Metamath Proof Explorer

Theorem btwnexch2

Description: Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 of Schwabhauser p. 30. (Contributed by Scott Fenton, 14-Jun-2013)

Ref Expression
Assertion btwnexch2 
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) -> C Btwn <. A , D >. ) )

Proof

Step Hyp Ref Expression
1 breq1 
 |-  ( B = C -> ( B Btwn <. A , D >. <-> C Btwn <. A , D >. ) )
2 1 biimpd 
 |-  ( B = C -> ( B Btwn <. A , D >. -> C Btwn <. A , D >. ) )
3 2 adantrd 
 |-  ( B = C -> ( ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) -> C Btwn <. A , D >. ) )
4 3 a1i 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( B = C -> ( ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) -> C Btwn <. A , D >. ) ) )
5 simprl 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( B =/= C /\ ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) ) ) -> B =/= C )
6 simprr 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( B =/= C /\ ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) ) ) -> ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) )
7 btwnintr 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) -> B Btwn <. A , C >. ) )
8 7 adantr 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( B =/= C /\ ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) ) ) -> ( ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) -> B Btwn <. A , C >. ) )
9 6 8 mpd 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( B =/= C /\ ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) ) ) -> B Btwn <. A , C >. )
10 simprrr 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( B =/= C /\ ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) ) ) -> C Btwn <. B , D >. )
11 btwnouttr2 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) -> C Btwn <. A , D >. ) )
12 11 adantr 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( B =/= C /\ ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) ) ) -> ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) -> C Btwn <. A , D >. ) )
13 5 9 10 12 mp3and 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( B =/= C /\ ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) ) ) -> C Btwn <. A , D >. )
14 13 exp32 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( B =/= C -> ( ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) -> C Btwn <. A , D >. ) ) )
15 4 14 pm2.61dne 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) -> C Btwn <. A , D >. ) )