Theorem btwnconn1lem2 32195

Description: Lemma for btwnconn1 32208. Now, we show that two of the hypotheticals we introduced in the first lemma are identical. (Contributed by Scott Fenton, 8-Oct-2013.)

Assertion

Ref

Expression

btwnconn1lem2

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

Proof of Theorem btwnconn1lem2

Step	Hyp	Ref	Expression
1		simp1ll 1124	. . 3 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) -> A =/= B )
2	1	adantl 482	. 2 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> A =/= B )
3		simp11 1091	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> N e. NN )
4		simp12 1092	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
5		simp13 1093	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
6		simp21 1094	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
7		simp33 1099	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> X e. ( EE ` N ) )
8		simp1rl 1126	. . . . 5 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) -> B Btwn <. A , C >. )
9	8	adantl 482	. . . 4 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> B Btwn <. A , C >. )
10		simp2rl 1130	. . . . . . 7 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) -> C Btwn <. A , d >. )
11		simp3rl 1134	. . . . . . 7 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) -> d Btwn <. A , X >. )
12	10, 11	jca 554	. . . . . 6 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) -> ( C Btwn <. A , d >. /\ d Btwn <. A , X >. ) )
13	12	adantl 482	. . . . 5 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> ( C Btwn <. A , d >. /\ d Btwn <. A , X >. ) )
14		simp31 1097	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> d e. ( EE ` N ) )
15		btwnexch 32132	. . . . . . 7 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , d >. /\ d Btwn <. A , X >. ) -> C Btwn <. A , X >. ) )
16	3, 4, 6, 14, 7, 15	syl122anc 1335	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , d >. /\ d Btwn <. A , X >. ) -> C Btwn <. A , X >. ) )
17	16	adantr 481	. . . . 5 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> ( ( C Btwn <. A , d >. /\ d Btwn <. A , X >. ) -> C Btwn <. A , X >. ) )
18	13, 17	mpd 15	. . . 4 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> C Btwn <. A , X >. )
19	3, 4, 5, 6, 7, 9, 18	btwnexchand 32133	. . 3 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> B Btwn <. A , X >. )
20	3, 5, 7	cgrrflx2d 32091	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> <. B , X >.Cgr <. X , B >. )
21	20	adantr 481	. . 3 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> <. B , X >.Cgr <. X , B >. )
22	19, 21	jca 554	. 2 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> ( B Btwn <. A , X >. /\ <. B , X >.Cgr <. X , B >. ) )
23		simp23 1096	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> c e. ( EE ` N ) )
24		simp32 1098	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> b e. ( EE ` N ) )
25		simp1rr 1127	. . . . . . 7 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) -> B Btwn <. A , D >. )
26		simp2ll 1128	. . . . . . 7 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) -> D Btwn <. A , c >. )
27	25, 26	jca 554	. . . . . 6 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) -> ( B Btwn <. A , D >. /\ D Btwn <. A , c >. ) )
28	27	adantl 482	. . . . 5 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> ( B Btwn <. A , D >. /\ D Btwn <. A , c >. ) )
29		simp22 1095	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
30		btwnexch 32132	. . . . . . 7 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ c e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , D >. /\ D Btwn <. A , c >. ) -> B Btwn <. A , c >. ) )
31	3, 4, 5, 29, 23, 30	syl122anc 1335	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , D >. /\ D Btwn <. A , c >. ) -> B Btwn <. A , c >. ) )
32	31	adantr 481	. . . . 5 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> ( ( B Btwn <. A , D >. /\ D Btwn <. A , c >. ) -> B Btwn <. A , c >. ) )
33	28, 32	mpd 15	. . . 4 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> B Btwn <. A , c >. )
34		simp3ll 1132	. . . . 5 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) -> c Btwn <. A , b >. )
35	34	adantl 482	. . . 4 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> c Btwn <. A , b >. )
36	3, 4, 5, 23, 24, 33, 35	btwnexchand 32133	. . 3 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> B Btwn <. A , b >. )
37	3, 4, 5, 23, 24, 33, 35	btwnexch3and 32128	. . . 4 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> c Btwn <. B , b >. )
38	3, 4, 5, 6, 7, 9, 18	btwnexch3and 32128	. . . . 5 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> C Btwn <. B , X >. )
39	3, 6, 5, 7, 38	btwncomand 32122	. . . 4 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> C Btwn <. X , B >. )
40		btwnconn1lem1 32194	. . . 4 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> <. B , c >.Cgr <. X , C >. )
41		simp3lr 1133	. . . . 5 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) -> <. c , b >.Cgr <. C , B >. )
42	41	adantl 482	. . . 4 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> <. c , b >.Cgr <. C , B >. )
43	3, 5, 23, 24, 7, 6, 5, 37, 39, 40, 42	cgrextendand 32116	. . 3 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> <. B , b >.Cgr <. X , B >. )
44	36, 43	jca 554	. 2 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> ( B Btwn <. A , b >. /\ <. B , b >.Cgr <. X , B >. ) )
45		segconeq 32117	. . . 4 |- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ X e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ X e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) -> ( ( A =/= B /\ ( B Btwn <. A , X >. /\ <. B , X >.Cgr <. X , B >. ) /\ ( B Btwn <. A , b >. /\ <. B , b >.Cgr <. X , B >. ) ) -> X = b ) )
46	3, 5, 7, 5, 4, 7, 24, 45	syl133anc 1349	. . 3 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) -> ( ( A =/= B /\ ( B Btwn <. A , X >. /\ <. B , X >.Cgr <. X , B >. ) /\ ( B Btwn <. A , b >. /\ <. B , b >.Cgr <. X , B >. ) ) -> X = b ) )
47	46	adantr 481	. 2 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> ( ( A =/= B /\ ( B Btwn <. A , X >. /\ <. B , X >.Cgr <. X , B >. ) /\ ( B Btwn <. A , b >. /\ <. B , b >.Cgr <. X , B >. ) ) -> X = b ) )
48	2, 22, 44, 47	mp3and 1427	1 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ X e. ( EE ` N ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , X >. /\ <. d , X >.Cgr <. D , B >. ) ) ) ) -> X = b )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ee 25771  df-btwn 25772  df-cgr 25773  df-ofs 32090

This theorem is referenced by:  btwnconn1lem14  32207