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
|- ( 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 trgcgrcom
|- ( ph -> <" 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
 |-  ( 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 cgr3simp1
 |-  ( ph -> ( A .- B ) = ( D .- E ) )
14 13 eqcomd
 |-  ( ph -> ( D .- E ) = ( A .- B ) )
15 1 2 3 4 5 6 7 8 9 10 11 12 cgr3simp2
 |-  ( ph -> ( B .- C ) = ( E .- F ) )
16 15 eqcomd
 |-  ( ph -> ( E .- F ) = ( B .- C ) )
17 1 2 3 4 5 6 7 8 9 10 11 12 cgr3simp3
 |-  ( ph -> ( C .- A ) = ( F .- D ) )
18 17 eqcomd
 |-  ( ph -> ( F .- D ) = ( C .- A ) )
19 1 2 4 5 9 10 11 6 7 8 14 16 18 trgcgr
 |-  ( ph -> <" D E F "> .~ <" A B C "> )