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Definition df-inag 26629
 Description: Definition of the geometrical "in angle" relation. (Contributed by Thierry Arnoux, 15-Aug-2020.)
Assertion
Ref Expression
df-inag inA = (𝑔 ∈ V ↦ {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))})
Distinct variable group:   𝑔,𝑝,𝑡,𝑥

Detailed syntax breakdown of Definition df-inag
StepHypRef Expression
1 cinag 26627 . 2 class inA
2 vg . . 3 setvar 𝑔
3 cvv 3469 . . 3 class V
4 vp . . . . . . . 8 setvar 𝑝
54cv 1537 . . . . . . 7 class 𝑝
62cv 1537 . . . . . . . 8 class 𝑔
7 cbs 16474 . . . . . . . 8 class Base
86, 7cfv 6334 . . . . . . 7 class (Base‘𝑔)
95, 8wcel 2114 . . . . . 6 wff 𝑝 ∈ (Base‘𝑔)
10 vt . . . . . . . 8 setvar 𝑡
1110cv 1537 . . . . . . 7 class 𝑡
12 cc0 10526 . . . . . . . . 9 class 0
13 c3 11681 . . . . . . . . 9 class 3
14 cfzo 13028 . . . . . . . . 9 class ..^
1512, 13, 14co 7140 . . . . . . . 8 class (0..^3)
16 cmap 8393 . . . . . . . 8 class m
178, 15, 16co 7140 . . . . . . 7 class ((Base‘𝑔) ↑m (0..^3))
1811, 17wcel 2114 . . . . . 6 wff 𝑡 ∈ ((Base‘𝑔) ↑m (0..^3))
199, 18wa 399 . . . . 5 wff (𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑m (0..^3)))
2012, 11cfv 6334 . . . . . . . 8 class (𝑡‘0)
21 c1 10527 . . . . . . . . 9 class 1
2221, 11cfv 6334 . . . . . . . 8 class (𝑡‘1)
2320, 22wne 3011 . . . . . . 7 wff (𝑡‘0) ≠ (𝑡‘1)
24 c2 11680 . . . . . . . . 9 class 2
2524, 11cfv 6334 . . . . . . . 8 class (𝑡‘2)
2625, 22wne 3011 . . . . . . 7 wff (𝑡‘2) ≠ (𝑡‘1)
275, 22wne 3011 . . . . . . 7 wff 𝑝 ≠ (𝑡‘1)
2823, 26, 27w3a 1084 . . . . . 6 wff ((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1))
29 vx . . . . . . . . . 10 setvar 𝑥
3029cv 1537 . . . . . . . . 9 class 𝑥
31 citv 26228 . . . . . . . . . . 11 class Itv
326, 31cfv 6334 . . . . . . . . . 10 class (Itv‘𝑔)
3320, 25, 32co 7140 . . . . . . . . 9 class ((𝑡‘0)(Itv‘𝑔)(𝑡‘2))
3430, 33wcel 2114 . . . . . . . 8 wff 𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2))
3530, 22wceq 1538 . . . . . . . . 9 wff 𝑥 = (𝑡‘1)
36 chlg 26392 . . . . . . . . . . . 12 class hlG
376, 36cfv 6334 . . . . . . . . . . 11 class (hlG‘𝑔)
3822, 37cfv 6334 . . . . . . . . . 10 class ((hlG‘𝑔)‘(𝑡‘1))
3930, 5, 38wbr 5042 . . . . . . . . 9 wff 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝
4035, 39wo 844 . . . . . . . 8 wff (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝)
4134, 40wa 399 . . . . . . 7 wff (𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))
4241, 29, 8wrex 3131 . . . . . 6 wff 𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))
4328, 42wa 399 . . . . 5 wff (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝)))
4419, 43wa 399 . . . 4 wff ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))
4544, 4, 10copab 5104 . . 3 class {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))}
462, 3, 45cmpt 5122 . 2 class (𝑔 ∈ V ↦ {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))})
471, 46wceq 1538 1 wff inA = (𝑔 ∈ V ↦ {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))})
 Colors of variables: wff setvar class This definition is referenced by:  isinag  26630
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