Metamath Proof Explorer

Theorem btwnintr

Description: Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of Schwabhauser p. 30. (Contributed by Scott Fenton, 12-Jun-2013)

Ref Expression
Assertion 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 >. ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> N e. NN )
2 simp2l 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
3 simp2r 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
4 simp3r 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
5 simp3l 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
6 axpasch 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , D >. /\ C Btwn <. B , D >. ) -> E. x e. ( EE ` N ) ( x Btwn <. B , B >. /\ x Btwn <. C , A >. ) ) )
7 1 2 3 4 3 5 6 syl132anc 
 |-  ( ( 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 >. ) -> E. x e. ( EE ` N ) ( x Btwn <. B , B >. /\ x Btwn <. C , A >. ) ) )
8 simpl1 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> N e. NN )
9 simpr 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> x e. ( EE ` N ) )
10 simpl2r 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> B e. ( EE ` N ) )
11 axbtwnid 
 |-  ( ( N e. NN /\ x e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( x Btwn <. B , B >. -> x = B ) )
12 8 9 10 11 syl3anc 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( x Btwn <. B , B >. -> x = B ) )
13 breq1 
 |-  ( x = B -> ( x Btwn <. C , A >. <-> B Btwn <. C , A >. ) )
14 13 biimpa 
 |-  ( ( x = B /\ x Btwn <. C , A >. ) -> B Btwn <. C , A >. )
15 simpl3l 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> C e. ( EE ` N ) )
16 simpl2l 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> A e. ( EE ` N ) )
17 btwncom 
 |-  ( ( N e. NN /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. <-> B Btwn <. A , C >. ) )
18 8 10 15 16 17 syl13anc 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( B Btwn <. C , A >. <-> B Btwn <. A , C >. ) )
19 14 18 syl5ib 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( ( x = B /\ x Btwn <. C , A >. ) -> B Btwn <. A , C >. ) )
20 12 19 syland 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( ( x Btwn <. B , B >. /\ x Btwn <. C , A >. ) -> B Btwn <. A , C >. ) )
21 20 rexlimdva 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( E. x e. ( EE ` N ) ( x Btwn <. B , B >. /\ x Btwn <. C , A >. ) -> B Btwn <. A , C >. ) )
22 7 21 syld 
 |-  ( ( 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 >. ) )