Metamath Proof Explorer


Theorem trgcgrcom

Description: Commutative law for three-place congruence. (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 trgcgrcom ( 𝜑 → ⟨“ 𝐷 𝐸 𝐹 ”⟩ ⟨“ 𝐴 𝐵 𝐶 ”⟩ )

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 3 4 5 6 7 8 9 10 11 12 cgr3simp1 ( 𝜑 → ( 𝐴 𝐵 ) = ( 𝐷 𝐸 ) )
14 13 eqcomd ( 𝜑 → ( 𝐷 𝐸 ) = ( 𝐴 𝐵 ) )
15 1 2 3 4 5 6 7 8 9 10 11 12 cgr3simp2 ( 𝜑 → ( 𝐵 𝐶 ) = ( 𝐸 𝐹 ) )
16 15 eqcomd ( 𝜑 → ( 𝐸 𝐹 ) = ( 𝐵 𝐶 ) )
17 1 2 3 4 5 6 7 8 9 10 11 12 cgr3simp3 ( 𝜑 → ( 𝐶 𝐴 ) = ( 𝐹 𝐷 ) )
18 17 eqcomd ( 𝜑 → ( 𝐹 𝐷 ) = ( 𝐶 𝐴 ) )
19 1 2 4 5 9 10 11 6 7 8 14 16 18 trgcgr ( 𝜑 → ⟨“ 𝐷 𝐸 𝐹 ”⟩ ⟨“ 𝐴 𝐵 𝐶 ”⟩ )