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Theorem btwnconn1lem8 29949
Description: Lemma for btwnconn1 29956. Now, we introduce the last three points used in the construction:  P,  Q, and  R will turn out to be equal further down, and will provide us with the key to the final statement. We begin by establishing congruence of  R P and  E d. (Contributed by Scott Fenton, 8-Oct-2013.)
Assertion
Ref Expression
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 >.
)

Proof of Theorem btwnconn1lem8
StepHypRef Expression
1 simpr2l 1055 . . . 4  |-  ( ( ( 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 >. )
21ad2antll 728 . . 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 )  /\  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 >. )
3 simpr1r 1054 . . . . . 6  |-  ( ( ( 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
>. )
43ad2antll 728 . . . . 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 )  /\  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 >.
)
5 simp11 1026 . . . . . . . 8  |-  ( ( ( 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 )
6 simp2l1 1095 . . . . . . . 8  |-  ( ( ( 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
) )
7 simp31 1032 . . . . . . . 8  |-  ( ( ( 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 simp2r1 1098 . . . . . . . 8  |-  ( ( ( 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
) )
9 cgrcomlr 29853 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  ( <. C ,  P >.Cgr <. C ,  d
>. 
<-> 
<. P ,  C >.Cgr <.
d ,  C >. ) )
105, 6, 7, 6, 8, 9syl122anc 1237 . . . . . . 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
)  /\  E  e.  ( EE `  N ) ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) ) )  ->  ( <. C ,  P >.Cgr <. C ,  d >.  <->  <. P ,  C >.Cgr <. d ,  C >. ) )
11 cgrcom 29845 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. P ,  C >.Cgr <.
d ,  C >.  <->  <. d ,  C >.Cgr <. P ,  C >. ) )
125, 7, 6, 8, 6, 11syl122anc 1237 . . . . . . 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
)  /\  E  e.  ( EE `  N ) ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) ) )  ->  ( <. P ,  C >.Cgr <.
d ,  C >.  <->  <. d ,  C >.Cgr <. P ,  C >. ) )
1310, 12bitrd 253 . . . . . 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
)  /\  E  e.  ( EE `  N ) ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) ) )  ->  ( <. C ,  P >.Cgr <. C ,  d >.  <->  <. d ,  C >.Cgr <. P ,  C >. ) )
1413adantr 465 . . . . 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 )  /\  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
>. 
<-> 
<. d ,  C >.Cgr <. P ,  C >. ) )
154, 14mpbid 210 . . . 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 )  /\  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
<. P ,  C >. )
16 simp33 1034 . . . . 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
)  /\  E  e.  ( EE `  N ) ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) ) )  ->  R  e.  ( EE `  N
) )
17 simp2r3 1100 . . . . 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
)  /\  E  e.  ( EE `  N ) ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) ) )  ->  E  e.  ( EE `  N
) )
18 simp2l3 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
)  /\  E  e.  ( EE `  N ) ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) ) )  ->  c  e.  ( EE `  N
) )
19 simpr1l 1053 . . . . . . . 8  |-  ( ( ( 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 >. )
2019ad2antll 728 . . . . . . 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 )  /\  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 >. )
215, 6, 18, 7, 20btwncomand 29870 . . . . . 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 )  /\  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  <. P , 
c >. )
22 simprll 763 . . . . . . 7  |-  ( ( ( ( ( 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
>. )
2322adantl 466 . . . . . 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 )  /\  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  Btwn  <. C , 
c >. )
24 btwnintr 29874 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) ) )  ->  ( ( C 
Btwn  <. P ,  c
>.  /\  E  Btwn  <. C , 
c >. )  ->  C  Btwn  <. P ,  E >. ) )
255, 7, 6, 17, 18, 24syl122anc 1237 . . . . . . 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
)  /\  E  e.  ( EE `  N ) ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) ) )  ->  (
( C  Btwn  <. P , 
