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Theorem segconeq 29822
Description: Two points that satsify the conclusion of axsegcon 24356 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.)
Assertion
Ref Expression
segconeq  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( A 
Btwn  <. Q ,  Y >.  /\  <. A ,  Y >.Cgr
<. B ,  C >. ) )  ->  X  =  Y ) )

Proof of Theorem segconeq
StepHypRef Expression
1 simpr2l 1055 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  A  Btwn  <. Q ,  X >. )
21, 1jca 532 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  ( A  Btwn  <. Q ,  X >.  /\  A  Btwn  <. Q ,  X >. ) )
3 simpl1 999 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  N  e.  NN )
4 simpl31 1077 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  Q  e.  ( EE `  N
) )
5 simpl21 1074 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  A  e.  ( EE `  N
) )
63, 4, 5cgrrflxd 29800 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. Q ,  A >.Cgr <. Q ,  A >. )
7 simpl32 1078 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  X  e.  ( EE `  N
) )
83, 5, 7cgrrflxd 29800 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  X >.Cgr <. A ,  X >. )
96, 8jca 532 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  ( <. Q ,  A >.Cgr <. Q ,  A >.  /\ 
<. A ,  X >.Cgr <. A ,  X >. ) )
10 simpl33 1079 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  Y  e.  ( EE `  N
) )
114, 5, 103jca 1176 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )
124, 5, 73jca 1176 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  X  e.  ( EE `  N ) ) )
133, 11, 123jca 1176 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  X  e.  ( EE `  N ) ) ) )
14 simpr3l 1057 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  A  Btwn  <. Q ,  Y >. )
1514, 1jca 532 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  ( A  Btwn  <. Q ,  Y >.  /\  A  Btwn  <. Q ,  X >. ) )
16 simpl22 1075 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  B  e.  ( EE `  N
) )
17 simpl23 1076 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  C  e.  ( EE `  N
) )
18 simpr3r 1058 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  Y >.Cgr <. B ,  C >. )
19 cgrcom 29802 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  Y  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. A ,  Y >.Cgr <. B ,  C >.  <->  <. B ,  C >.Cgr <. A ,  Y >. ) )
203, 5, 10, 16, 17, 19syl122anc 1237 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  ( <. A ,  Y >.Cgr <. B ,  C >.  <->  <. B ,  C >.Cgr <. A ,  Y >. ) )
2118, 20mpbid 210 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. B ,  C >.Cgr <. A ,  Y >. )
22 simpr2r 1056 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  X >.Cgr <. B ,  C >. )
23 cgrcom 29802 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  X  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. A ,  X >.Cgr <. B ,  C >.  <->  <. B ,  C >.Cgr <. A ,  X >. ) )
243, 5, 7, 16, 17, 23syl122anc 1237 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  ( <. A ,  X >.Cgr <. B ,  C >.  <->  <. B ,  C >.Cgr <. A ,  X >. ) )
2522, 24mpbid 210 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. B ,  C >.Cgr <. A ,  X >. )
263, 16, 17, 5, 10, 5, 7, 21, 25cgrtr4d 29797 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  Y >.Cgr <. A ,  X >. )
2715, 6, 26jca32 535 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  (
( A  Btwn  <. Q ,  Y >.  /\  A  Btwn  <. Q ,  X >. )  /\  ( <. Q ,  A >.Cgr <. Q ,  A >.  /\  <. A ,  Y >.Cgr
<. A ,  X >. ) ) )
28 cgrextend 29820 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  Y  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  X  e.  ( EE `  N ) ) )  ->  (
( ( A  Btwn  <. Q ,  Y >.  /\  A  Btwn  <. Q ,  X >. )  /\  ( <. Q ,  A >.Cgr <. Q ,  A >.  /\ 
<. A ,  Y >.Cgr <. A ,  X >. ) )  ->  <. Q ,  Y >.Cgr <. Q ,  X >. ) )
2913, 27, 28sylc 60 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. Q ,  Y >.Cgr <. Q ,  X >. )
3029, 26jca 532 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  ( <. Q ,  Y >.Cgr <. Q ,  X >.  /\ 
<. A ,  Y >.Cgr <. A ,  X >. ) )
312, 9, 303jca 1176 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  /\  ( A  Btwn  <. Q ,  Y >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  (
( A  Btwn  <. Q ,  X >.  /\  A  Btwn  <. Q ,  X >. )  /\  ( <. Q ,  A >.Cgr <. Q ,  A >.  /\  <. A ,  X >.Cgr
<. A ,  X >. )  /\  ( <. Q ,  Y >.Cgr <. Q ,  X >.  /\  <. A ,  Y >.Cgr
<. A ,  X >. ) ) )
3231ex 434 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( A 
Btwn  <. Q ,  Y >.  /\  <. A ,  Y >.Cgr
<. B ,  C >. ) )  ->  ( ( A  Btwn  <. Q ,  X >.  /\  A  Btwn  <. Q ,  X >. )  /\  ( <. Q ,  A >.Cgr <. Q ,  A >.  /\ 
<. A ,  X >.Cgr <. A ,  X >. )  /\  ( <. Q ,  Y >.Cgr <. Q ,  X >.  /\  <. A ,  Y >.Cgr
<. A ,  X >. ) ) ) )
33 simp1 996 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  N  e.  NN )
34 simp31 1032 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  Q  e.  ( EE `  N
) )
35 simp21 1029 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
36 simp32 1033 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  X  e.  ( EE `  N
) )
37 simp33 1034 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  Y  e.  ( EE `  N
) )
38 brofs 29817 . . . . 5  |-  ( ( ( N  e.  NN  /\  Q  e.  ( EE
`  N )  /\  A  e.  ( EE `  N ) )  /\  ( X  e.  ( EE `  N )  /\  Y  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) )  /\  ( A  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  X  e.  ( EE `  N ) ) )  ->  ( <. <. Q ,  A >. ,  <. X ,  Y >. >. 
