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Theorem btwnconn1lem5 26017
Description: Lemma for btwnconn1 26027. Now, we introduce  E, the intersection of  C c and  D d. We begin by showing that it is the midpoint of  C and  c (Contributed by Scott Fenton, 8-Oct-2013.)
Assertion
Ref Expression
btwnconn1lem5  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
)  /\  E  e.  ( EE `  N ) ) )  /\  (
( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) ) )  ->  <. E ,  C >.Cgr <. E ,  c
>. )

Proof of Theorem btwnconn1lem5
StepHypRef Expression
1 simprrr 742 . 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 ) ) )  /\  (
( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) ) )  ->  E  Btwn  <. D ,  d
>. )
2 simp11 987 . . . 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
) ) )  ->  N  e.  NN )
3 simp22 991 . . . 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
) ) )  ->  D  e.  ( EE `  N ) )
4 simp33 995 . . . 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
) ) )  ->  E  e.  ( EE `  N ) )
5 simp31 993 . . . 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
) ) )  -> 
d  e.  ( EE
`  N ) )
6 cgr3rflx 25980 . . . 4  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  ->  <. D ,  <. E , 
d >. >.Cgr3 <. D ,  <. E ,  d >. >. )
72, 3, 4, 5, 6syl13anc 1186 . . 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
) ) )  ->  <. D ,  <. E , 
d >. >.Cgr3 <. D ,  <. E ,  d >. >. )
87adantr 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
)  /\  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 >. ) ) )  ->  <. D ,  <. E ,  d >. >.Cgr3
<. D ,  <. E , 
d >. >. )
9 simp2lr 1025 . . . . 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 >. ) ) )  ->  <. D ,  c >.Cgr <. C ,  D >. )
109ad2antrl 709 . . . 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 >. ) ) )  ->  <. D , 
c >.Cgr <. C ,  D >. )
11 simp23 992 . . . . . . 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
) ) )  -> 
c  e.  ( EE
`  N ) )
12 simp21 990 . . . . . . 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
) ) )  ->  C  e.  ( EE `  N ) )
13 cgrcomr 25923 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. D , 
c >.Cgr <. C ,  D >.  <->  <. D ,  c >.Cgr <. D ,  C >. ) )
142, 3, 11, 12, 3, 13syl122anc 1193 . . . . . 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
) ) )  -> 
( <. D ,  c
>.Cgr <. C ,  D >.  <->  <. D ,  c >.Cgr <. D ,  C >. ) )
15 cgrcom 25916 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. D , 
c >.Cgr <. D ,  C >.  <->  <. D ,  C >.Cgr <. D ,  c >. ) )
162, 3, 11, 3, 12, 15syl122anc 1193 . . . . . 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
) ) )  -> 
( <. D ,  c
>.Cgr <. D ,  C >.  <->  <. D ,  C >.Cgr <. D ,  c >. ) )
1714, 16bitrd 245 . . . . 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
) ) )  -> 
( <. D ,  c
>.Cgr <. C ,  D >.  <->  <. D ,  C >.Cgr <. D ,  c >. ) )
1817adantr 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
)  /\  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 >. ) ) )  ->  ( <. D ,  c >.Cgr <. C ,  D >.  <->  <. D ,  C >.Cgr <. D , 
c >. ) )
1910, 18mpbid 202 . . 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 ) ) )  /\  (
( ( ( 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 >. ) ) )  ->  <. D ,  C >.Cgr <. D ,  c
>. )
20 simp2rr 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 ,  d >.Cgr <. C ,  D >. )
2120ad2antrl 709 . . . . 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 ) ) )  /\  (
( ( ( 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 >.Cgr <. C ,  D >. )
222, 12, 5, 12, 3, 21cgrcomlrand 25927 . . . 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 >. ) ) )  ->  <. d ,  C >.Cgr <. D ,  C >. )
23 3simpa 954 . . . . . 6  |-  ( ( d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) )  ->  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) )
24233anim3i 1141 . . . . 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
) ) )  -> 
( ( 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
) ) ) )
25 simpl 444 . . . . 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 >. ) )  ->  ( (
( 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 >. ) ) ) )
26 btwnconn1lem4 26016 . . . . 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 >. ) ) ) )  ->  <. d ,  c
>.Cgr <. D ,  C >. )
2724, 25, 26syl2an 464 . . . 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 >. ) ) )  ->  <. d ,  c >.Cgr <. D ,  C >. )
282, 5, 12, 5, 11, 3, 12, 22, 27cgrtr3and 25921 . . 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 ) ) )  /\  (
( ( ( 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 >. ) ) )  ->  <. d ,  C >.Cgr <. d ,  c
>. )
2919, 28jca 519 . 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 ) ) )  /\  (
( ( ( 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 >. ) ) )  ->  ( <. D ,  C >.Cgr <. D ,  c >.  /\ 
<. d ,  C >.Cgr <.
