Theorem tgsegconeq

Description: Two points that satisfy the conclusion of axtgsegcon are identical. Uniqueness portion of Theorem 2.12 of Schwabhauser p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019)

	Ref	Expression
Hypotheses	tkgeom.p	|- P = ( Base ` G )
	tkgeom.d	|- .- = ( dist ` G )
	tkgeom.i	|- I = ( Itv ` G )
	tkgeom.g	|- ( ph -> G e. TarskiG )
	tgcgrextend.a	|- ( ph -> A e. P )
	tgcgrextend.b	|- ( ph -> B e. P )
	tgcgrextend.c	|- ( ph -> C e. P )
	tgcgrextend.d	|- ( ph -> D e. P )
	tgcgrextend.e	|- ( ph -> E e. P )
	tgcgrextend.f	|- ( ph -> F e. P )
	tgsegconeq.1	|- ( ph -> D =/= A )
	tgsegconeq.2	|- ( ph -> A e. ( D I E ) )
	tgsegconeq.3	|- ( ph -> A e. ( D I F ) )
	tgsegconeq.4	|- ( ph -> ( A .- E ) = ( B .- C ) )
	tgsegconeq.5	|- ( ph -> ( A .- F ) = ( B .- C ) )
Assertion	tgsegconeq	|- ( ph -> E = F )

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	|- P = ( Base ` G )
2		tkgeom.d	|- .- = ( dist ` G )
3		tkgeom.i	|- I = ( Itv ` G )
4		tkgeom.g	|- ( ph -> G e. TarskiG )
5		tgcgrextend.a	|- ( ph -> A e. P )
6		tgcgrextend.b	|- ( ph -> B e. P )
7		tgcgrextend.c	|- ( ph -> C e. P )
8		tgcgrextend.d	|- ( ph -> D e. P )
9		tgcgrextend.e	|- ( ph -> E e. P )
10		tgcgrextend.f	|- ( ph -> F e. P )
11		tgsegconeq.1	|- ( ph -> D =/= A )
12		tgsegconeq.2	|- ( ph -> A e. ( D I E ) )
13		tgsegconeq.3	|- ( ph -> A e. ( D I F ) )
14		tgsegconeq.4	|- ( ph -> ( A .- E ) = ( B .- C ) )
15		tgsegconeq.5	|- ( ph -> ( A .- F ) = ( B .- C ) )
16		eqidd	|- ( ph -> ( D .- A ) = ( D .- A ) )
17		eqidd	|- ( ph -> ( A .- E ) = ( A .- E ) )
18	14 15	eqtr4d	|- ( ph -> ( A .- E ) = ( A .- F ) )
19	1 2 3 4 8 5 9 8 5 10 12 13 16 18	tgcgrextend	|- ( ph -> ( D .- E ) = ( D .- F ) )
20	1 2 3 4 8 5 9 8 5 9 9 10 11 12 12 16 17 19 18	axtg5seg	|- ( ph -> ( E .- E ) = ( E .- F ) )
21	20	eqcomd	|- ( ph -> ( E .- F ) = ( E .- E ) )
22	1 2 3 4 9 10 9 21	axtgcgrid	|- ( ph -> E = F )