Theorem cgr3simp3

Description: Deduce segment congruence from a triangle congruence. This is a portion of CPCTC, focusing on a specific segment. (Contributed by Thierry Arnoux, 27-Apr-2019)

	Ref	Expression
Hypotheses	tgcgrxfr.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tgcgrxfr.m	⊢ − = ( dist ‘ 𝐺 )
	tgcgrxfr.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tgcgrxfr.r	⊢ ∼ = ( cgrG ‘ 𝐺 )
	tgcgrxfr.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgbtwnxfr.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgbtwnxfr.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgbtwnxfr.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgbtwnxfr.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
	tgbtwnxfr.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
	tgbtwnxfr.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑃 )
	tgbtwnxfr.2	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∼ ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
Assertion	cgr3simp3	⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		tgcgrxfr.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tgcgrxfr.m	⊢ − = ( dist ‘ 𝐺 )
3		tgcgrxfr.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tgcgrxfr.r	⊢ ∼ = ( cgrG ‘ 𝐺 )
5		tgcgrxfr.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
6		tgbtwnxfr.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
7		tgbtwnxfr.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
8		tgbtwnxfr.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
9		tgbtwnxfr.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
10		tgbtwnxfr.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
11		tgbtwnxfr.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑃 )
12		tgbtwnxfr.2	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∼ ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
13	1 2 4 5 6 7 8 9 10 11	trgcgrg	⊢ ( 𝜑 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∼ ⟨“ 𝐷 𝐸 𝐹 ”⟩ ↔ ( ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐸 ) ∧ ( 𝐵 − 𝐶 ) = ( 𝐸 − 𝐹 ) ∧ ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) ) ) )
14	12 13	mpbid	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐸 ) ∧ ( 𝐵 − 𝐶 ) = ( 𝐸 − 𝐹 ) ∧ ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) ) )
15	14	simp3d	⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) )