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Theorem segconeq 24041
Description: Two points that satsify the conclusion of axsegcon 23963 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 1016 . . . . . . 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 520 . . . . . 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 960 . . . . . . . 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 1038 . . . . . . . 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 1035 . . . . . . . 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 24019 . . . . . . 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 1039 . . . . . . . 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 24019 . . . . . . 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 520 . . . . . 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 1040 . . . . . . . . . 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 1134 . . . . . . . . 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 1134 . . . . . . . . 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 1134 . . . . . . . 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 1018 . . . . . . . . . 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 520 . . . . . . . . 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 1036 . . . . . . . . . 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 1037 . . . . . . . . . 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 1019 . . . . . . . . . . 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 24021 . . . . . . . . . . . 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 1193 . . . . . . . . . . 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 203 . . . . . . . . . 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 1017 . . . . . . . . . . 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 24021 . . . . . . . . . . . 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 1193 . . . . . . . . . . 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 203 . . . . . . . . . 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 24016 . . . . . . . . 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 523 . . . . . . . 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 24039 . . . . . . . 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 58 . . . . . . 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 520 . . . . . 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 1134 . . . . 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 425 . . . 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 957 . . . . 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 993 . . . . 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 990 . . . . 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 994 . . . . 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 995 . . . . 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 24036 . . . . 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 1216 . . . 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 227 . . 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 957 . . . 4  |-  ( ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( A 
Btwn  <. Q ,  Y >.  /\  <. A ,  Y >.Cgr
<. B ,  C >. ) )  ->  Q  =/=  A )
4241a1i 12 . . 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 521 . 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 24037 . . 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 1216 . 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 23952 . . 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 1186 . 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 53 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 set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2448   <.cop 3645   class class class wbr 4025   ` cfv 5222   NNcn 9742   EEcee 23924    Btwn cbtwn 23925  Cgrccgr 23926    OuterFiveSeg cofs 24013
This theorem is referenced by:  segconeu  24042  btwnouttr2  24053  cgrxfr  24086  btwnconn1lem2  24119
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-ico 10657  df-icc 10658  df-fz 10778  df-fzo 10866  df-seq 11042  df-exp 11100  df-hash 11333  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-clim 11957  df-sum 12154  df-ee 23927  df-btwn 23928  df-cgr 23929  df-ofs 24014
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