Step |
Hyp |
Ref |
Expression |
1 |
|
colperpex.p |
|- P = ( Base ` G ) |
2 |
|
colperpex.d |
|- .- = ( dist ` G ) |
3 |
|
colperpex.i |
|- I = ( Itv ` G ) |
4 |
|
colperpex.l |
|- L = ( LineG ` G ) |
5 |
|
colperpex.g |
|- ( ph -> G e. TarskiG ) |
6 |
|
perpdrag.1 |
|- ( ph -> A e. D ) |
7 |
|
perpdrag.2 |
|- ( ph -> B e. D ) |
8 |
|
perpdrag.3 |
|- ( ph -> C e. P ) |
9 |
|
perpdrag.4 |
|- ( ph -> D ( perpG ` G ) ( B L C ) ) |
10 |
5
|
ad2antrr |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> G e. TarskiG ) |
11 |
9
|
ad2antrr |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> D ( perpG ` G ) ( B L C ) ) |
12 |
4 10 11
|
perpln1 |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> D e. ran L ) |
13 |
6
|
ad2antrr |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> A e. D ) |
14 |
1 4 3 10 12 13
|
tglnpt |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> A e. P ) |
15 |
|
simplr |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> x e. D ) |
16 |
1 4 3 10 12 15
|
tglnpt |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> x e. P ) |
17 |
7
|
ad2antrr |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> B e. D ) |
18 |
|
simpr |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> A =/= x ) |
19 |
1 3 4 10 14 16 18 18 12 13 15
|
tglinethru |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> D = ( A L x ) ) |
20 |
17 19
|
eleqtrd |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> B e. ( A L x ) ) |
21 |
8
|
ad2antrr |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> C e. P ) |
22 |
19 11
|
eqbrtrrd |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> ( A L x ) ( perpG ` G ) ( B L C ) ) |
23 |
1 2 3 4 10 14 16 20 21 22
|
perprag |
|- ( ( ( ph /\ x e. D ) /\ A =/= x ) -> <" A B C "> e. ( raG ` G ) ) |
24 |
4 5 9
|
perpln1 |
|- ( ph -> D e. ran L ) |
25 |
1 3 4 5 24 6
|
tglnpt2 |
|- ( ph -> E. x e. D A =/= x ) |
26 |
23 25
|
r19.29a |
|- ( ph -> <" A B C "> e. ( raG ` G ) ) |