Metamath Proof Explorer

Theorem colinearperm1

Description: Permutation law for colinearity. Part of theorem 4.11 of Schwabhauser p. 36. (Contributed by Scott Fenton, 5-Oct-2013)

Ref Expression
Assertion colinearperm1 
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. <-> A Colinear <. C , B >. ) )

Proof

Step Hyp Ref Expression
1 btwncom 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Btwn <. B , C >. <-> A Btwn <. C , B >. ) )
2 3anrot 
 |-  ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) <-> ( B e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) )
3 btwncom 
 |-  ( ( N e. NN /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. <-> B Btwn <. A , C >. ) )
4 2 3 sylan2b 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. <-> B Btwn <. A , C >. ) )
5 3anrot 
 |-  ( ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) <-> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) )
6 btwncom 
 |-  ( ( N e. NN /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. <-> C Btwn <. B , A >. ) )
7 5 6 sylan2br 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. <-> C Btwn <. B , A >. ) )
8 1 4 7 3orbi123d 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) <-> ( A Btwn <. C , B >. \/ B Btwn <. A , C >. \/ C Btwn <. B , A >. ) ) )
9 3orcomb 
 |-  ( ( A Btwn <. C , B >. \/ B Btwn <. A , C >. \/ C Btwn <. B , A >. ) <-> ( A Btwn <. C , B >. \/ C Btwn <. B , A >. \/ B Btwn <. A , C >. ) )
10 8 9 syl6bb 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) <-> ( A Btwn <. C , B >. \/ C Btwn <. B , A >. \/ B Btwn <. A , C >. ) ) )
11 brcolinear 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. <-> ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) ) )
12 3ancomb 
 |-  ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) <-> ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
13 brcolinear 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A Colinear <. C , B >. <-> ( A Btwn <. C , B >. \/ C Btwn <. B , A >. \/ B Btwn <. A , C >. ) ) )
14 12 13 sylan2b 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. C , B >. <-> ( A Btwn <. C , B >. \/ C Btwn <. B , A >. \/ B Btwn <. A , C >. ) ) )
15 10 11 14 3bitr4d 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. <-> A Colinear <. C , B >. ) )