Theorem lncom

Description: Swapping the points defining a line keeps it unchanged. Part of Theorem 4.11 of Schwabhauser p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019)

	Ref	Expression
Hypotheses	btwnlng1.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	btwnlng1.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	btwnlng1.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	btwnlng1.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	btwnlng1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	btwnlng1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
	btwnlng1.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
	btwnlng1.d	⊢ ( 𝜑 → 𝑋 ≠ 𝑌 )
	lncom.1	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑌 𝐿 𝑋 ) )
Assertion	lncom	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		btwnlng1.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		btwnlng1.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		btwnlng1.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
4		btwnlng1.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		btwnlng1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
6		btwnlng1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
7		btwnlng1.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
8		btwnlng1.d	⊢ ( 𝜑 → 𝑋 ≠ 𝑌 )
9		lncom.1	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑌 𝐿 𝑋 ) )
10		3orcomb	⊢ ( ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ↔ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) )
11		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
12	1 11 2 4 5 7 6	tgbtwncomb	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ↔ 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ) )
13	1 11 2 4 5 6 7	tgbtwncomb	⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ↔ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) )
14	1 11 2 4 7 5 6	tgbtwncomb	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ↔ 𝑋 ∈ ( 𝑌 𝐼 𝑍 ) ) )
15	12 13 14	3orbi123d	⊢ ( 𝜑 → ( ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) ↔ ( 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ∨ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ∨ 𝑋 ∈ ( 𝑌 𝐼 𝑍 ) ) ) )
16	10 15	syl5bb	⊢ ( 𝜑 → ( ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ↔ ( 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ∨ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ∨ 𝑋 ∈ ( 𝑌 𝐼 𝑍 ) ) ) )
17	1 3 2 4 5 6 8 7	tgellng	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ↔ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) )
18	8	necomd	⊢ ( 𝜑 → 𝑌 ≠ 𝑋 )
19	1 3 2 4 6 5 18 7	tgellng	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑌 𝐿 𝑋 ) ↔ ( 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ∨ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ∨ 𝑋 ∈ ( 𝑌 𝐼 𝑍 ) ) ) )
20	16 17 19	3bitr4d	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ↔ 𝑍 ∈ ( 𝑌 𝐿 𝑋 ) ) )
21	9 20	mpbird	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) )