d ,  c >.
) )
30 brifs2 26004 . . . . 5  |-  ( ( ( N  e.  NN  /\  D  e.  ( EE
`  N )  /\  E  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  d  e.  ( EE `  N
)  /\  c  e.  ( EE `  N ) ) )  ->  ( <. <. D ,  E >. ,  <. d ,  C >. >. 
InnerFiveSeg  <. <. D ,  E >. ,  <. d ,  c
>. >. 
<->  ( E  Btwn  <. D , 
d >.  /\  <. D ,  <. E ,  d >. >.Cgr3
<. D ,  <. E , 
d >. >.  /\  ( <. D ,  C >.Cgr <. D , 
c >.  /\  <. d ,  C >.Cgr <. d ,  c
>. ) ) ) )
31 ifscgr 25970 . . . . 5  |-  ( ( ( N  e.  NN  /\  D  e.  ( EE
`  N )  /\  E  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  d  e.  ( EE `  N
)  /\  c  e.  ( EE `  N ) ) )  ->  ( <. <. D ,  E >. ,  <. d ,  C >. >. 
InnerFiveSeg  <. <. D ,  E >. ,  <. d ,  c
>. >.  ->  <. E ,  C >.Cgr <. E ,  c
>. ) )
3230, 31sylbird 227 . . . 4  |-  ( ( ( N  e.  NN  /\  D  e.  ( EE
`  N )  /\  E  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  d  e.  ( EE `  N
)  /\  c  e.  ( EE `  N ) ) )  ->  (
( E  Btwn  <. D , 
d >.  /\  <. D ,  <. E ,  d >. >.Cgr3
<. D ,  <. E , 
d >. >.  /\  ( <. D ,  C >.Cgr <. D , 
c >.  /\  <. d ,  C >.Cgr <. d ,  c
>. ) )  ->  <. E ,  C >.Cgr <. E ,  c
>. ) )
332, 3, 4, 5, 12, 3, 4, 5, 11, 32syl333anc 1216 . . 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
) ) )  -> 
( ( E  Btwn  <. D ,  d >.  /\ 
<. D ,  <. E , 
d >. >.Cgr3 <. D ,  <. E ,  d >. >.  /\  ( <. D ,  C >.Cgr <. D ,  c >.  /\ 
<. d ,  C >.Cgr <.
d ,  c >.
) )  ->  <. E ,  C >.Cgr <. E ,  c
>. ) )
3433adantr 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
)  /\  E  e.  ( EE `  N ) ) )  /\  (
( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) ) )  ->  (
( E  Btwn  <. D , 
d >.  /\  <. D ,  <. E ,  d >. >.Cgr3
<. D ,  <. E , 
d >. >.  /\  ( <. D ,  C >.Cgr <. D , 
c >.  /\  <. d ,  C >.Cgr <. d ,  c
>. ) )  ->  <. E ,  C >.Cgr <. E ,  c
>. ) )
351, 8, 29, 34mp3and 1282 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 ) ) )  /\  (
( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  c )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) ) )  ->  <. E ,  C >.Cgr <. E ,  c
>. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2598   <.cop 3809   class class class wbr 4204   ` cfv 5446   NNcn 9992   EEcee 25819    Btwn cbtwn 25820  Cgrccgr 25821    InnerFiveSeg cifs 25961  Cgr3ccgr3 25962
This theorem is referenced by:  btwnconn1lem12  26024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-ee 25822  df-btwn 25823  df-cgr 25824  df-ofs 25909  df-ifs 25965  df-cgr3 25966
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