Theorem lnrot2

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	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 )
	lnrot2.1	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑌 𝐿 𝑍 ) )
	lnrot2.2	⊢ ( 𝜑 → 𝑌 ≠ 𝑍 )
Assertion	lnrot2	⊢ ( 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		lnrot2.1	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑌 𝐿 𝑍 ) )
10		lnrot2.2	⊢ ( 𝜑 → 𝑌 ≠ 𝑍 )
11		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
12	1 11 2 4 6 5 7	tgbtwncomb	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑌 𝐼 𝑍 ) ↔ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) )
13		biidd	⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ↔ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) )
14	1 11 2 4 6 7 5	tgbtwncomb	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ↔ 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ) )
15	12 13 14	3orbi123d	⊢ ( 𝜑 → ( ( 𝑋 ∈ ( 𝑌 𝐼 𝑍 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ∨ 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ) ↔ ( 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ∨ 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ) ) )
16		3orrot	⊢ ( ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ↔ ( 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ∨ 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ) )
17	15 16	bitr4di	⊢ ( 𝜑 → ( ( 𝑋 ∈ ( 𝑌 𝐼 𝑍 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ∨ 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ) ↔ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) )
18	1 3 2 4 6 7 10 5	tgellng	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑌 𝐿 𝑍 ) ↔ ( 𝑋 ∈ ( 𝑌 𝐼 𝑍 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ∨ 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ) ) )
19	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 ) ) ) )
20	17 18 19	3bitr4d	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑌 𝐿 𝑍 ) ↔ 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ) )
21	9 20	mpbid	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) )