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 |
|
eqid |
|- ( dist ` G ) = ( dist ` G ) |
13 |
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 ) |
14 |
|
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 ) |
15 |
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 ) |
16 |
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 ) |
17 |
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 ) |
18 |
|
eqid |
|- ( cgrG ` G ) = ( cgrG ` G ) |
19 |
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 ) |
20 |
|
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 ) |
21 |
|
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 "> ) |
22 |
1 12 2 18 13 16 17 19 14 15 20 21
|
cgr3simp1 |
|- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( A ( dist ` G ) B ) = ( x ( dist ` G ) E ) ) |
23 |
22
|
eqcomd |
|- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( x ( dist ` G ) E ) = ( A ( dist ` G ) B ) ) |
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 |
1 2 3 14 24 15 13 25
|
hlne1 |
|- ( ( ( ( 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 ) |
27 |
1 12 2 13 14 15 16 17 23 26
|
tgcgrneq |
|- ( ( ( ( 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 ) |
28 |
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 ) ) ) |
29 |
11 28
|
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 ) ) |
30 |
27 29
|
r19.29vva |
|- ( ph -> A =/= B ) |