Metamath Proof Explorer


Theorem btwnconn1lem2

Description: Lemma for btwnconn1 . Now, we show that two of the hypotheticals we introduced in the first lemma are identical. (Contributed by Scott Fenton, 8-Oct-2013)

Ref Expression
Assertion btwnconn1lem2
|- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> X = b )

Proof

Step Hyp Ref Expression
1 simp1ll
 |-  ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) -> A =/= B )
2 1 adantl
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> A =/= B )
3 simp11
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> N e. NN )
4 simp12
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
5 simp13
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
6 simp21
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
7 simp33
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> X e. ( EE ` N ) )
8 simp1rl
 |-  ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) -> B Btwn <. A , C >. )
9 8 adantl
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> B Btwn <. A , C >. )
10 simp2rl
 |-  ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) -> C Btwn <. A , d >. )
11 simp3rl
 |-  ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) -> d Btwn <. A , X >. )
12 10 11 jca
 |-  ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) -> ( C Btwn <. A , d >. /\ d Btwn <. A , X >. ) )
13 12 adantl
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> ( C Btwn <. A , d >. /\ d Btwn <. A , X >. ) )
14 simp31
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> d e. ( EE ` N ) )
15 btwnexch
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , d >. /\ d Btwn <. A , X >. ) -> C Btwn <. A , X >. ) )
16 3 4 6 14 7 15 syl122anc
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , d >. /\ d Btwn <. A , X >. ) -> C Btwn <. A , X >. ) )
17 16 adantr
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> ( ( C Btwn <. A , d >. /\ d Btwn <. A , X >. ) -> C Btwn <. A , X >. ) )
18 13 17 mpd
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> C Btwn <. A , X >. )
19 3 4 5 6 7 9 18 btwnexchand
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> B Btwn <. A , X >. )
20 3 5 7 cgrrflx2d
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> <. B , X >. Cgr <. X , B >. )
21 20 adantr
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> <. B , X >. Cgr <. X , B >. )
22 19 21 jca
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> ( B Btwn <. A , X >. /\ <. B , X >. Cgr <. X , B >. ) )
23 simp23
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> c e. ( EE ` N ) )
24 simp32
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> b e. ( EE ` N ) )
25 simp1rr
 |-  ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) -> B Btwn <. A , D >. )
26 simp2ll
 |-  ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) -> D Btwn <. A , c >. )
27 25 26 jca
 |-  ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) -> ( B Btwn <. A , D >. /\ D Btwn <. A , c >. ) )
28 27 adantl
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> ( B Btwn <. A , D >. /\ D Btwn <. A , c >. ) )
29 simp22
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
30 btwnexch
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ c e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , D >. /\ D Btwn <. A , c >. ) -> B Btwn <. A , c >. ) )
31 3 4 5 29 23 30 syl122anc
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , D >. /\ D Btwn <. A , c >. ) -> B Btwn <. A , c >. ) )
32 31 adantr
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> ( ( B Btwn <. A , D >. /\ D Btwn <. A , c >. ) -> B Btwn <. A , c >. ) )
33 28 32 mpd
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> B Btwn <. A , c >. )
34 simp3ll
 |-  ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) -> c Btwn <. A , b >. )
35 34 adantl
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> c Btwn <. A , b >. )
36 3 4 5 23 24 33 35 btwnexchand
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> B Btwn <. A , b >. )
37 3 4 5 23 24 33 35 btwnexch3and
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> c Btwn <. B , b >. )
38 3 4 5 6 7 9 18 btwnexch3and
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> C Btwn <. B , X >. )
39 3 6 5 7 38 btwncomand
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> C Btwn <. X , B >. )
40 btwnconn1lem1
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> <. B , c >. Cgr <. X , C >. )
41 simp3lr
 |-  ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) -> <. c , b >. Cgr <. C , B >. )
42 41 adantl
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> <. c , b >. Cgr <. C , B >. )
43 3 5 23 24 7 6 5 37 39 40 42 cgrextendand
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> <. B , b >. Cgr <. X , B >. )
44 36 43 jca
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> ( B Btwn <. A , b >. /\ <. B , b >. Cgr <. X , B >. ) )
45 segconeq
 |-  ( ( N e. NN /\ ( B e. ( EE ` N ) /\ X e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ X e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) -> ( ( A =/= B /\ ( B Btwn <. A , X >. /\ <. B , X >. Cgr <. X , B >. ) /\ ( B Btwn <. A , b >. /\ <. B , b >. Cgr <. X , B >. ) ) -> X = b ) )
46 3 5 7 5 4 7 24 45 syl133anc
 |-  ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( ( A =/= B /\ ( B Btwn <. A , X >. /\ <. B , X >. Cgr <. X , B >. ) /\ ( B Btwn <. A , b >. /\ <. B , b >. Cgr <. X , B >. ) ) -> X = b ) )
47 46 adantr
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> ( ( A =/= B /\ ( B Btwn <. A , X >. /\ <. B , X >. Cgr <. X , B >. ) /\ ( B Btwn <. A , b >. /\ <. B , b >. Cgr <. X , B >. ) ) -> X = b ) )
48 2 22 44 47 mp3and
 |-  ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >. Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >. Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >. Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >. Cgr <. D , B >. ) ) ) ) -> X = b )