Metamath Proof Explorer

Theorem btwnconn1lem1

Description: Lemma for btwnconn1 . The next several lemmas introduce various properties of hypothetical points that end up eliminating alternatives to connectivity. We begin by showing a congruence property of those hypothetical points. (Contributed by Scott Fenton, 8-Oct-2013)

Ref Expression
Assertion 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 >. )

Proof

Step Hyp Ref Expression
1 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 )
2 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 ) )
3 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 ) )
4 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 ) )
5 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 ) )
6 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 ) )
7 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 ) )
8 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 ) )
9 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 >. )
10 9 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 >. )
11 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 >. )
12 11 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 >. ) ) ) ) -> D Btwn <. A , c >. )
13 1 8 2 3 4 10 12 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 >. ) ) ) ) -> D Btwn <. B , c >. )
14 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 >. )
15 14 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 >. )
16 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 >. )
17 16 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 >. ) ) ) ) -> d Btwn <. A , X >. )
18 1 8 7 6 5 15 17 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 >. ) ) ) ) -> d Btwn <. C , X >. )
19 1 6 7 5 18 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 >. ) ) ) ) -> d Btwn <. X , C >. )
20 simp3rr 
 |-  ( ( ( ( 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 , X >. Cgr <. D , B >. )
21 20 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 >. ) ) ) ) -> <. d , X >. Cgr <. D , B >. )
22 1 6 5 3 2 21 cgrcomand 
 |-  ( ( ( ( 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 >. ) ) ) ) -> <. D , B >. Cgr <. d , X >. )
23 1 3 2 6 5 22 cgrcomlrand 
 |-  ( ( ( ( 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 , D >. Cgr <. X , d >. )
24 simp2lr 
 |-  ( ( ( ( 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 , c >. Cgr <. C , D >. )
25 24 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 >. ) ) ) ) -> <. D , c >. Cgr <. C , D >. )
26 simp2rr 
 |-  ( ( ( ( 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 , d >. Cgr <. C , D >. )
27 26 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 , d >. Cgr <. C , D >. )
28 1 7 6 7 3 27 cgrcomland 
 |-  ( ( ( ( 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 >. ) ) ) ) -> <. d , C >. Cgr <. C , D >. )
29 1 3 4 6 7 7 3 25 28 cgrtr3and 
 |-  ( ( ( ( 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 >. ) ) ) ) -> <. D , c >. Cgr <. d , C >. )
30 1 2 3 4 5 6 7 13 19 23 29 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 , c >. Cgr <. X , C >. )