OuterFiveSeg  <. <. Q ,  A >. ,  <. X ,  X >. >. 
<->  ( ( A  Btwn  <. Q ,  X >.  /\  A  Btwn  <. Q ,  X >. )  /\  ( <. Q ,  A >.Cgr <. Q ,  A >.  /\ 
<. A ,  X >.Cgr <. A ,  X >. )  /\  ( <. Q ,  Y >.Cgr <. Q ,  X >.  /\  <. A ,  Y >.Cgr
<. A ,  X >. ) ) ) )
3933, 34, 35, 36, 37, 34, 35, 36, 36, 38syl333anc 1260 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  ( <. <. Q ,  A >. ,  <. X ,  Y >. >. 
OuterFiveSeg  <. <. Q ,  A >. ,  <. X ,  X >. >. 
<->  ( ( A  Btwn  <. Q ,  X >.  /\  A  Btwn  <. Q ,  X >. )  /\  ( <. Q ,  A >.Cgr <. Q ,  A >.  /\ 
<. A ,  X >.Cgr <. A ,  X >. )  /\  ( <. Q ,  Y >.Cgr <. Q ,  X >.  /\  <. A ,  Y >.Cgr
<. A ,  X >. ) ) ) )
4032, 39sylibrd 234 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( A 
Btwn  <. Q ,  Y >.  /\  <. A ,  Y >.Cgr
<. B ,  C >. ) )  ->  <. <. Q ,  A >. ,  <. X ,  Y >. >. 
OuterFiveSeg  <. <. Q ,  A >. ,  <. X ,  X >. >. ) )
41 simp1 996 . . . 4  |-  ( ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( A 
Btwn  <. Q ,  Y >.  /\  <. A ,  Y >.Cgr
<. B ,  C >. ) )  ->  Q  =/=  A )
4241a1i 11 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( A 
Btwn  <. Q ,  Y >.  /\  <. A ,  Y >.Cgr
<. B ,  C >. ) )  ->  Q  =/=  A ) )
4340, 42jcad 533 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( A 
Btwn  <. Q ,  Y >.  /\  <. A ,  Y >.Cgr
<. B ,  C >. ) )  ->  ( <. <. Q ,  A >. , 
<. X ,  Y >. >.  OuterFiveSeg  <. <. Q ,  A >. , 
<. X ,  X >. >.  /\  Q  =/=  A
) ) )
44 5segofs 29818 . . 3  |-  ( ( ( N  e.  NN  /\  Q  e.  ( EE
`  N )  /\  A  e.  ( EE `  N ) )  /\  ( X  e.  ( EE `  N )  /\  Y  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) )  /\  ( A  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  X  e.  ( EE `  N ) ) )  ->  (
( <. <. Q ,  A >. ,  <. X ,  Y >. >. 
OuterFiveSeg  <. <. Q ,  A >. ,  <. X ,  X >. >.  /\  Q  =/=  A )  ->  <. X ,  Y >.Cgr <. X ,  X >. ) )
4533, 34, 35, 36, 37, 34, 35, 36, 36, 44syl333anc 1260 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( <. <. Q ,  A >. ,  <. X ,  Y >. >. 
OuterFiveSeg  <. <. Q ,  A >. ,  <. X ,  X >. >.  /\  Q  =/=  A )  ->  <. X ,  Y >.Cgr <. X ,  X >. ) )
46 axcgrid 24345 . . 3  |-  ( ( N  e.  NN  /\  ( X  e.  ( EE `  N )  /\  Y  e.  ( EE `  N )  /\  X  e.  ( EE `  N
) ) )  -> 
( <. X ,  Y >.Cgr
<. X ,  X >.  ->  X  =  Y )
)
4733, 36, 37, 36, 46syl13anc 1230 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  ( <. X ,  Y >.Cgr <. X ,  X >.  ->  X  =  Y )
)
4843, 45, 473syld 55 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( Q  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( A 
Btwn  <. Q ,  Y >.  /\  <. A ,  Y >.Cgr
<. B ,  C >. ) )  ->  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   <.cop 4038   class class class wbr 4456   ` cfv 5594   NNcn 10556   EEcee 24317    Btwn cbtwn 24318  Cgrccgr 24319    OuterFiveSeg cofs 29794
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 11821  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520  df-ee 24320  df-btwn 24321  df-cgr 24322  df-ofs 29795
This theorem is referenced by:  segconeu  29823  btwnouttr2  29834  cgrxfr  29867  btwnconn1lem2  29900
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