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Theorem segconeq 24008
Description: Two points that satsify the conclusion of axsegcon 23930 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 1019 . . . . . . 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 963 . . . . . . . 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 1041 . . . . . . . 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 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 >. ) ) )  ->  A  e.  ( EE `  N
) )
63, 4, 5cgrrflxd 23986 . . . . . . 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 1042 . . . . . . . 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 23986 . . . . . . 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 1043 . . . . . . . . . 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 1137 . . . . . . . . 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 1137 . . . . . . . . 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 1137 . . . . . . . 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 1021 . . . . . . . . . 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 1039 . . . . . . . . . 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 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 >. ) ) )  ->  C  e.  ( EE `  N
) )
18 simpr3r 1022 . . . . . . . . . . 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 23988 . . . . . . . . . . . 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 1196 . . . . . . . . . . 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 1020 . . . . . . . . . . 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 23988 . . . . . . . . . . . 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 1196 . . . . . . . . . . 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 23983 . . . . . . . . 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 24006 . . . . . . . 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 1137 . . . . 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 960 . . . . 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 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 ) ) )  ->  Q  e.  ( EE `  N
) )
35 simp21 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 ) ) )  ->  A  e.  ( EE `  N
) )
36 simp32 997 . . . . 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 998 . . . . 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 24003 . . . . 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 1219 . . . 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 960 . . . 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 24004 . . 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 1219 . 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 23919 . . 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 1189 . 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 939    = wceq 1619    e. wcel 1621    =/= wne 2421   <.cop 3617   class class class wbr 3997   ` cfv 4673   NNcn 9714   EEcee 23891    Btwn cbtwn 23892  Cgrccgr 23893    OuterFiveSeg cofs 23980
This theorem is referenced by:  segconeu  24009  btwnouttr2  24020  cgrxfr  24053  btwnconn1lem2  24086
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-ico 10628  df-icc 10629  df-fz 10749  df-fzo 10837  df-seq 11013  df-exp 11071  df-hash 11304  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-clim 11927  df-sum 12124  df-ee 23894  df-btwn 23895  df-cgr 23896  df-ofs 23981
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