Theorem lnid

Description: Identity law for points on lines. Theorem 4.18 of Schwabhauser p. 38. (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 )
	lnid.1	|- ( ph -> X =/= Y )
	lnid.2	|- ( ph -> ( Y e. ( X L Z ) \/ X = Z ) )
	lnid.3	|- ( ph -> ( X .- Z ) = ( X .- A ) )
	lnid.4	|- ( ph -> ( Y .- Z ) = ( Y .- A ) )
Assertion	lnid	|- ( ph -> Z = A )

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		lnid.1	|- ( ph -> X =/= Y )
13		lnid.2	|- ( ph -> ( Y e. ( X L Z ) \/ X = Z ) )
14		lnid.3	|- ( ph -> ( X .- Z ) = ( X .- A ) )
15		lnid.4	|- ( ph -> ( Y .- Z ) = ( Y .- A ) )
16	1 2 3 4 5 6 7 8 7 9 11 12 13 14 15	lncgr	|- ( ph -> ( Z .- Z ) = ( Z .- A ) )
17	16	eqcomd	|- ( ph -> ( Z .- A ) = ( Z .- Z ) )
18	1 11 3 4 7 9 7 17	axtgcgrid	|- ( ph -> Z = A )