Theorem colrot2

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

	Ref	Expression
Hypotheses	tglngval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tglngval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	tglngval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tglngval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tglngval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	tglngval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
	tgcolg.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
	colrot	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ∨ 𝑋 = 𝑌 ) )
Assertion	colrot2	⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑍 𝐿 𝑋 ) ∨ 𝑍 = 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		tglngval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tglngval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
3		tglngval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tglngval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tglngval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
6		tglngval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
7		tgcolg.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
8		colrot	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ∨ 𝑋 = 𝑌 ) )
9	1 2 3 4 5 6 7 8	colrot1	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑌 𝐿 𝑍 ) ∨ 𝑌 = 𝑍 ) )
10	1 2 3 4 6 7 5 9	colrot1	⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑍 𝐿 𝑋 ) ∨ 𝑍 = 𝑋 ) )