Metamath Proof Explorer


Theorem cgr3tr

Description: Transitivity 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 "> )
cgr3tr.j
|- ( ph -> J e. P )
cgr3tr.k
|- ( ph -> K e. P )
cgr3tr.l
|- ( ph -> L e. P )
cgr3tr.1
|- ( ph -> <" D E F "> .~ <" J K L "> )
Assertion cgr3tr
|- ( ph -> <" A B C "> .~ <" J K L "> )

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 cgr3tr.j
 |-  ( ph -> J e. P )
14 cgr3tr.k
 |-  ( ph -> K e. P )
15 cgr3tr.l
 |-  ( ph -> L e. P )
16 cgr3tr.1
 |-  ( ph -> <" D E F "> .~ <" J K L "> )
17 1 2 3 4 5 6 7 8 9 10 11 12 cgr3simp1
 |-  ( ph -> ( A .- B ) = ( D .- E ) )
18 1 2 3 4 5 9 10 11 13 14 15 16 cgr3simp1
 |-  ( ph -> ( D .- E ) = ( J .- K ) )
19 17 18 eqtrd
 |-  ( ph -> ( A .- B ) = ( J .- K ) )
20 1 2 3 4 5 6 7 8 9 10 11 12 cgr3simp2
 |-  ( ph -> ( B .- C ) = ( E .- F ) )
21 1 2 3 4 5 9 10 11 13 14 15 16 cgr3simp2
 |-  ( ph -> ( E .- F ) = ( K .- L ) )
22 20 21 eqtrd
 |-  ( ph -> ( B .- C ) = ( K .- L ) )
23 1 2 3 4 5 6 7 8 9 10 11 12 cgr3simp3
 |-  ( ph -> ( C .- A ) = ( F .- D ) )
24 1 2 3 4 5 9 10 11 13 14 15 16 cgr3simp3
 |-  ( ph -> ( F .- D ) = ( L .- J ) )
25 23 24 eqtrd
 |-  ( ph -> ( C .- A ) = ( L .- J ) )
26 1 2 4 5 6 7 8 13 14 15 19 22 25 trgcgr
 |-  ( ph -> <" A B C "> .~ <" J K L "> )