Metamath Proof Explorer


Theorem hpgne2

Description: Points on the open half plane cannot lie on its border. (Contributed by Thierry Arnoux, 4-Mar-2020)

Ref Expression
Hypotheses ishpg.p
|- P = ( Base ` G )
ishpg.i
|- I = ( Itv ` G )
ishpg.l
|- L = ( LineG ` G )
ishpg.o
|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
ishpg.g
|- ( ph -> G e. TarskiG )
ishpg.d
|- ( ph -> D e. ran L )
hpgbr.a
|- ( ph -> A e. P )
hpgbr.b
|- ( ph -> B e. P )
hpgne1.1
|- ( ph -> A ( ( hpG ` G ) ` D ) B )
Assertion hpgne2
|- ( ph -> -. B e. D )

Proof

Step Hyp Ref Expression
1 ishpg.p
 |-  P = ( Base ` G )
2 ishpg.i
 |-  I = ( Itv ` G )
3 ishpg.l
 |-  L = ( LineG ` G )
4 ishpg.o
 |-  O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
5 ishpg.g
 |-  ( ph -> G e. TarskiG )
6 ishpg.d
 |-  ( ph -> D e. ran L )
7 hpgbr.a
 |-  ( ph -> A e. P )
8 hpgbr.b
 |-  ( ph -> B e. P )
9 hpgne1.1
 |-  ( ph -> A ( ( hpG ` G ) ` D ) B )
10 eqid
 |-  ( dist ` G ) = ( dist ` G )
11 6 ad2antrr
 |-  ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> D e. ran L )
12 5 ad2antrr
 |-  ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> G e. TarskiG )
13 8 ad2antrr
 |-  ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> B e. P )
14 simplr
 |-  ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> c e. P )
15 simprr
 |-  ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> B O c )
16 1 10 2 4 3 11 12 13 14 15 oppne1
 |-  ( ( ( ph /\ c e. P ) /\ ( A O c /\ B O c ) ) -> -. B e. D )
17 1 2 3 4 5 6 7 8 hpgbr
 |-  ( ph -> ( A ( ( hpG ` G ) ` D ) B <-> E. c e. P ( A O c /\ B O c ) ) )
18 9 17 mpbid
 |-  ( ph -> E. c e. P ( A O c /\ B O c ) )
19 16 18 r19.29a
 |-  ( ph -> -. B e. D )