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Theorem btwnconn1lem4 25272
Description: Lemma for btwnconn1 25283. Assuming  C  =/=  c, we now attempt to force  D  =  d from here out via a series of congruences. (Contributed by Scott Fenton, 8-Oct-2013.)
Assertion
Ref Expression
btwnconn1lem4  |-  ( ( ( ( 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
) ) )  /\  ( ( ( 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 ,  c
>.Cgr <. D ,  C >. )

Proof of Theorem btwnconn1lem4
StepHypRef Expression
1 simp1rl 1020 . . . . . 6  |-  ( ( ( ( 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 >. )
2 simp2rl 1024 . . . . . 6  |-  ( ( ( ( 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
>. )
31, 2jca 518 . . . . 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 >. ) ) )  -> 
( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  d >. ) )
43adantl 452 . . . 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
) ) )  /\  ( ( ( 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 >.  /\  C  Btwn  <. A , 
d >. ) )
5 simp11 985 . . . . . 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 ) ) )  ->  N  e.  NN )
6 simp12 986 . . . . . 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 ) ) )  ->  A  e.  ( EE `  N ) )
7 simp13 987 . . . . . 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 ) ) )  ->  B  e.  ( EE `  N ) )
8 simp21 988 . . . . . 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 ) ) )  ->  C  e.  ( EE `  N ) )
9 simp3l 983 . . . . . 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 ) ) )  ->  d  e.  ( EE `  N ) )
10 btwnexch3 25202 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  ( ( B 
Btwn  <. A ,  C >.  /\  C  Btwn  <. A , 
d >. )  ->  C  Btwn  <. B ,  d
>. ) )
115, 6, 7, 8, 9, 10syl122anc 1191 . . . . 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 ) ) )  ->  ( ( B 
Btwn  <. A ,  C >.  /\  C  Btwn  <. A , 
d >. )  ->  C  Btwn  <. B ,  d
>. ) )
1211adantr 451 . . . 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
) ) )  /\  ( ( ( 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 >.  /\  C  Btwn  <. A , 
d >. )  ->  C  Btwn  <. B ,  d
>. ) )
134, 12mpd 14 . . 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
) ) )  /\  ( ( ( 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  <. B , 
d >. )
14 simp3lr 1027 . . . . . 6  |-  ( ( ( ( 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 ,  b >.Cgr <. C ,  B >. )
1514adantl 452 . . . . 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
) ) )  /\  ( ( ( 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 ,  b
>.Cgr <. C ,  B >. )
16 simp23 990 . . . . . . . 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 ) ) )  ->  c  e.  ( EE `  N ) )
17 simp3r 984 . . . . . . . 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 ) ) )  ->  b  e.  ( EE `  N ) )
18 cgrcomlr 25180 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( c  e.  ( EE `  N )  /\  b  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( <. c ,  b
>.Cgr <. C ,  B >.  <->  <. b ,  c >.Cgr <. B ,  C >. ) )
195, 16, 17, 8, 7, 18syl122anc 1191 . . . . . . 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 ) ) )  ->  ( <. c ,  b >.Cgr <. C ,  B >. 
<-> 
<. b ,  c >.Cgr <. B ,  C >. ) )
20 cgrcom 25172 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( b  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. b ,  c
>.Cgr <. B ,  C >.  <->  <. B ,  C >.Cgr <.
b ,  c >.
) )
215, 17, 16, 7, 8, 20syl122anc 1191 . . . . . . 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 ) ) )  ->  ( <. b ,  c >.Cgr <. B ,  C >. 
<-> 
<. B ,  C >.Cgr <.
b ,  c >.
) )
2219, 21bitrd 244 . . . . . 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 ) ) )  ->  ( <. c ,  b >.Cgr <. C ,  B >. 
<-> 
<. B ,  C >.Cgr <.
b ,  c >.
) )
2322adantr 451 . . . . 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
) ) )  /\  ( ( ( 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 ,  b >.Cgr <. C ,  B >. 
<-> 
<. B ,  C >.Cgr <.
b ,  c >.
) )
2415, 23mpbid 201 . . . 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
) ) )  /\  ( ( ( 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 ,  C >.Cgr
<. b ,  c >.
