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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> N e. NN ) |
2 |
|
simp2l1 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> C e. ( EE ` N ) ) |
3 |
|
simp2l3 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> c e. ( EE ` N ) ) |
4 |
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> P e. ( EE ` N ) ) |
5 |
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> Q e. ( EE ` N ) ) |
6 |
|
simp1l3 |
|- ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) -> C =/= c ) |
7 |
6
|
ad2antrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> C =/= c ) |
8 |
|
simpr1l |
|- ( ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) -> C Btwn <. c , P >. ) |
9 |
8
|
ad2antll |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> C Btwn <. c , P >. ) |
10 |
|
btwncolinear5 |
|- ( ( N e. NN /\ ( c e. ( EE ` N ) /\ P e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. c , P >. -> C Colinear <. c , P >. ) ) |
11 |
1 3 4 2 10
|
syl13anc |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( C Btwn <. c , P >. -> C Colinear <. c , P >. ) ) |
12 |
11
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( C Btwn <. c , P >. -> C Colinear <. c , P >. ) ) |
13 |
9 12
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> C Colinear <. c , P >. ) |
14 |
|
simp2l2 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> D e. ( EE ` N ) ) |
15 |
|
simp2r1 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> d e. ( EE ` N ) ) |
16 |
|
simpr1r |
|- ( ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) -> <. C , P >. Cgr <. C , d >. ) |
17 |
16
|
ad2antll |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. C , P >. Cgr <. C , d >. ) |
18 |
|
simp2rr |
|- ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) -> <. C , d >. Cgr <. C , D >. ) |
19 |
18
|
ad2antrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. C , d >. Cgr <. C , D >. ) |
20 |
1 2 4 2 15 2 14 17 19
|
cgrtrand |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. C , P >. Cgr <. C , D >. ) |
21 |
|
btwnconn1lem11 |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. D , C >. Cgr <. Q , C >. ) |
22 |
1 14 2 5 2 21
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. C , D >. Cgr <. C , Q >. ) |
23 |
1 2 4 2 14 2 5 20 22
|
cgrtrand |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. C , P >. Cgr <. C , Q >. ) |
24 |
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> A e. ( EE ` N ) ) |
25 |
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> B e. ( EE ` N ) ) |
26 |
|
simp2r2 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> b e. ( EE ` N ) ) |
27 |
|
simp1rr |
|- ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) -> B Btwn <. A , D >. ) |
28 |
27
|
ad2antrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> B Btwn <. A , D >. ) |
29 |
|
simp2ll |
|- ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) -> D Btwn <. A , c >. ) |
30 |
29
|
ad2antrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> D Btwn <. A , c >. ) |
31 |
1 24 25 14 3 28 30
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> B Btwn <. A , c >. ) |
32 |
|
simp3ll |
|- ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) -> c Btwn <. A , b >. ) |
33 |
32
|
ad2antrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> c Btwn <. A , b >. ) |
34 |
1 24 25 3 26 31 33
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> c Btwn <. B , b >. ) |
35 |
|
opeq1 |
|- ( B = b -> <. B , b >. = <. b , b >. ) |
36 |
35
|
breq2d |
|- ( B = b -> ( c Btwn <. B , b >. <-> c Btwn <. b , b >. ) ) |
37 |
36
|
biimpac |
|- ( ( c Btwn <. B , b >. /\ B = b ) -> c Btwn <. b , b >. ) |
38 |
|
axbtwnid |
|- ( ( N e. NN /\ c e. ( EE ` N ) /\ b e. ( EE ` N ) ) -> ( c Btwn <. b , b >. -> c = b ) ) |
39 |
1 3 26 38
|
syl3anc |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( c Btwn <. b , b >. -> c = b ) ) |
40 |
37 39
|
syl5 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( ( c Btwn <. B , b >. /\ B = b ) -> c = b ) ) |
41 |
40
|
