Theorem btwnconn1lem9

Description: Lemma for btwnconn1 . Now, a quick use of transitivity to establish congruence on R Q and E D . (Contributed by Scott Fenton, 8-Oct-2013)

	Ref	Expression
Assertion	btwnconn1lem9	|- ( ( ( ( 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 <. E , D >. )

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 ) /\ E e. ( EE ` N ) ) ) /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> N e. NN )
2		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 ) )
3		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 ) )
4		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 ) )
5		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 ) )
6		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 ) )
7		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 ) )
8		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 >. )
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 >. ) ) ) ) ) -> <. R , Q >. Cgr <. R , P >. )
10		btwnconn1lem8	|- ( ( ( ( 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 <. E , d >. )
11	1 2 3 2 7 4 6 9 10	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 >. ) ) ) ) ) -> <. R , Q >. Cgr <. E , d >. )
12		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 ) ) )
13		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 ) ) )
14		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 ) ) )
15	12 13 14	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 ) ) ) )
16		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 >. ) ) ) )
17		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 >. ) )
18	16 17	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 >. ) ) )
19		btwnconn1lem6	|- ( ( ( ( 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 , D >. Cgr <. E , d >. )
20	15 18 19	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 , D >. Cgr <. E , d >. )
21	1 2 3 4 5 4 6 11 20	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 ) /\ 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 <. E , D >. )