c >.  /\  E  Btwn  <. C ,  c >. )  ->  C  Btwn  <. P ,  E >. ) )
2625adantr 465 . . . . . 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 )  /\  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  <. P ,  c
>.  /\  E  Btwn  <. C , 
c >. )  ->  C  Btwn  <. P ,  E >. ) )
2721, 23, 26mp2and 679 . . . . 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 )  /\  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  <. P ,  E >. )
28 simpr2r 1056 . . . . . 6  |-  ( ( ( 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 >. )
2928ad2antll 728 . . . . 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 )  /\  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 >. )
305, 8, 6, 16, 7, 6, 17, 2, 27, 15, 29cgrextendand 29864 . . . 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 )  /\  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 ,  R >.Cgr
<. P ,  E >. )
31 brcgr3 29901 . . . . . 6  |-  ( ( N  e.  NN  /\  ( d  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  e.  ( EE `  N )  /\  C  e.  ( EE `  N
)  /\  E  e.  ( EE `  N ) ) )  ->  ( <. d ,  <. C ,  R >. >.Cgr3 <. P ,  <. C ,  E >. >.  <->  ( <. d ,  C >.Cgr <. P ,  C >.  /\  <. d ,  R >.Cgr <. P ,  E >.  /\  <. C ,  R >.Cgr
<. C ,  E >. ) ) )
325, 8, 6, 16, 7, 6, 17, 31syl133anc 1251 . . . . 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
)  /\  E  e.  ( EE `  N ) ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) ) )  ->  ( <. d ,  <. C ,  R >. >.Cgr3 <. P ,  <. C ,  E >. >.  <->  ( <. d ,  C >.Cgr <. P ,  C >.  /\  <. d ,  R >.Cgr <. P ,  E >.  /\  <. C ,  R >.Cgr
<. C ,  E >. ) ) )
3332adantr 465 . . . 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 )  /\  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 ,  R >. >.Cgr3 <. P ,  <. C ,  E >. >.  <->  ( <. d ,  C >.Cgr <. P ,  C >.  /\  <. d ,  R >.Cgr <. P ,  E >.  /\  <. C ,  R >.Cgr
<. C ,  E >. ) ) )
3415, 30, 29, 33mpbir3and 1179 . . 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 )  /\  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 ,  R >. >.Cgr3 <. P ,  <. C ,  E >. >.
)
355, 8, 7cgrrflx2d 29839 . . . . 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
)  /\  E  e.  ( EE `  N ) ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) ) )  ->  <. d ,  P >.Cgr <. P ,  d
>. )
3635adantr 465 . . . 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 )  /\  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
<. P ,  d >.
)
3736, 4jca 532 . . 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 )  /\  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 <. P ,  d
>.  /\  <. C ,  P >.Cgr
<. C ,  d >.
) )
382, 34, 373jca 1176 . 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 )  /\  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 >.  /\ 
<. d ,  <. C ,  R >. >.Cgr3 <. P ,  <. C ,  E >. >.  /\  ( <. d ,  P >.Cgr <. P ,  d >.  /\ 
<. C ,  P >.Cgr <. C ,  d >. ) ) )
39 simp1 996 . . . . 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
)  /\  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 ) ) )
40 simp2l 1022 . . . . 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
)  /\  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 ) ) )
41 simp2r 1023 . . . . 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
)  /\  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
) ) )
4239, 40, 413jca 1176 . . . 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
)  /\  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
) ) ) )
43 simpl 457 . . . . 5  |-  ( ( ( ( ( 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 >. ) ) ) )
44 simprl 756 . . . . 5  |-  ( ( ( ( ( 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 >. ) )
4543, 44jca 532 . . . 4  |-  ( ( ( ( ( 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 >. ) ) )
46 btwnconn1lem7 29948 . . . 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
)  /\  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 )
4742, 45, 46syl2an 477 . . 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 )  /\  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
)
4847necomd 2728 . 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 )  /\  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
)
49 brofs2 29932 . . . . . 6  |-  ( ( ( N  e.  NN  /\  d  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( R  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  d  e.  ( EE `  N ) ) )  ->  ( <. <. d ,  C >. ,  <. R ,  P >. >. 