)
25 3simpa 952 . . . . . 6  |-  ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  ->  ( A  =/=  B  /\  B  =/=  C ) )
2625anim1i 551 . . . . 5  |-  ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  -> 
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )
27 btwnconn1lem3 25271 . . . . 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
) ) )  /\  ( ( ( 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 ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )  ->  <. B ,  d
>.Cgr <. b ,  D >. )
2826, 27syl3anr1 1234 . . . 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
) ) )  /\  ( ( ( 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 ,  d
>.Cgr <. b ,  D >. )
29 simp22 989 . . . . 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 ) ) )  ->  D  e.  ( EE `  N ) )
30 simp2rr 1025 . . . . . 6  |-  ( ( ( ( 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 >. )
3130adantl 452 . . . . 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
) ) )  /\  ( ( ( 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 >. )
32 simp2lr 1023 . . . . . . 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 >. ) ) )  ->  <. D ,  c >.Cgr <. C ,  D >. )
3332adantl 452 . . . . . 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
) ) )  /\  ( ( ( 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 ,  c
>.Cgr <. C ,  D >. )
345, 29, 16, 8, 29, 33cgrcomland 25181 . . . . 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
) ) )  /\  ( ( ( 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 >. )
355, 8, 9, 16, 29, 8, 29, 31, 34cgrtr3and 25177 . . . 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
) ) )  /\  ( ( ( 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 >. )
36 brcgr3 25228 . . . . . 6  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) )  /\  (
b  e.  ( EE
`  N )  /\  c  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( <. B ,  <. C ,  d >. >.Cgr3 <. b ,  <. c ,  D >. >. 
<->  ( <. B ,  C >.Cgr
<. b ,  c >.  /\  <. B ,  d
>.Cgr <. b ,  D >.  /\  <. C ,  d
>.Cgr <. c ,  D >. ) ) )
375, 7, 8, 9, 17, 16, 29, 36syl133anc 1205 . . . . 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 ) ) )  ->  ( <. B ,  <. C ,  d >. >.Cgr3
<. b ,  <. c ,  D >. >. 
<->  ( <. B ,  C >.Cgr
<. b ,  c >.  /\  <. B ,  d
>.Cgr <. b ,  D >.  /\  <. C ,  d
>.Cgr <. c ,  D >. ) ) )
3837adantr 451 . . . 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
) ) )  /\  ( ( ( 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 ,  <. C ,  d >. >.Cgr3
<. b ,  <. c ,  D >. >. 
<->  ( <. B ,  C >.Cgr
<. b ,  c >.  /\  <. B ,  d
>.Cgr <. b ,  D >.  /\  <. C ,  d
>.Cgr <. c ,  D >. ) ) )
3924, 28, 35, 38mpbir3and 1135 . . 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
) ) )  /\  ( ( ( 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 ,  <. C ,  d >. >.Cgr3 <. b ,  <. c ,  D >. >. )
40 simpl 443 . . . . . . 7  |-  ( ( d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) )  -> 
d  e.  ( EE
`  N ) )
41 simpr 447 . . . . . . 7  |-  ( ( d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) )  -> 
b  e.  ( EE
`  N ) )
4240, 41, 413jca 1132 . . . . . 6  |-  ( ( d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) )  -> 
( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N )  /\  b  e.  ( EE `  N ) ) )
43423anim3i 1139 . . . . 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 ) ) )  ->  ( ( 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
)  /\  b  e.  ( EE `  N ) ) ) )
44263anim1i 1138 . . . . 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 >. ) ) )  -> 
( ( ( 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 ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )
45 btwnconn1lem1 25269 . . . . 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
)  /\  b  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 ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )  ->  <. B ,  c
>.Cgr <. b ,  C >. )
4643, 44, 45syl2an 463 . . . 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
) ) )  /\  ( ( ( 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 ,  c
>.Cgr <. b ,  C >. )
475, 8, 16cgrrflx2d 25166 . . . . 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 ) ) )  ->  <. C ,  c
>.Cgr <. c ,  C >. )
4847adantr 451 . . . 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
) ) )  /\  ( ( ( 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
>.Cgr <. c ,  C >. )
4946, 48jca 518 . . 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
) ) )  /\  ( ( ( 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 , 
c >.Cgr <. b ,  C >.  /\  <. C ,  c
>.Cgr <. c ,  C >. ) )
5013, 39, 493jca 1132 . 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
) ) )  /\  ( ( ( 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  <. B ,  d >.  /\ 
<. B ,  <. C , 
d >. >.Cgr3 <. b ,  <. c ,  D >. >.  /\  ( <. B ,  c >.Cgr <. b ,  C >.  /\ 
<. C ,  c >.Cgr <. c ,  C >. ) ) )
51 simp1l2 1049 . . 3  |-  ( ( ( ( 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  =/=  C )
5251adantl 452 . 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
) ) )  /\  ( ( ( 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  =/=  C
)
53 brofs2 25259 . . . . . 6  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  c  e.  ( EE `  N )  /\  b  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  ->  ( <. <. B ,  C >. ,  <. d ,  c
>. >. 
