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Theorem segconeq 30826
Description: Two points that satsify the conclusion of axsegcon 25006 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 1073 . . . . . . 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 539 . . . . . 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 1017 . . . . . . . 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 1095 . . . . . . . 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 1092 . . . . . . . 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 30804 . . . . . . 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 1096 . . . . . . . 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 30804 . . . . . . 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 539 . . . . . 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 1097 . . . . . . . . . 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 1194 . . . . . . . . 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 1194 . . . . . . . . 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 1194 . . . . . . . 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 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 >. ) ) )  ->  A  Btwn  <. Q ,  Y >. )
1514, 1jca 539 . . . . . . . . 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 1093 . . . . . . . . . 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 1094 . . . . . . . . . 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 1076 . . . . . . . . . . 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 30806 . . . . . . . . . . . 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 1285 . . . . . . . . . . 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 215 . . . . . . . . . 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 1074 . . . . . . . . . . 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 30806 . . . . . . . . . . . 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 1285 . . . . . . . . . . 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 215 . . . . . . . . . 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 30801 . . . . . . . . 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 542 . . . . . . . 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 30824 . . . . . . . 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 62 . . . . . . 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 539 . . . . . 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 1194 . . . . 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 440 . . . 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 1014 . . . . 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 1050 . . . . 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 1047 . . . . 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 1051 . . . . 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 1052 . . . . 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 30821 . . . . 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 1308 . . . 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 242 . . 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 1014 . . . 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 540 . 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 30822 . . 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 1308 . 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 24995 . . 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 1278 . 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 57 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   <.cop 3986   class class class wbr 4416   ` cfv 5601   NNcn 10637   EEcee 24967    Btwn cbtwn 24968  Cgrccgr 24969    OuterFiveSeg cofs 30798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-sup 7982  df-oi 8051  df-card 8399  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-rp 11332  df-ico 11670  df-icc 11671  df-fz 11814  df-fzo 11947  df-seq 12246  df-exp 12305  df-hash 12548  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-clim 13601  df-sum 13802  df-ee 24970  df-btwn 24971  df-cgr 24972  df-ofs 30799
This theorem is referenced by:  segconeu  30827  btwnouttr2  30838  cgrxfr  30871  btwnconn1lem2  30904
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