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Theorem ishlg 28529
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 481 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑎 = 𝐴)
21neeq1d 2990 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝐶𝐴𝐶))
3 simpr 483 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑏 = 𝐵)
43neeq1d 2990 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑏𝐶𝐵𝐶))
53oveq2d 7440 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝐶𝐼𝑏) = (𝐶𝐼𝐵))
61, 5eleq12d 2820 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 ∈ (𝐶𝐼𝑏) ↔ 𝐴 ∈ (𝐶𝐼𝐵)))
71oveq2d 7440 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝐶𝐼𝑎) = (𝐶𝐼𝐴))
83, 7eleq12d 2820 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑏 ∈ (𝐶𝐼𝑎) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))
96, 8orbi12d 916 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)) ↔ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))
102, 4, 93anbi123d 1433 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))) ↔ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))
11 eqid 2726 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}
1210, 11brab2a 5775 . . 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 3482 . . . . . 6 (𝐺𝑉𝐺 ∈ V)
17 fveq2 6901 . . . . . . . . 9 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
18 ishlg.p . . . . . . . . 9 𝑃 = (Base‘𝐺)
1917, 18eqtr4di 2784 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
2019eleq2d 2812 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑎 ∈ (Base‘𝑔) ↔ 𝑎𝑃))
2119eleq2d 2812 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑏 ∈ (Base‘𝑔) ↔ 𝑏𝑃))
2220, 21anbi12d 630 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ↔ (𝑎𝑃𝑏𝑃)))
23 fveq2 6901 . . . . . . . . . . . . . . 15 (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
24 ishlg.i . . . . . . . . . . . . . . 15 𝐼 = (Itv‘𝐺)
2523, 24eqtr4di 2784 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
2625oveqd 7441 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑏) = (𝑐𝐼𝑏))
2726eleq2d 2812 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ↔ 𝑎 ∈ (𝑐𝐼𝑏)))
2825oveqd 7441 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑎) = (𝑐𝐼𝑎))
2928eleq2d 2812 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (𝑏 ∈ (𝑐(Itv‘𝑔)𝑎) ↔ 𝑏 ∈ (𝑐𝐼𝑎)))
3027, 29orbi12d 916 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎)) ↔ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))
31303anbi3d 1439 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))) ↔ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))))
3222, 31anbi12d 630 . . . . . . . . 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 5219 . . . . . . . 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 5244 . . . . . . 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 28528 . . . . . . 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 7245 . . . . . 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 2778 . . . 4 (𝜑𝐾 = (𝑐𝑃 ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}))
39 neeq2 2994 . . . . . . . 8 (𝑐 = 𝐶 → (𝑎𝑐𝑎𝐶))
40 neeq2 2994 . . . . . . . 8 (𝑐 = 𝐶 → (𝑏𝑐𝑏𝐶))
41 oveq1 7431 . . . . . . . . . 10 (𝑐 = 𝐶 → (𝑐𝐼𝑏) = (𝐶𝐼𝑏))
4241eleq2d 2812 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑎 ∈ (𝑐𝐼𝑏) ↔ 𝑎 ∈ (𝐶𝐼𝑏)))
43 oveq1 7431 . . . . . . . . . 10 (𝑐 = 𝐶 → (𝑐𝐼𝑎) = (𝐶𝐼𝑎))
4443eleq2d 2812 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑏 ∈ (𝑐𝐼𝑎) ↔ 𝑏 ∈ (𝐶𝐼𝑎)))
4542, 44orbi12d 916 . . . . . . . 8 (𝑐 = 𝐶 → ((𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)) ↔ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))
4639, 40, 453anbi123d 1433 . . . . . . 7 (𝑐 = 𝐶 → ((𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))) ↔ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)))))
4746anbi2d 628 . . . . . 6 (𝑐 = 𝐶 → (((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))) ↔ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))))
4847opabbidv 5219 . . . . 5 (𝑐 = 𝐶 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))})
4948adantl 480 . . . 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 6915 . . . . . . 7 𝑃 ∈ V
5251, 51xpex 7761 . . . . . 6 (𝑃 × 𝑃) ∈ V
53 opabssxp 5774 . . . . . 6 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ⊆ (𝑃 × 𝑃)
5452, 53ssexi 5327 . . . . 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 7016 . . 3 (𝜑 → (𝐾𝐶) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))})
5756breqd 5164 . 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 510 . . 3 (𝜑 → (𝐴𝑃𝐵𝑃))
6160biantrurd 531 . 2 (𝜑 → ((𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) ↔ ((𝐴𝑃𝐵𝑃) ∧ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))))
6213, 57, 613bitr4d 310 1 (𝜑 → (𝐴(𝐾𝐶)𝐵 ↔ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1534  wcel 2099  wne 2930  Vcvv 3462   class class class wbr 5153  {copab 5215  cmpt 5236   × cxp 5680  cfv 6554  (class class class)co 7424  Basecbs 17213  Itvcitv 28360  hlGchlg 28527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-hlg 28528
This theorem is referenced by:  hlcomb  28530  hlne1  28532  hlne2  28533  hlln  28534  hlid  28536  hltr  28537  hlbtwn  28538  btwnhl1  28539  btwnhl2  28540  btwnhl  28541  lnhl  28542  hlcgrex  28543  mirhl  28606  mirbtwnhl  28607  mirhl2  28608  opphllem4  28677  opphl  28681  hlpasch  28683  lnopp2hpgb  28690  cgracgr  28745  cgraswap  28747  flatcgra  28751  cgrahl  28754  cgracol  28755
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