Metamath Proof Explorer

Theorem cgrahl1

Description: Angle congruence is independent of the choice of points on the rays. Proposition 11.10 of Schwabhauser p. 95. (Contributed by Thierry Arnoux, 1-Aug-2020)

Ref Expression
Hypotheses iscgra.p 
|- P = ( Base ` G )
iscgra.i 
|- I = ( Itv ` G )
iscgra.k 
|- K = ( hlG ` G )
iscgra.g 
|- ( ph -> G e. TarskiG )
iscgra.a 
|- ( ph -> A e. P )
iscgra.b 
|- ( ph -> B e. P )
iscgra.c 
|- ( ph -> C e. P )
iscgra.d 
|- ( ph -> D e. P )
iscgra.e 
|- ( ph -> E e. P )
iscgra.f 
|- ( ph -> F e. P )
cgrahl1.2 
|- ( ph -> <" A B C "> ( cgrA ` G ) <" D E F "> )
cgrahl1.x 
|- ( ph -> X e. P )
cgrahl1.1 
|- ( ph -> X ( K ` E ) D )
Assertion cgrahl1 
|- ( ph -> <" A B C "> ( cgrA ` G ) <" X E F "> )

Proof

Step Hyp Ref Expression
1 iscgra.p 
 |-  P = ( Base ` G )
2 iscgra.i 
 |-  I = ( Itv ` G )
3 iscgra.k 
 |-  K = ( hlG ` G )
4 iscgra.g 
 |-  ( ph -> G e. TarskiG )
5 iscgra.a 
 |-  ( ph -> A e. P )
6 iscgra.b 
 |-  ( ph -> B e. P )
7 iscgra.c 
 |-  ( ph -> C e. P )
8 iscgra.d 
 |-  ( ph -> D e. P )
9 iscgra.e 
 |-  ( ph -> E e. P )
10 iscgra.f 
 |-  ( ph -> F e. P )
11 cgrahl1.2 
 |-  ( ph -> <" A B C "> ( cgrA ` G ) <" D E F "> )
12 cgrahl1.x 
 |-  ( ph -> X e. P )
13 cgrahl1.1 
 |-  ( ph -> X ( K ` E ) D )
14 4 ad3antrrr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> G e. TarskiG )
15 5 ad3antrrr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> A e. P )
16 6 ad3antrrr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> B e. P )
17 7 ad3antrrr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> C e. P )
18 12 ad3antrrr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> X e. P )
19 9 ad3antrrr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> E e. P )
20 10 ad3antrrr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> F e. P )
21 simpllr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> x e. P )
22 simplr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> y e. P )
23 simpr1 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> <" A B C "> ( cgrG ` G ) <" x E y "> )
24 8 ad3antrrr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> D e. P )
25 simpr2 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> x ( K ` E ) D )
26 13 ad3antrrr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> X ( K ` E ) D )
27 1 2 3 18 24 19 14 26 hlcomd 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> D ( K ` E ) X )
28 1 2 3 21 24 18 14 19 25 27 hltr 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> x ( K ` E ) X )
29 simpr3 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> y ( K ` E ) F )
30 1 2 3 14 15 16 17 18 19 20 21 22 23 28 29 iscgrad 
 |-  ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> <" A B C "> ( cgrA ` G ) <" X E F "> )
31 1 2 3 4 5 6 7 8 9 10 iscgra 
 |-  ( ph -> ( <" A B C "> ( cgrA ` G ) <" D E F "> <-> E. x e. P E. y e. P ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) )
32 11 31 mpbid 
 |-  ( ph -> E. x e. P E. y e. P ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) )
33 30 32 r19.29vva 
 |-  ( ph -> <" A B C "> ( cgrA ` G ) <" X E F "> )