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Definition df-perpg 25309
Description: Define the "perpendicular" relation. Definition 8.11 of [Schwabhauser] p. 59. See isperp 25325. (Contributed by Thierry Arnoux, 8-Sep-2019.)
Assertion
Ref Expression
df-perpg ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
Distinct variable group:   𝑎,𝑏,𝑢,𝑣,𝑥,𝑔

Detailed syntax breakdown of Definition df-perpg
StepHypRef Expression
1 cperpg 25308 . 2 class ⟂G
2 vg . . 3 setvar 𝑔
3 cvv 3172 . . 3 class V
4 va . . . . . . . 8 setvar 𝑎
54cv 1473 . . . . . . 7 class 𝑎
62cv 1473 . . . . . . . . 9 class 𝑔
7 clng 25053 . . . . . . . . 9 class LineG
86, 7cfv 5790 . . . . . . . 8 class (LineG‘𝑔)
98crn 5029 . . . . . . 7 class ran (LineG‘𝑔)
105, 9wcel 1976 . . . . . 6 wff 𝑎 ∈ ran (LineG‘𝑔)
11 vb . . . . . . . 8 setvar 𝑏
1211cv 1473 . . . . . . 7 class 𝑏
1312, 9wcel 1976 . . . . . 6 wff 𝑏 ∈ ran (LineG‘𝑔)
1410, 13wa 382 . . . . 5 wff (𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔))
15 vu . . . . . . . . . . 11 setvar 𝑢
1615cv 1473 . . . . . . . . . 10 class 𝑢
17 vx . . . . . . . . . . 11 setvar 𝑥
1817cv 1473 . . . . . . . . . 10 class 𝑥
19 vv . . . . . . . . . . 11 setvar 𝑣
2019cv 1473 . . . . . . . . . 10 class 𝑣
2116, 18, 20cs3 13384 . . . . . . . . 9 class ⟨“𝑢𝑥𝑣”⟩
22 crag 25306 . . . . . . . . . 10 class ∟G
236, 22cfv 5790 . . . . . . . . 9 class (∟G‘𝑔)
2421, 23wcel 1976 . . . . . . . 8 wff ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔)
2524, 19, 12wral 2895 . . . . . . 7 wff 𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔)
2625, 15, 5wral 2895 . . . . . 6 wff 𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔)
275, 12cin 3538 . . . . . 6 class (𝑎𝑏)
2826, 17, 27wrex 2896 . . . . 5 wff 𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔)
2914, 28wa 382 . . . 4 wff ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))
3029, 4, 11copab 4636 . . 3 class {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))}
312, 3, 30cmpt 4637 . 2 class (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
321, 31wceq 1474 1 wff ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
Colors of variables: wff setvar class
This definition is referenced by:  perpln1  25323  perpln2  25324  isperp  25325
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