Metamath Proof Explorer


Theorem tgsas2

Description: First congruence theorem: SAS. Theorem 11.49 of Schwabhauser p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020)

Ref Expression
Hypotheses tgsas.p
|- P = ( Base ` G )
tgsas.m
|- .- = ( dist ` G )
tgsas.i
|- I = ( Itv ` G )
tgsas.g
|- ( ph -> G e. TarskiG )
tgsas.a
|- ( ph -> A e. P )
tgsas.b
|- ( ph -> B e. P )
tgsas.c
|- ( ph -> C e. P )
tgsas.d
|- ( ph -> D e. P )
tgsas.e
|- ( ph -> E e. P )
tgsas.f
|- ( ph -> F e. P )
tgsas.1
|- ( ph -> ( A .- B ) = ( D .- E ) )
tgsas.2
|- ( ph -> <" A B C "> ( cgrA ` G ) <" D E F "> )
tgsas.3
|- ( ph -> ( B .- C ) = ( E .- F ) )
tgsas2.4
|- ( ph -> A =/= C )
Assertion tgsas2
|- ( ph -> <" C A B "> ( cgrA ` G ) <" F D E "> )

Proof

Step Hyp Ref Expression
1 tgsas.p
 |-  P = ( Base ` G )
2 tgsas.m
 |-  .- = ( dist ` G )
3 tgsas.i
 |-  I = ( Itv ` G )
4 tgsas.g
 |-  ( ph -> G e. TarskiG )
5 tgsas.a
 |-  ( ph -> A e. P )
6 tgsas.b
 |-  ( ph -> B e. P )
7 tgsas.c
 |-  ( ph -> C e. P )
8 tgsas.d
 |-  ( ph -> D e. P )
9 tgsas.e
 |-  ( ph -> E e. P )
10 tgsas.f
 |-  ( ph -> F e. P )
11 tgsas.1
 |-  ( ph -> ( A .- B ) = ( D .- E ) )
12 tgsas.2
 |-  ( ph -> <" A B C "> ( cgrA ` G ) <" D E F "> )
13 tgsas.3
 |-  ( ph -> ( B .- C ) = ( E .- F ) )
14 tgsas2.4
 |-  ( ph -> A =/= C )
15 eqid
 |-  ( hlG ` G ) = ( hlG ` G )
16 eqid
 |-  ( cgrG ` G ) = ( cgrG ` G )
17 1 2 3 4 5 6 7 8 9 10 11 12 13 tgsas
 |-  ( ph -> <" A B C "> ( cgrG ` G ) <" D E F "> )
18 1 2 3 16 4 5 6 7 8 9 10 17 cgr3rotr
 |-  ( ph -> <" C A B "> ( cgrG ` G ) <" F D E "> )
19 1 2 3 4 5 6 7 8 9 10 11 12 13 tgsas1
 |-  ( ph -> ( C .- A ) = ( F .- D ) )
20 14 necomd
 |-  ( ph -> C =/= A )
21 1 2 3 4 7 5 10 8 19 20 tgcgrneq
 |-  ( ph -> F =/= D )
22 1 3 15 10 5 8 4 21 hlid
 |-  ( ph -> F ( ( hlG ` G ) ` D ) F )
23 1 3 15 4 5 6 7 8 9 10 12 cgrane3
 |-  ( ph -> E =/= D )
24 1 3 15 9 5 8 4 23 hlid
 |-  ( ph -> E ( ( hlG ` G ) ` D ) E )
25 1 3 15 4 7 5 6 10 8 9 10 9 18 22 24 iscgrad
 |-  ( ph -> <" C A B "> ( cgrA ` G ) <" F D E "> )