Metamath Proof Explorer


Theorem cgr3simp2

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 ( 𝜑𝐺 ∈ TarskiG )
tgbtwnxfr.a ( 𝜑𝐴𝑃 )
tgbtwnxfr.b ( 𝜑𝐵𝑃 )
tgbtwnxfr.c ( 𝜑𝐶𝑃 )
tgbtwnxfr.d ( 𝜑𝐷𝑃 )
tgbtwnxfr.e ( 𝜑𝐸𝑃 )
tgbtwnxfr.f ( 𝜑𝐹𝑃 )
tgbtwnxfr.2 ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
Assertion cgr3simp2 ( 𝜑 → ( 𝐵 𝐶 ) = ( 𝐸 𝐹 ) )

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 ( 𝜑𝐺 ∈ TarskiG )
6 tgbtwnxfr.a ( 𝜑𝐴𝑃 )
7 tgbtwnxfr.b ( 𝜑𝐵𝑃 )
8 tgbtwnxfr.c ( 𝜑𝐶𝑃 )
9 tgbtwnxfr.d ( 𝜑𝐷𝑃 )
10 tgbtwnxfr.e ( 𝜑𝐸𝑃 )
11 tgbtwnxfr.f ( 𝜑𝐹𝑃 )
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 simp2d ( 𝜑 → ( 𝐵 𝐶 ) = ( 𝐸 𝐹 ) )