . . . . . 8 class (𝑚(dist‘𝑔)𝑎)
1714, 16wceq 1480 . . . . . . 7 wff (𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎)
18 citv 25248 . . . . . . . . . 10 class Itv
195, 18cfv 5852 . . . . . . . . 9 class (Itv‘𝑔)
2011, 15, 19co 6610 . . . . . . . 8 class (𝑏(Itv‘𝑔)𝑎)
219, 20wcel 1987 . . . . . . 7 wff 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎)
2217, 21wa 384 . . . . . 6 wff ((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎))
2322, 10, 7crio 6570 . . . . 5 class (𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎)))
248, 7, 23cmpt 4678 . . . 4 class (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎))))
254, 7, 24cmpt 4678 . . 3 class (𝑚 ∈ (Base‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎)))))
262, 3, 25cmpt 4678 . 2 class (𝑔 ∈ V ↦ (𝑚 ∈ (Base‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎))))))
271, 26wceq 1480 1 wff pInvG = (𝑔 ∈ V ↦ (𝑚 ∈ (Base‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎))))))
Colors of variables: wff setvar class
This definition is referenced by:  mirval  25463
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