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	|- ( ph -> G e. TarskiG )
	tglngval.x	|- ( ph -> X e. P )
	tglngval.y	|- ( ph -> Y e. P )
	tgcolg.z	|- ( ph -> Z e. P )
	colrot	|- ( ph -> ( Z e. ( X L Y ) \/ X = Y ) )
Assertion	colrot2	|- ( ph -> ( Y e. ( 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	|- ( ph -> G e. TarskiG )
5		tglngval.x	|- ( ph -> X e. P )
6		tglngval.y	|- ( ph -> Y e. P )
7		tgcolg.z	|- ( ph -> Z e. P )
8		colrot	|- ( ph -> ( Z e. ( X L Y ) \/ X = Y ) )
9	1 2 3 4 5 6 7 8	colrot1	|- ( ph -> ( X e. ( Y L Z ) \/ Y = Z ) )
10	1 2 3 4 6 7 5 9	colrot1	|- ( ph -> ( Y e. ( Z L X ) \/ Z = X ) )