Metamath Proof Explorer

Theorem cgr3rotl

Description: Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 1-Aug-2020)

Ref Expression
Hypotheses tgcgrxfr.p 
|- P = ( Base ` G )
tgcgrxfr.m 
|- .- = ( dist ` G )
tgcgrxfr.i 
|- I = ( Itv ` G )
tgcgrxfr.r 
|- .~ = ( cgrG ` G )
tgcgrxfr.g 
|- ( ph -> G e. TarskiG )
tgbtwnxfr.a 
|- ( ph -> A e. P )
tgbtwnxfr.b 
|- ( ph -> B e. P )
tgbtwnxfr.c 
|- ( ph -> C e. P )
tgbtwnxfr.d 
|- ( ph -> D e. P )
tgbtwnxfr.e 
|- ( ph -> E e. P )
tgbtwnxfr.f 
|- ( ph -> F e. P )
tgbtwnxfr.2 
|- ( ph -> <" A B C "> .~ <" D E F "> )
Assertion cgr3rotl 
|- ( ph -> <" B C A "> .~ <" E 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 
 |-  ( ph -> G e. TarskiG )
6 tgbtwnxfr.a 
 |-  ( ph -> A e. P )
7 tgbtwnxfr.b 
 |-  ( ph -> B e. P )
8 tgbtwnxfr.c 
 |-  ( ph -> C e. P )
9 tgbtwnxfr.d 
 |-  ( ph -> D e. P )
10 tgbtwnxfr.e 
 |-  ( ph -> E e. P )
11 tgbtwnxfr.f 
 |-  ( ph -> F e. P )
12 tgbtwnxfr.2 
 |-  ( ph -> <" A B C "> .~ <" D E F "> )
13 1 2 3 4 5 6 7 8 9 10 11 12 cgr3swap12 
 |-  ( ph -> <" B A C "> .~ <" E D F "> )
14 1 2 3 4 5 7 6 8 10 9 11 13 cgr3swap23 
 |-  ( ph -> <" B C A "> .~ <" E F D "> )