OuterFiveSeg  <. <. b ,  c
>. ,  <. D ,  C >. >. 
<->  ( C  Btwn  <. B , 
d >.  /\  <. B ,  <. C ,  d >. >.Cgr3
<. b ,  <. c ,  D >. >.  /\  ( <. B ,  c >.Cgr <. b ,  C >.  /\  <. C , 
c >.Cgr <. c ,  C >. ) ) ) )
5453anbi1d 685 . . . . 5  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  c  e.  ( EE `  N )  /\  b  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  ->  (
( <. <. B ,  C >. ,  <. d ,  c
>. >. 
OuterFiveSeg  <. <. b ,  c
>. ,  <. D ,  C >. >.  /\  B  =/=  C )  <->  ( ( C 
Btwn  <. B ,  d
>.  /\  <. B ,  <. C ,  d >. >.Cgr3 <. b ,  <. c ,  D >. >.  /\  ( <. B ,  c >.Cgr <. b ,  C >.  /\  <. C , 
c >.Cgr <. c ,  C >. ) )  /\  B  =/=  C ) ) )
55 5segofs 25188 . . . . 5  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  c  e.  ( EE `  N )  /\  b  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  ->  (
( <. <. B ,  C >. ,  <. d ,  c
>. >. 
OuterFiveSeg  <. <. b ,  c
>. ,  <. D ,  C >. >.  /\  B  =/=  C )  ->  <. d ,  c >.Cgr <. D ,  C >. ) )
5654, 55sylbird 226 . . . 4  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  c  e.  ( EE `  N )  /\  b  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  ->  (
( ( C  Btwn  <. B ,  d >.  /\ 
<. B ,  <. C , 
d >. >.Cgr3 <. b ,  <. c ,  D >. >.  /\  ( <. B ,  c >.Cgr <. b ,  C >.  /\ 
<. C ,  c >.Cgr <. c ,  C >. ) )  /\  B  =/= 
C )  ->  <. d ,  c >.Cgr <. D ,  C >. ) )
575, 7, 8, 9, 16, 17, 16, 29, 8, 56syl333anc 1214 . . 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 ) ) )  ->  ( ( ( C  Btwn  <. B , 
d >.  /\  <. B ,  <. C ,  d >. >.Cgr3
<. b ,  <. c ,  D >. >.  /\  ( <. B ,  c >.Cgr <. b ,  C >.  /\  <. C , 
c >.Cgr <. c ,  C >. ) )  /\  B  =/=  C )  ->  <. d ,  c >.Cgr <. D ,  C >. ) )
5857adantr 451 . 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
) ) )  /\  ( ( ( 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  <. B , 
d >.  /\  <. B ,  <. C ,  d >. >.Cgr3
<. b ,  <. c ,  D >. >.  /\  ( <. B ,  c >.Cgr <. b ,  C >.  /\  <. C , 
c >.Cgr <. c ,  C >. ) )  /\  B  =/=  C )  ->  <. d ,  c >.Cgr <. D ,  C >. ) )
5950, 52, 58mp2and 660 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
) ) )  /\  ( ( ( 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 ,  c
>.Cgr <. D ,  C >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1710    =/= wne 2521   <.cop 3719   class class class wbr 4102   ` cfv 5334   NNcn 9833   EEcee 25075    Btwn cbtwn 25076  Cgrccgr 25077    OuterFiveSeg cofs 25164  Cgr3ccgr3 25218
This theorem is referenced by:  btwnconn1lem5  25273  btwnconn1lem6  25274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-ico 10751  df-icc 10752  df-fz 10872  df-fzo 10960  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-sum 12250  df-ee 25078  df-btwn 25079  df-cgr 25080  df-ofs 25165  df-ifs 25221  df-cgr3 25222
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