expd |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( c Btwn <. B , b >. -> ( B = b -> c = b ) ) ) |
42 |
41
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( c Btwn <. B , b >. -> ( B = b -> c = b ) ) ) |
43 |
34 42
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( B = b -> c = b ) ) |
44 |
|
simp1 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) |
45 |
|
simp2l |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) ) |
46 |
|
simp2r |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) |
47 |
44 45 46
|
3jca |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( ( 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 ) /\ E e. ( EE ` N ) ) ) ) |
48 |
|
simpl |
|- ( ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) -> ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) ) |
49 |
|
simprl |
|- ( ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) -> ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) |
50 |
48 49
|
jca |
|- ( ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) -> ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) ) |
51 |
|
btwnconn1lem7 |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) ) -> C =/= d ) |
52 |
47 50 51
|
syl2an |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> C =/= d ) |
53 |
|
simp2rl |
|- ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) -> C Btwn <. A , d >. ) |
54 |
53
|
ad2antrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> C Btwn <. A , d >. ) |
55 |
|
simp3rl |
|- ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) -> d Btwn <. A , b >. ) |
56 |
55
|
ad2antrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> d Btwn <. A , b >. ) |
57 |
1 24 2 15 26 54 56
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> d Btwn <. C , b >. ) |
58 |
|
btwncolinear2 |
|- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ b e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> ( d Btwn <. C , b >. -> C Colinear <. d , b >. ) ) |
59 |
1 2 26 15 58
|
syl13anc |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( d Btwn <. C , b >. -> C Colinear <. d , b >. ) ) |
60 |
59
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( d Btwn <. C , b >. -> C Colinear <. d , b >. ) ) |
61 |
57 60
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> C Colinear <. d , b >. ) |
62 |
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> R e. ( EE ` N ) ) |
63 |
|
simpr2r |
|- ( ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) -> <. C , R >. Cgr <. C , E >. ) |
64 |
63
|
ad2antll |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. C , R >. Cgr <. C , E >. ) |
65 |
|
btwnconn1lem5 |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) ) -> <. E , C >. Cgr <. E , c >. ) |
66 |
47 50 65
|
syl2an |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. E , C >. Cgr <. E , c >. ) |
67 |
|
opeq2 |
|- ( R = C -> <. C , R >. = <. C , C >. ) |
68 |
67
|
breq1d |
|- ( R = C -> ( <. C , R >. Cgr <. C , E >. <-> <. C , C >. Cgr <. C , E >. ) ) |
69 |
68
|
anbi1d |
|- ( R = C -> ( ( <. C , R >. Cgr <. C , E >. /\ <. E , C >. Cgr <. E , c >. ) <-> ( <. C , C >. Cgr <. C , E >. /\ <. E , C >. Cgr <. E , c >. ) ) ) |
70 |
69
|
biimpac |
|- ( ( ( <. C , R >. Cgr <. C , E >. /\ <. E , C >. Cgr <. E , c >. ) /\ R = C ) -> ( <. C , C >. Cgr <. C , E >. /\ <. E , C >. Cgr <. E , c >. ) ) |
71 |
|
simp2r3 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> E e. ( EE ` N ) ) |
72 |
|
cgrid2 |
|- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ C e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( <. C , C >. Cgr <. C , E >. -> C = E ) ) |
73 |
1 2 2 71 72
|
syl13anc |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( <. C , C >. Cgr <. C , E >. -> C = E ) ) |
74 |
|
opeq1 |
|- ( C = E -> <. C , C >. = <. E , C >. ) |
75 |
|
opeq1 |
|- ( C = E -> <. C , c >. = <. E , c >. ) |
76 |
74 75
|
breq12d |
|- ( C = E -> ( <. C , C >. Cgr <. C , c >. <-> <. E , C >. Cgr <. E , c >. ) ) |
77 |
76
|
biimpar |
|- ( ( C = E /\ <. E , C >. Cgr <. E , c >. ) -> <. C , C >. Cgr <. C , c >. ) |
78 |
|
cgrid2 |
|- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ C e. ( EE ` N ) /\ c e. ( EE ` N ) ) ) -> ( <. C , C >. Cgr <. C , c >. -> C = c ) ) |
79 |
1 2 2 3 78
|
syl13anc |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( <. C , C >. Cgr <. C , c >. -> C = c ) ) |
80 |
77 79
|
syl5 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( ( C = E /\ <. E , C >. Cgr <. E , c >. ) -> C = c ) ) |
81 |
73 80
|
syland |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( ( <. C , C >. Cgr <. C , E >. /\ <. E , C >. Cgr <. E , c >. ) -> C = c ) ) |
82 |
70 81
|
syl5 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( ( ( <. C , R >. Cgr <. C , E >. /\ <. E , C >. Cgr <. E , c >. ) /\ R = C ) -> C = c ) ) |
83 |
82
|
expd |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( ( <. C , R >. Cgr <. C , E >. /\ <. E , C >. Cgr <. E , c >. ) -> ( R = C -> C = c ) ) ) |
84 |
83
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( ( <. C , R >. Cgr <. C , E >. /\ <. E , C >. Cgr <. E , c >. ) -> ( R = C -> C = c ) ) ) |
85 |
64 66 84
|
mp2and |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( R = C -> C = c ) ) |
86 |
85
|
necon3d |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( C =/= c -> R =/= C ) ) |
87 |
7 86
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> R =/= C ) |
88 |
|
simpr2l |
|- ( ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) -> C Btwn <. d , R >. ) |
89 |
88
|
ad2antll |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> C Btwn <. d , R >. ) |
90 |
|
btwncolinear4 |
|- ( ( N e. NN /\ ( d e. ( EE ` N ) /\ R e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. d , R >. -> R Colinear <. C , d >. ) ) |
91 |
1 15 62 2 90
|
syl13anc |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( C Btwn <. d , R >. -> R Colinear <. C , d >. ) ) |
92 |
91
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( C Btwn <. d , R >. -> R Colinear <. C , d >. ) ) |
93 |
89 92
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> R Colinear <. C , d >. ) |
94 |
|
simpr3r |
|- ( ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) -> <. R , Q >. Cgr <. R , P >. ) |
95 |
94
|
ad2antll |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. R , Q >. Cgr <. R , P >. ) |
96 |
1 62 5 62 4 95
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. R , P >. Cgr <. R , Q >. ) |
97 |
1 62 2 15 4 5 87 93 96 23
|
linecgrand |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. d , P >. Cgr <. d , Q >. ) |
98 |
1 2 15 26 4 5 52 61 23 97
|
linecgrand |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. b , P >. Cgr <. b , Q >. ) |
99 |
|
opeq1 |
|- ( c = b -> <. c , P >. = <. b , P >. ) |
100 |
|
opeq1 |
|- ( c = b -> <. c , Q >. = <. b , Q >. ) |
101 |
99 100
|
breq12d |
|- ( c = b -> ( <. c , P >. Cgr <. c , Q >. <-> <. b , P >. Cgr <. b , Q >. ) ) |
102 |
101
|
biimprd |
|- ( c = b -> ( <. b , P >. Cgr <. b , Q >. -> <. c , P >. Cgr <. c , Q >. ) ) |
103 |
43 98 102
|
syl6ci |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( B = b -> <. c , P >. Cgr <. c , Q >. ) ) |
104 |
103
|
com12 |
|- ( B = b -> ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. c , P >. Cgr <. c , Q >. ) ) |
105 |
|
simprl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( B =/= b /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) ) -> B =/= b ) |
106 |
34
|
adantrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( B =/= b /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) ) -> c Btwn <. B , b >. ) |
107 |
|
btwncolinear1 |
|- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ b e. ( EE ` N ) /\ c e. ( EE ` N ) ) ) -> ( c Btwn <. B , b >. -> B Colinear <. b , c >. ) ) |
108 |
1 25 26 3 107
|
syl13anc |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( c Btwn <. B , b >. -> B Colinear <. b , c >. ) ) |
109 |
108
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( B =/= b /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) ) -> ( c Btwn <. B , b >. -> B Colinear <. b , c >. ) ) |
110 |
106 109
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( B =/= b /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) ) -> B Colinear <. b , c >. ) |
