Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ishlg Structured version   Visualization version   GIF version

Theorem ishlg 26406
 Description: Rays : Definition 6.1 of [Schwabhauser] p. 43. With this definition, 𝐴(𝐾‘𝐶)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. ((𝐾‘𝐶) “ {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 486 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑎 = 𝐴)
21neeq1d 3046 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝐶𝐴𝐶))
3 simpr 488 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑏 = 𝐵)
43neeq1d 3046 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑏𝐶𝐵𝐶))
53oveq2d 7152 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝐶𝐼𝑏) = (𝐶𝐼𝐵))
61, 5eleq12d 2884 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 ∈ (𝐶𝐼𝑏) ↔ 𝐴 ∈ (𝐶𝐼𝐵)))
71oveq2d 7152 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝐶𝐼𝑎) = (𝐶𝐼𝐴))
83, 7eleq12d 2884 . . . . . 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 2798 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}
1210, 11brab2a 5609 . . 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 3459 . . . . . 6 (𝐺𝑉𝐺 ∈ V)
17 fveq2 6646 . . . . . . . . 9 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
18 ishlg.p . . . . . . . . 9 𝑃 = (Base‘𝐺)
1917, 18eqtr4di 2851 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
2019eleq2d 2875 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑎 ∈ (Base‘𝑔) ↔ 𝑎𝑃))
2119eleq2d 2875 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑏 ∈ (Base‘𝑔) ↔ 𝑏𝑃))
2220, 21anbi12d 633 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ↔ (𝑎𝑃𝑏𝑃)))
23 fveq2 6646 . . . . . . . . . . . . . . 15 (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
24 ishlg.i . . . . . . . . . . . . . . 15 𝐼 = (Itv‘𝐺)
2523, 24eqtr4di 2851 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
2625oveqd 7153 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑏) = (𝑐𝐼𝑏))
2726eleq2d 2875 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ↔ 𝑎 ∈ (𝑐𝐼𝑏)))
2825oveqd 7153 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑎) = (𝑐𝐼𝑎))
2928eleq2d 2875 . . . . . . . . . . . 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 633 . . . . . . . . 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 5097 . . . . . . . 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 5116 . . . . . . 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 26405 . . . . . . 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 6969 . . . . . 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, 37syl5eq 2845 . . . 4 (𝜑𝐾 = (𝑐𝑃 ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}))
39 neeq2 3050 . . . . . . . 8 (𝑐 = 𝐶 → (𝑎𝑐𝑎𝐶))
40 neeq2 3050 . . . . . . . 8 (𝑐 = 𝐶 → (𝑏𝑐𝑏𝐶))
41 oveq1 7143 . . . . . . . . . 10 (𝑐 = 𝐶 → (𝑐𝐼𝑏) = (𝐶𝐼𝑏))
4241eleq2d 2875 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑎 ∈ (𝑐𝐼𝑏) ↔ 𝑎 ∈ (𝐶𝐼𝑏)))
43 oveq1 7143 . . . . . . . . . 10 (𝑐 = 𝐶 → (𝑐𝐼𝑎) = (𝐶𝐼𝑎))
4443eleq2d 2875 . . . . . . . . 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 631 . . . . . 6 (𝑐 = 𝐶 → (((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))) ↔ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))))
4847opabbidv 5097 . . . . 5 (𝑐 = 𝐶 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))})
4948adantl 485 . . . 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 6660 . . . . . . 7 𝑃 ∈ V
5251, 51xpex 7459 . . . . . 6 (𝑃 × 𝑃) ∈ V
53 opabssxp 5608 . . . . . 6 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ⊆ (𝑃 × 𝑃)
5452, 53ssexi 5191 . . . . 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 6753 . . 3 (𝜑 → (𝐾𝐶) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))})
5756breqd 5042 . 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 515 . . 3 (𝜑 → (𝐴𝑃𝐵𝑃))
6160biantrurd 536 . 2 (𝜑 → ((𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) ↔ ((𝐴𝑃𝐵𝑃) ∧ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))))
6213, 57, 613bitr4d 314 1 (𝜑 → (𝐴(𝐾𝐶)𝐵 ↔ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441   class class class wbr 5031  {copab 5093   ↦ cmpt 5111   × cxp 5518  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  Itvcitv 26240  hlGchlg 26404 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-hlg 26405 This theorem is referenced by:  hlcomb  26407  hlne1  26409  hlne2  26410  hlln  26411  hlid  26413  hltr  26414  hlbtwn  26415  btwnhl1  26416  btwnhl2  26417  btwnhl  26418  lnhl  26419  hlcgrex  26420  mirhl  26483  mirbtwnhl  26484  mirhl2  26485  opphllem4  26554  opphl  26558  hlpasch  26560  lnopp2hpgb  26567  cgracgr  26622  cgraswap  26624  flatcgra  26628  cgrahl  26631  cgracol  26632
 Copyright terms: Public domain W3C validator