Step |
Hyp |
Ref |
Expression |
1 |
|
hpgid.p |
|- P = ( Base ` G ) |
2 |
|
hpgid.i |
|- I = ( Itv ` G ) |
3 |
|
hpgid.l |
|- L = ( LineG ` G ) |
4 |
|
hpgid.g |
|- ( ph -> G e. TarskiG ) |
5 |
|
hpgid.d |
|- ( ph -> D e. ran L ) |
6 |
|
hpgid.a |
|- ( ph -> A e. P ) |
7 |
|
hpgid.o |
|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) } |
8 |
|
hpgcom.b |
|- ( ph -> B e. P ) |
9 |
|
hpgcom.1 |
|- ( ph -> A ( ( hpG ` G ) ` D ) B ) |
10 |
|
ancom |
|- ( ( A O c /\ B O c ) <-> ( B O c /\ A O c ) ) |
11 |
10
|
a1i |
|- ( ph -> ( ( A O c /\ B O c ) <-> ( B O c /\ A O c ) ) ) |
12 |
11
|
rexbidv |
|- ( ph -> ( E. c e. P ( A O c /\ B O c ) <-> E. c e. P ( B O c /\ A O c ) ) ) |
13 |
1 2 3 7 4 5 6 8
|
hpgbr |
|- ( ph -> ( A ( ( hpG ` G ) ` D ) B <-> E. c e. P ( A O c /\ B O c ) ) ) |
14 |
1 2 3 7 4 5 8 6
|
hpgbr |
|- ( ph -> ( B ( ( hpG ` G ) ` D ) A <-> E. c e. P ( B O c /\ A O c ) ) ) |
15 |
12 13 14
|
3bitr4d |
|- ( ph -> ( A ( ( hpG ` G ) ` D ) B <-> B ( ( hpG ` G ) ` D ) A ) ) |
16 |
9 15
|
mpbid |
|- ( ph -> B ( ( hpG ` G ) ` D ) A ) |