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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgcgrextend.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgcgrextend.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgcgrextend.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgcgrextend.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
	tgcgrextend.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
	tgcgrextend.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑃 )
	tgsegconeq.1	⊢ ( 𝜑 → 𝐷 ≠ 𝐴 )
	tgsegconeq.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐷 𝐼 𝐸 ) )
	tgsegconeq.3	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐷 𝐼 𝐹 ) )
	tgsegconeq.4	⊢ ( 𝜑 → ( 𝐴 − 𝐸 ) = ( 𝐵 − 𝐶 ) )
	tgsegconeq.5	⊢ ( 𝜑 → ( 𝐴 − 𝐹 ) = ( 𝐵 − 𝐶 ) )
Assertion	tgsegconeq	⊢ ( 𝜑 → 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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgcgrextend.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgcgrextend.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		tgcgrextend.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		tgcgrextend.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
9		tgcgrextend.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
10		tgcgrextend.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑃 )
11		tgsegconeq.1	⊢ ( 𝜑 → 𝐷 ≠ 𝐴 )
12		tgsegconeq.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐷 𝐼 𝐸 ) )
13		tgsegconeq.3	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐷 𝐼 𝐹 ) )
14		tgsegconeq.4	⊢ ( 𝜑 → ( 𝐴 − 𝐸 ) = ( 𝐵 − 𝐶 ) )
15		tgsegconeq.5	⊢ ( 𝜑 → ( 𝐴 − 𝐹 ) = ( 𝐵 − 𝐶 ) )
16		eqidd	⊢ ( 𝜑 → ( 𝐷 − 𝐴 ) = ( 𝐷 − 𝐴 ) )
17		eqidd	⊢ ( 𝜑 → ( 𝐴 − 𝐸 ) = ( 𝐴 − 𝐸 ) )
18	14 15	eqtr4d	⊢ ( 𝜑 → ( 𝐴 − 𝐸 ) = ( 𝐴 − 𝐹 ) )
19	1 2 3 4 8 5 9 8 5 10 12 13 16 18	tgcgrextend	⊢ ( 𝜑 → ( 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	⊢ ( 𝜑 → ( 𝐸 − 𝐸 ) = ( 𝐸 − 𝐹 ) )
21	20	eqcomd	⊢ ( 𝜑 → ( 𝐸 − 𝐹 ) = ( 𝐸 − 𝐸 ) )
22	1 2 3 4 9 10 9 21	axtgcgrid	⊢ ( 𝜑 → 𝐸 = 𝐹 )