Theorem lncgr

Description: Congruence rule for lines. Theorem 4.17 of Schwabhauser p. 37. (Contributed by Thierry Arnoux, 28-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 )
	lnxfr.r	|- .~ = ( cgrG ` G )
	lnxfr.a	|- ( ph -> A e. P )
	lnxfr.b	|- ( ph -> B e. P )
	lnxfr.d	|- .- = ( dist ` G )
	lncgr.1	|- ( ph -> X =/= Y )
	lncgr.2	|- ( ph -> ( Y e. ( X L Z ) \/ X = Z ) )
	lncgr.3	|- ( ph -> ( X .- A ) = ( X .- B ) )
	lncgr.4	|- ( ph -> ( Y .- A ) = ( Y .- B ) )
Assertion	lncgr	|- ( ph -> ( Z .- A ) = ( Z .- B ) )

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		lnxfr.r	|- .~ = ( cgrG ` G )
9		lnxfr.a	|- ( ph -> A e. P )
10		lnxfr.b	|- ( ph -> B e. P )
11		lnxfr.d	|- .- = ( dist ` G )
12		lncgr.1	|- ( ph -> X =/= Y )
13		lncgr.2	|- ( ph -> ( Y e. ( X L Z ) \/ X = Z ) )
14		lncgr.3	|- ( ph -> ( X .- A ) = ( X .- B ) )
15		lncgr.4	|- ( ph -> ( Y .- A ) = ( Y .- B ) )
16	1 11 3 8 4 5 6 7	cgr3id	|- ( ph -> <" X Y Z "> .~ <" X Y Z "> )
17	1 2 3 4 5 6 7 8 5 6 11 9 7 10 13 16 14 15 12	tgfscgr	|- ( ph -> ( Z .- A ) = ( Z .- B ) )