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Definition df-lmi 26553
Description: Define the line mirroring function. Definition 10.3 of [Schwabhauser] p. 89. See islmib 26565. (Contributed by Thierry Arnoux, 1-Dec-2019.)
Assertion
Ref Expression
df-lmi lInvG = (𝑔 ∈ V ↦ (𝑚 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))))))
Distinct variable group:   𝑎,𝑏,𝑔,𝑚

Detailed syntax breakdown of Definition df-lmi
StepHypRef Expression
1 clmi 26551 . 2 class lInvG
2 vg . . 3 setvar 𝑔
3 cvv 3493 . . 3 class V
4 vm . . . 4 setvar 𝑚
52cv 1529 . . . . . 6 class 𝑔
6 clng 26215 . . . . . 6 class LineG
75, 6cfv 6348 . . . . 5 class (LineG‘𝑔)
87crn 5549 . . . 4 class ran (LineG‘𝑔)
9 va . . . . 5 setvar 𝑎
10 cbs 16475 . . . . . 6 class Base
115, 10cfv 6348 . . . . 5 class (Base‘𝑔)
129cv 1529 . . . . . . . . 9 class 𝑎
13 vb . . . . . . . . . 10 setvar 𝑏
1413cv 1529 . . . . . . . . 9 class 𝑏
15 cmid 26550 . . . . . . . . . 10 class midG
165, 15cfv 6348 . . . . . . . . 9 class (midG‘𝑔)
1712, 14, 16co 7148 . . . . . . . 8 class (𝑎(midG‘𝑔)𝑏)
184cv 1529 . . . . . . . 8 class 𝑚
1917, 18wcel 2107 . . . . . . 7 wff (𝑎(midG‘𝑔)𝑏) ∈ 𝑚
2012, 14, 7co 7148 . . . . . . . . 9 class (𝑎(LineG‘𝑔)𝑏)
21 cperpg 26473 . . . . . . . . . 10 class ⟂G
225, 21cfv 6348 . . . . . . . . 9 class (⟂G‘𝑔)
2318, 20, 22wbr 5057 . . . . . . . 8 wff 𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏)
249, 13weq 1957 . . . . . . . 8 wff 𝑎 = 𝑏
2523, 24wo 843 . . . . . . 7 wff (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏)
2619, 25wa 398 . . . . . 6 wff ((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))
2726, 13, 11crio 7105 . . . . 5 class (𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏)))
289, 11, 27cmpt 5137 . . . 4 class (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))))
294, 8, 28cmpt 5137 . . 3 class (𝑚 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏)))))
302, 3, 29cmpt 5137 . 2 class (𝑔 ∈ V ↦ (𝑚 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))))))
311, 30wceq 1530 1 wff lInvG = (𝑔 ∈ V ↦ (𝑚 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))))))
Colors of variables: wff setvar class
This definition is referenced by:  lmif  26563  islmib  26565
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