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Theorem ishlg 28625
Description: Rays : Definition 6.1 of [Schwabhauser] p. 43. With this definition, 𝐴(𝐾𝐶)𝐵 means that 𝐴 and 𝐵 are on the same ray with initial point 𝐶. This follows the same notation as Schwabhauser where rays are first defined as a relation. It is possible to recover the ray itself using e.g., ((𝐾𝐶) “ {𝐴}). (Contributed by Thierry Arnoux, 21-Dec-2019.)
Hypotheses
Ref Expression
ishlg.p 𝑃 = (Base‘𝐺)
ishlg.i 𝐼 = (Itv‘𝐺)
ishlg.k 𝐾 = (hlG‘𝐺)
ishlg.a (𝜑𝐴𝑃)
ishlg.b (𝜑𝐵𝑃)
ishlg.c (𝜑𝐶𝑃)
ishlg.g (𝜑𝐺𝑉)
Assertion
Ref Expression
ishlg (𝜑 → (𝐴(𝐾𝐶)𝐵 ↔ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))

Proof of Theorem ishlg
Dummy variables 𝑎 𝑏 𝑐 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑎 = 𝐴)
21neeq1d 2998 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝐶𝐴𝐶))
3 simpr 484 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑏 = 𝐵)
43neeq1d 2998 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑏𝐶𝐵𝐶))
53oveq2d 7447 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝐶𝐼𝑏) = (𝐶𝐼𝐵))
61, 5eleq12d 2833 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 ∈ (𝐶𝐼𝑏) ↔ 𝐴 ∈ (𝐶𝐼𝐵)))
71oveq2d 7447 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝐶𝐼𝑎) = (𝐶𝐼𝐴))
83, 7eleq12d 2833 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑏 ∈ (𝐶𝐼𝑎) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))
96, 8orbi12d 918 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)) ↔ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))
102, 4, 93anbi123d 1435 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))) ↔ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))
11 eqid 2735 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}
1210, 11brab2a 5782 . . 3 (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}𝐵 ↔ ((𝐴𝑃𝐵𝑃) ∧ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))
1312a1i 11 . 2 (𝜑 → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}𝐵 ↔ ((𝐴𝑃𝐵𝑃) ∧ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))))
14 ishlg.k . . . . 5 𝐾 = (hlG‘𝐺)
15 ishlg.g . . . . . 6 (𝜑𝐺𝑉)
16 elex 3499 . . . . . 6 (𝐺𝑉𝐺 ∈ V)
17 fveq2 6907 . . . . . . . . 9 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
18 ishlg.p . . . . . . . . 9 𝑃 = (Base‘𝐺)
1917, 18eqtr4di 2793 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
2019eleq2d 2825 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑎 ∈ (Base‘𝑔) ↔ 𝑎𝑃))
2119eleq2d 2825 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑏 ∈ (Base‘𝑔) ↔ 𝑏𝑃))
2220, 21anbi12d 632 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ↔ (𝑎𝑃𝑏𝑃)))
23 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
24 ishlg.i . . . . . . . . . . . . . . 15 𝐼 = (Itv‘𝐺)
2523, 24eqtr4di 2793 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
2625oveqd 7448 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑏) = (𝑐𝐼𝑏))
2726eleq2d 2825 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ↔ 𝑎 ∈ (𝑐𝐼𝑏)))
2825oveqd 7448 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑎) = (𝑐𝐼𝑎))
2928eleq2d 2825 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (𝑏 ∈ (𝑐(Itv‘𝑔)𝑎) ↔ 𝑏 ∈ (𝑐𝐼𝑎)))
3027, 29orbi12d 918 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎)) ↔ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))
31303anbi3d 1441 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))) ↔ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))))