OuterFiveSeg  <. <. P ,  C >. ,  <. E ,  d
>. >. 
<->  ( C  Btwn  <. d ,  R >.  /\  <. d ,  <. C ,  R >. >.Cgr3 <. P ,  <. C ,  E >. >.  /\  ( <. d ,  P >.Cgr <. P ,  d >.  /\ 
<. C ,  P >.Cgr <. C ,  d >. ) ) ) )
5049anbi1d 704 . . . . 5  |-  ( ( ( N  e.  NN  /\  d  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( R  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  d  e.  ( EE `  N ) ) )  ->  (
( <. <. d ,  C >. ,  <. R ,  P >. >. 
OuterFiveSeg  <. <. P ,  C >. ,  <. E ,  d
>. >.  /\  d  =/=  C )  <->  ( ( C 
Btwn  <. d ,  R >.  /\  <. d ,  <. C ,  R >. >.Cgr3 <. P ,  <. C ,  E >. >.  /\  ( <. d ,  P >.Cgr
<. P ,  d >.  /\  <. C ,  P >.Cgr
<. C ,  d >.
) )  /\  d  =/=  C ) ) )
51 5segofs 29861 . . . . 5  |-  ( ( ( N  e.  NN  /\  d  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( R  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  d  e.  ( EE `  N ) ) )  ->  (
( <. <. d ,  C >. ,  <. R ,  P >. >. 
OuterFiveSeg  <. <. P ,  C >. ,  <. E ,  d
>. >.  /\  d  =/=  C )  ->  <. R ,  P >.Cgr <. E ,  d
>. ) )
5250, 51sylbird 235 . . . 4  |-  ( ( ( N  e.  NN  /\  d  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( R  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  d  e.  ( EE `  N ) ) )  ->  (
( ( C  Btwn  <.
d ,  R >.  /\ 
<. d ,  <. C ,  R >. >.Cgr3 <. P ,  <. C ,  E >. >.  /\  ( <. d ,  P >.Cgr <. P ,  d >.  /\ 
<. C ,  P >.Cgr <. C ,  d >. ) )  /\  d  =/= 
C )  ->  <. R ,  P >.Cgr <. E ,  d
>. ) )
535, 8, 6, 16, 7, 7, 6, 17, 8, 52syl333anc 1260 . . 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
)  /\  E  e.  ( EE `  N ) ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) ) )  ->  (
( ( C  Btwn  <.
d ,  R >.  /\ 
<. d ,  <. C ,  R >. >.Cgr3 <. P ,  <. C ,  E >. >.  /\  ( <. d ,  P >.Cgr <. P ,  d >.  /\ 
<. C ,  P >.Cgr <. C ,  d >. ) )  /\  d  =/= 
C )  ->  <. R ,  P >.Cgr <. E ,  d
>. ) )
5453adantr 465 . 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 )  /\  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 >.  /\  <. d ,  <. C ,  R >. >.Cgr3 <. P ,  <. C ,  E >. >.  /\  ( <. d ,  P >.Cgr <. P ,  d >.  /\ 
<. C ,  P >.Cgr <. C ,  d >. ) )  /\  d  =/= 
C )  ->  <. R ,  P >.Cgr <. E ,  d
>. ) )
5538, 48, 54mp2and 679 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 )  /\  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 >.
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1819    =/= wne 2652   <.cop 4038   class class class wbr 4456   ` cfv 5594   NNcn 10556   EEcee 24318    Btwn cbtwn 24319  Cgrccgr 24320    OuterFiveSeg cofs 29837  Cgr3ccgr3 29891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-ee 24321  df-btwn 24322  df-cgr 24323  df-ofs 29838  df-ifs 29895  df-cgr3 29896
This theorem is referenced by:  btwnconn1lem9  29950  btwnconn1lem10  29951  btwnconn1lem11  29952
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