111 |
|
simp1rl |
|- ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) -> B Btwn <. A , C >. ) |
112 |
111
|
ad2antrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> B Btwn <. A , C >. ) |
113 |
1 24 25 2 15 112 54
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> C Btwn <. B , d >. ) |
114 |
|
btwncolinear6 |
|- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ d e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. B , d >. -> C Colinear <. d , B >. ) ) |
115 |
1 25 15 2 114
|
syl13anc |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( C Btwn <. B , d >. -> C Colinear <. d , B >. ) ) |
116 |
115
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( C Btwn <. B , d >. -> C Colinear <. d , B >. ) ) |
117 |
113 116
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> C Colinear <. d , B >. ) |
118 |
1 2 15 25 4 5 52 117 23 97
|
linecgrand |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. B , P >. Cgr <. B , Q >. ) |
119 |
118
|
adantrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( B =/= b /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) ) -> <. B , P >. Cgr <. B , Q >. ) |
120 |
98
|
adantrl |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( B =/= b /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) ) -> <. b , P >. Cgr <. b , Q >. ) |
121 |
1 25 26 3 4 5 105 110 119 120
|
linecgrand |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( B =/= b /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) ) -> <. c , P >. Cgr <. c , Q >. ) |
122 |
121
|
an12s |
|- ( ( B =/= b /\ ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) ) -> <. c , P >. Cgr <. c , Q >. ) |
123 |
122
|
ex |
|- ( B =/= b -> ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. c , P >. Cgr <. c , Q >. ) ) |
124 |
104 123
|
pm2.61ine |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. c , P >. Cgr <. c , Q >. ) |
125 |
1 2 3 4 4 5 7 13 23 124
|
linecgrand |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. P , P >. Cgr <. P , Q >. ) |
126 |
|
cgrid2 |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ P e. ( EE ` N ) /\ Q e. ( EE ` N ) ) ) -> ( <. P , P >. Cgr <. P , Q >. -> P = Q ) ) |
127 |
1 4 4 5 126
|
syl13anc |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( <. P , P >. Cgr <. P , Q >. -> P = Q ) ) |
128 |
127
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( <. P , P >. Cgr <. P , Q >. -> P = Q ) ) |
129 |
125 128
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> P = Q ) |
130 |
|
btwnconn1lem10 |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> <. d , D >. Cgr <. P , Q >. ) |
131 |
|
opeq1 |
|- ( P = Q -> <. P , Q >. = <. Q , Q >. ) |
132 |
131
|
breq2d |
|- ( P = Q -> ( <. d , D >. Cgr <. P , Q >. <-> <. d , D >. Cgr <. Q , Q >. ) ) |
133 |
132
|
biimpa |
|- ( ( P = Q /\ <. d , D >. Cgr <. P , Q >. ) -> <. d , D >. Cgr <. Q , Q >. ) |
134 |
|
axcgrid |
|- ( ( N e. NN /\ ( d e. ( EE ` N ) /\ D e. ( EE ` N ) /\ Q e. ( EE ` N ) ) ) -> ( <. d , D >. Cgr <. Q , Q >. -> d = D ) ) |
135 |
1 15 14 5 134
|
syl13anc |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( <. d , D >. Cgr <. Q , Q >. -> d = D ) ) |
136 |
133 135
|
syl5 |
|- ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( ( P = Q /\ <. d , D >. Cgr <. P , Q >. ) -> d = D ) ) |
137 |
136
|
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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> ( ( P = Q /\ <. d , D >. Cgr <. P , Q >. ) -> d = D ) ) |
138 |
129 130 137
|
mp2and |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> d = D ) |
139 |
138
|
eqcomd |
|- ( ( ( ( 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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= 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 , b >. /\ <. d , b >. Cgr <. D , B >. ) ) ) /\ ( ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) /\ ( ( C Btwn <. c , P >. /\ <. C , P >. Cgr <. C , d >. ) /\ ( C Btwn <. d , R >. /\ <. C , R >. Cgr <. C , E >. ) /\ ( R Btwn <. P , Q >. /\ <. R , Q >. Cgr <. R , P >. ) ) ) ) ) -> D = d ) |