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Theorem ishlg 25404
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 473 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑎 = 𝐴)
21neeq1d 2849 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝐶𝐴𝐶))
3 simpr 477 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑏 = 𝐵)
43neeq1d 2849 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑏𝐶𝐵𝐶))
53oveq2d 6623 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝐶𝐼𝑏) = (𝐶𝐼𝐵))
61, 5eleq12d 2692 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 ∈ (𝐶𝐼𝑏) ↔ 𝐴 ∈ (𝐶𝐼𝐵)))
71oveq2d 6623 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝐶𝐼𝑎) = (𝐶𝐼𝐴))
83, 7eleq12d 2692 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑏 ∈ (𝐶𝐼𝑎) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))
96, 8orbi12d 745 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)) ↔ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))
102, 4, 93anbi123d 1396 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))) ↔ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))
11 eqid 2621 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}
1210, 11brab2a 5131 . . 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 3198 . . . . . 6 (𝐺𝑉𝐺 ∈ V)
17 fveq2 6150 . . . . . . . . 9 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
18 ishlg.p . . . . . . . . 9 𝑃 = (Base‘𝐺)
1917, 18syl6eqr 2673 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
2019eleq2d 2684 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑎 ∈ (Base‘𝑔) ↔ 𝑎𝑃))
2119eleq2d 2684 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑏 ∈ (Base‘𝑔) ↔ 𝑏𝑃))
2220, 21anbi12d 746 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ↔ (𝑎𝑃𝑏𝑃)))
23 fveq2 6150 . . . . . . . . . . . . . . 15 (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
24 ishlg.i . . . . . . . . . . . . . . 15 𝐼 = (Itv‘𝐺)
2523, 24syl6eqr 2673 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
2625oveqd 6624 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑏) = (𝑐𝐼𝑏))
2726eleq2d 2684 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ↔ 𝑎 ∈ (𝑐𝐼𝑏)))
2825oveqd 6624 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (𝑐(Itv‘𝑔)𝑎) = (𝑐𝐼𝑎))
2928eleq2d 2684 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (𝑏 ∈ (𝑐(Itv‘𝑔)𝑎) ↔ 𝑏 ∈ (𝑐𝐼𝑎)))
3027, 29orbi12d 745 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎)) ↔ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))
31303anbi3d 1402 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))) ↔ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))))
3222, 31anbi12d 746 . . . . . . . . 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 4680 . . . . . . . 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 4695 . . . . . . 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 25403 . . . . . . 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))))}))
36 fvex 6160 . . . . . . . . 9 (Base‘𝐺) ∈ V
3718, 36eqeltri 2694 . . . . . . . 8 𝑃 ∈ V
3837mptex 6443 . . . . . . 7 (𝑐𝑃 ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}) ∈ V
3934, 35, 38fvmpt 6241 . . . . . 6 (𝐺 ∈ V → (hlG‘𝐺) = (𝑐𝑃 ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}))
4015, 16, 393syl 18 . . . . 5 (𝜑 → (hlG‘𝐺) = (𝑐𝑃 ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}))
4114, 40syl5eq 2667 . . . 4 (𝜑𝐾 = (𝑐𝑃 ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))}))
42 neeq2 2853 . . . . . . . 8 (𝑐 = 𝐶 → (𝑎𝑐𝑎𝐶))
43 neeq2 2853 . . . . . . . 8 (𝑐 = 𝐶 → (𝑏𝑐𝑏𝐶))
44 oveq1 6614 . . . . . . . . . 10 (𝑐 = 𝐶 → (𝑐𝐼𝑏) = (𝐶𝐼𝑏))
4544eleq2d 2684 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑎 ∈ (𝑐𝐼𝑏) ↔ 𝑎 ∈ (𝐶𝐼𝑏)))
46 oveq1 6614 . . . . . . . . . 10 (𝑐 = 𝐶 → (𝑐𝐼𝑎) = (𝐶𝐼𝑎))
4746eleq2d 2684 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑏 ∈ (𝑐𝐼𝑎) ↔ 𝑏 ∈ (𝐶𝐼𝑎)))
4845, 47orbi12d 745 . . . . . . . 8 (𝑐 = 𝐶 → ((𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)) ↔ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))
4942, 43, 483anbi123d 1396 . . . . . . 7 (𝑐 = 𝐶 → ((𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))) ↔ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎)))))
5049anbi2d 739 . . . . . 6 (𝑐 = 𝐶 → (((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎)))) ↔ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))))
5150opabbidv 4680 . . . . 5 (𝑐 = 𝐶 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))})
5251adantl 482 . . . 4 ((𝜑𝑐 = 𝐶) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑐𝐼𝑎))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))})
53 ishlg.c . . . 4 (𝜑𝐶𝑃)
5437, 37xpex 6918 . . . . . 6 (𝑃 × 𝑃) ∈ V
55 opabssxp 5156 . . . . . 6 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ⊆ (𝑃 × 𝑃)
5654, 55ssexi 4765 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ∈ V
5756a1i 11 . . . 4 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))} ∈ V)
5841, 52, 53, 57fvmptd 6247 . . 3 (𝜑 → (𝐾𝐶) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))})
5958breqd 4626 . 2 (𝜑 → (𝐴(𝐾𝐶)𝐵𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑃𝑏𝑃) ∧ (𝑎𝐶𝑏𝐶 ∧ (𝑎 ∈ (𝐶𝐼𝑏) ∨ 𝑏 ∈ (𝐶𝐼𝑎))))}𝐵))
60 ishlg.a . . . 4 (𝜑𝐴𝑃)
61 ishlg.b . . . 4 (𝜑𝐵𝑃)
6260, 61jca 554 . . 3 (𝜑 → (𝐴𝑃𝐵𝑃))
6362biantrurd 529 . 2 (𝜑 → ((𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) ↔ ((𝐴𝑃𝐵𝑃) ∧ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))))
6413, 59, 633bitr4d 300 1 (𝜑 → (𝐴(𝐾𝐶)𝐵 ↔ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186   class class class wbr 4615  {copab 4674  cmpt 4675   × cxp 5074  cfv 5849  (class class class)co 6607  Basecbs 15784  Itvcitv 25242  hlGchlg 25402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-hlg 25403
This theorem is referenced by:  hlcomb  25405  hlne1  25407  hlne2  25408  hlln  25409  hlid  25411  hltr  25412  hlbtwn  25413  btwnhl1  25414  btwnhl2  25415  btwnhl  25416  lnhl  25417  hlcgrex  25418  mirhl  25481  mirbtwnhl  25482  mirhl2  25483  hlperpnel  25524  opphllem4  25549  opphl  25553  hlpasch  25555  lnopp2hpgb  25562  cgracgr  25617  cgraswap  25619  cgrahl  25625  cgracol  25626
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