Theorem hlcomb

Description: The half-line relation commutes. Theorem 6.6 of Schwabhauser p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020)

	Ref	Expression
Hypotheses	ishlg.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	ishlg.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	ishlg.k	⊢ 𝐾 = ( hlG ‘ 𝐺 )
	ishlg.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	ishlg.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	ishlg.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	ishlg.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑉 )
Assertion	hlcomb	⊢ ( 𝜑 → ( 𝐴 ( 𝐾 ‘ 𝐶 ) 𝐵 ↔ 𝐵 ( 𝐾 ‘ 𝐶 ) 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ishlg.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		ishlg.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		ishlg.k	⊢ 𝐾 = ( hlG ‘ 𝐺 )
4		ishlg.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
5		ishlg.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
6		ishlg.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
7		ishlg.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑉 )
8		3ancoma	⊢ ( ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ ( 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ∨ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) ) ↔ ( 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶 ∧ ( 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ∨ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) ) )
9		orcom	⊢ ( ( 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ∨ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) ↔ ( 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ∨ 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ) )
10	9	a1i	⊢ ( 𝜑 → ( ( 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ∨ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) ↔ ( 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ∨ 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ) ) )
11	10	3anbi3d	⊢ ( 𝜑 → ( ( 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶 ∧ ( 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ∨ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) ) ↔ ( 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶 ∧ ( 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ∨ 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ) ) ) )
12	8 11	syl5bb	⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ ( 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ∨ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) ) ↔ ( 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶 ∧ ( 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ∨ 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ) ) ) )
13	1 2 3 4 5 6 7	ishlg	⊢ ( 𝜑 → ( 𝐴 ( 𝐾 ‘ 𝐶 ) 𝐵 ↔ ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ ( 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ∨ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) ) ) )
14	1 2 3 5 4 6 7	ishlg	⊢ ( 𝜑 → ( 𝐵 ( 𝐾 ‘ 𝐶 ) 𝐴 ↔ ( 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶 ∧ ( 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ∨ 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ) ) ) )
15	12 13 14	3bitr4d	⊢ ( 𝜑 → ( 𝐴 ( 𝐾 ‘ 𝐶 ) 𝐵 ↔ 𝐵 ( 𝐾 ‘ 𝐶 ) 𝐴 ) )