3222, 31anbi12d 632 . . . . . . . . 9 (𝑔 = 𝐺 → (((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎)))) ↔ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))))
3332opabbidv 5214 . . . . . . . 8 (𝑔 = 𝐺 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))})
3419, 33mpteq12dv 5239 . . . . . . 7 (𝑔 = 𝐺 → (𝑐 ∈ (Base‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}) = (𝑐𝑃 ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}))
35 df-hlg 28624 . . . . . . 7 hlG = (𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}))
3634, 35, 18mptfvmpt 7248 . . . . . 6 (𝐺 ∈ V → (hlG‘𝐺) = (𝑐𝑃 ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}))
3715, 16, 363syl 18 . . . . 5 (𝜑 → (hlG‘𝐺) = (𝑐𝑃 ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}))
3814, 37eqtrid 2787 . . . 4 (𝜑𝐾 = (𝑐𝑃 ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}))
39 neeq2 3002 . . . . . . . 8 (𝑐 = 𝐶 → (𝑎𝑐𝑎𝐶))
40 neeq2 3002 . . . . . . . 8 (𝑐 = 𝐶 → (𝑏𝑐𝑏𝐶))
41 oveq1 7438 . . . . . . . . . 10 (𝑐 = 𝐶 → (𝑐𝐼𝑏) = (𝐶𝐼𝑏))
4241eleq2d 2825 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑎 ∈ (𝑐𝐼𝑏) ↔ 𝑎 ∈ (𝐶𝐼𝑏)))
43 oveq1 7438 . . . . . . . . . 10 (𝑐 = 𝐶 → (𝑐𝐼𝑎) = (𝐶𝐼𝑎))
4443eleq2d 2825 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑏 ∈ (𝑐𝐼𝑎) ↔ 𝑏 ∈ (𝐶𝐼𝑎)))
4542, 44orbi12d 918 . . . . . . . 8 (𝑐 = 𝐶 → ((𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)) ↔ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))
4639, 40, 453anbi123d 1435 . . . . . . 7 (𝑐 = 𝐶 → ((𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))) ↔ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)))))
4746anbi2d 630 . . . . . 6 (𝑐 = 𝐶 → (((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))) ↔ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))))
4847opabbidv 5214 . . . . 5 (𝑐 = 𝐶 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))})
4948adantl 481 . . . 4 ((𝜑𝑐 = 𝐶) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))})
50 ishlg.c . . . 4 (𝜑𝐶𝑃)
5118fvexi 6921 . . . . . . 7 𝑃 ∈ V
5251, 51xpex 7772 . . . . . 6 (𝑃 × 𝑃) ∈ V
53 opabssxp 5781 . . . . . 6 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ⊆ (𝑃 × 𝑃)
5452, 53ssexi 5328 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ∈ V
5554a1i 11 . . . 4 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ∈ V)
5638, 49, 50, 55fvmptd 7023 . . 3 (𝜑 → (𝐾𝐶) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))})
5756breqd 5159 . 2 (𝜑 → (𝐴(𝐾𝐶)𝐵𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}𝐵))
58 ishlg.a . . . 4 (𝜑𝐴𝑃)
59 ishlg.b . . . 4 (𝜑𝐵𝑃)
6058, 59jca 511 . . 3 (𝜑 → (𝐴𝑃𝐵𝑃))
6160biantrurd 532 . 2 (𝜑 → ((𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) ↔ ((𝐴𝑃𝐵𝑃) ∧ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))))
6213, 57, 613bitr4d 311 1 (𝜑 → (𝐴(𝐾𝐶)𝐵 ↔ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478   class class class wbr 5148  {copab 5210  cmpt 5231   × cxp 5687  cfv 6563  (class class class)co 7431  Basecbs 17245  Itvcitv 28456  hlGchlg 28623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-hlg 28624
This theorem is referenced by:  hlcomb  28626  hlne1  28628  hlne2  28629  hlln  28630  hlid  28632  hltr  28633  hlbtwn  28634  btwnhl1  28635  btwnhl2  28636  btwnhl  28637  lnhl  28638  hlcgrex  28639  mirhl  28702  mirbtwnhl  28703  mirhl2  28704  opphllem4  28773  opphl  28777  hlpasch  28779  lnopp2hpgb  28786  cgracgr  28841  cgraswap  28843  flatcgra  28847  cgrahl  28850  cgracol  28851
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