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Theorem segconeq 24705
Description: Two points that satsify the conclusion of axsegcon 24627 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 1014 . . . . . . 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 518 . . . . . 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 958 . . . . . . . 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 1036 . . . . . . . 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 1033 . . . . . . . 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 24683 . . . . . . 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 1037 . . . . . . . 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 24683 . . . . . . 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 518 . . . . . 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 1038 . . . . . . . . . 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 1132 . . . . . . . . 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 1132 . . . . . . . . 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 1132 . . . . . . . 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 1016 . . . . . . . . . 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 518 . . . . . . . . 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 1034 . . . . . . . . . 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 1035 . . . . . . . . . 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 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 ,  Y >.Cgr <. B ,  C >. )
19 cgrcom 24685 . . . . . . . . . . . 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 1191 . . . . . . . . . . 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 201 . . . . . . . . . 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 1015 . . . . . . . . . . 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 24685 . . . . . . . . . . . 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 1191 . . . . . . . . . . 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 201 . . . . . . . . . 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 24680 . . . . . . . . 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 521 . . . . . . . 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 24703 . . . . . . . 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 56 . . . . . . 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 518 . . . . . 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 1132 . . . . 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 423 . . . 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 955 . . . . 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 991 . . . . 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 988 . . . . 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 992 . . . . 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 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 ) ) )  ->  Y  e.  ( EE `  N
) )
38 brofs 24700 . . . . 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 1214 . . . 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 225 . . 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 955 . . . 4  |-  ( ( Q  =/=  A  /\  ( A  Btwn  <. Q ,  X >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( A 
Btwn  <. Q ,  Y >.  /\  <. A ,  Y >.Cgr
<. B ,  C >. ) )  ->  Q  =/=  A )
4241a1i 10 . . 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 519 . 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 24701 . . 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 1214 . 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 24616 . . 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 1184 . 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 51 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 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   <.cop 3656   class class class wbr 4039   ` cfv 5271   NNcn 9762   EEcee 24588    Btwn cbtwn 24589  Cgrccgr 24590    OuterFiveSeg cofs 24677
This theorem is referenced by:  segconeu  24706  btwnouttr2  24717  cgrxfr  24750  btwnconn1lem2  24783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-ee 24591  df-btwn 24592  df-cgr 24593  df-ofs 24678
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