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Theorem trisegint 24026
Description: A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Scott Fenton, 24-Sep-2013.)
Assertion
Ref Expression
trisegint  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. )  ->  E. q  e.  ( EE `  N
) ( q  Btwn  <. P ,  C >.  /\  q  Btwn  <. B ,  E >. ) ) )
Distinct variable groups:    A, q    B, q    C, q    D, q    E, q    N, q    P, q

Proof of Theorem trisegint
StepHypRef Expression
1 simpl1 963 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  N  e.  NN )
2 simpl23 1040 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  C  e.  ( EE `  N ) )
3 simpl21 1038 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  A  e.  ( EE `  N ) )
4 simpl31 1041 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  D  e.  ( EE `  N ) )
52, 3, 43jca 1137 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  ( C  e.  ( EE `  N
)  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )
6 simpl32 1042 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  E  e.  ( EE `  N ) )
7 simpl33 1043 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  P  e.  ( EE `  N ) )
86, 7jca 520 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  ( E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )
91, 5, 83jca 1137 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) ) )
10 simpr2 967 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  E  Btwn  <. D ,  C >. )
11 btwncom 24012 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( E  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( E  Btwn  <. D ,  C >. 
<->  E  Btwn  <. C ,  D >. ) )
121, 6, 4, 2, 11syl13anc 1189 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  ( E  Btwn  <. D ,  C >.  <-> 
E  Btwn  <. C ,  D >. ) )
1310, 12mpbid 203 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  E  Btwn  <. C ,  D >. )
14 simpr3 968 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  P  Btwn  <. A ,  D >. )
1513, 14jca 520 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  ( E  Btwn  <. C ,  D >.  /\  P  Btwn  <. A ,  D >. ) )
16 axpasch 23944 . . . 4  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  -> 
( ( E  Btwn  <. C ,  D >.  /\  P  Btwn  <. A ,  D >. )  ->  E. r  e.  ( EE `  N
) ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) ) )
179, 15, 16sylc 58 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  E. r  e.  ( EE `  N
) ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )
18 simp1l1 1053 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  ->  N  e.  NN )
1963ad2ant1 981 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  ->  E  e.  ( EE `  N ) )
2023ad2ant1 981 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  ->  C  e.  ( EE `  N ) )
2133ad2ant1 981 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  ->  A  e.  ( EE `  N ) )
2219, 20, 213jca 1137 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  -> 
( E  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) ) )
23 simp2 961 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  -> 
r  e.  ( EE
`  N ) )
24 simpl22 1039 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  B  e.  ( EE `  N ) )
25243ad2ant1 981 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  ->  B  e.  ( EE `  N ) )
2623, 25jca 520 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  -> 
( r  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )
2718, 22, 263jca 1137 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  -> 
( N  e.  NN  /\  ( E  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( r  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) ) )
28 simp3l 988 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  -> 
r  Btwn  <. E ,  A >. )
29 simp1r1 1056 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  ->  B  Btwn  <. A ,  C >. )
30 btwncom 24012 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >. 
<->  B  Btwn  <. C ,  A >. ) )
3118, 25, 21, 20, 30syl13anc 1189 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  -> 
( B  Btwn  <. A ,  C >. 
<->  B  Btwn  <. C ,  A >. ) )
3229, 31mpbid 203 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  ->  B  Btwn  <. C ,  A >. )
3328, 32jca 520 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  -> 
( r  Btwn  <. E ,  A >.  /\  B  Btwn  <. C ,  A >. ) )
34 axpasch 23944 . . . . . 6  |-  ( ( N  e.  NN  /\  ( E  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  (
r  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( ( r 
Btwn  <. E ,  A >.  /\  B  Btwn  <. C ,  A >. )  ->  E. q  e.  ( EE `  N
) ( q  Btwn  <.
r ,  C >.  /\  q  Btwn  <. B ,  E >. ) ) )
3527, 33, 34sylc 58 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  ->  E. q  e.  ( EE `  N ) ( q  Btwn  <. r ,  C >.  /\  q  Btwn  <. B ,  E >. ) )
36 simpll1 999 . . . . . . . . . . 11  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) ) )
3736, 1syl 17 . . . . . . . . . 10  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  N  e.  NN )
3836, 7syl 17 . . . . . . . . . . 11  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  P  e.  ( EE `  N ) )
39 simpll2 1000 . . . . . . . . . . 11  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  r  e.  ( EE `  N ) )
4038, 39jca 520 . . . . . . . . . 10  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  ( P  e.  ( EE `  N
)  /\  r  e.  ( EE `  N ) ) )
41 simplr 734 . . . . . . . . . . 11  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  q  e.  ( EE `  N ) )
4236, 2syl 17 . . . . . . . . . . 11  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  C  e.  ( EE `  N ) )
4341, 42jca 520 . . . . . . . . . 10  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  ( q  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )
4437, 40, 433jca 1137 . . . . . . . . 9  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  r  e.  ( EE `  N
) )  /\  (
q  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) ) ) )
45 simpl3r 1016 . . . . . . . . . 10  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  -> 
r  Btwn  <. P ,  C >. )
4645anim1i 554 . . . . . . . . 9  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  ( r  Btwn  <. P ,  C >.  /\  q  Btwn  <. r ,  C >. ) )
47 btwnexch2 24021 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  r  e.  ( EE `  N ) )  /\  ( q  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  (
( r  Btwn  <. P ,  C >.  /\  q  Btwn  <.
r ,  C >. )  ->  q  Btwn  <. P ,  C >. ) )
4844, 46, 47sylc 58 . . . . . . . 8  |-  ( ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  /\  q  Btwn  <. r ,  C >. )  ->  q  Btwn  <. P ,  C >. )
4948ex 425 . . . . . . 7  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  -> 
( q  Btwn  <. r ,  C >.  ->  q  Btwn  <. P ,  C >. ) )
5049anim1d 549 . . . . . 6  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  /\  q  e.  ( EE `  N ) )  -> 
( ( q  Btwn  <.
r ,  C >.  /\  q  Btwn  <. B ,  E >. )  ->  (
q  Btwn  <. P ,  C >.  /\  q  Btwn  <. B ,  E >. ) ) )
5150reximdva 2630 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  -> 
( E. q  e.  ( EE `  N
) ( q  Btwn  <.
r ,  C >.  /\  q  Btwn  <. B ,  E >. )  ->  E. q  e.  ( EE `  N
) ( q  Btwn  <. P ,  C >.  /\  q  Btwn  <. B ,  E >. ) ) )
5235, 51mpd 16 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  /\  r  e.  ( EE `  N
)  /\  ( r  Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. ) )  ->  E. q  e.  ( EE `  N ) ( q  Btwn  <. P ,  C >.  /\  q  Btwn  <. B ,  E >. ) )
5352rexlimdv3a 2644 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  ( E. r  e.  ( EE `  N ) ( r 
Btwn  <. E ,  A >.  /\  r  Btwn  <. P ,  C >. )  ->  E. q  e.  ( EE `  N
) ( q  Btwn  <. P ,  C >.  /\  q  Btwn  <. B ,  E >. ) ) )
5417, 53mpd 16 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. ) )  ->  E. q  e.  ( EE `  N
) ( q  Btwn  <. P ,  C >.  /\  q  Btwn  <. B ,  E >. ) )
5554ex 425 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  P  e.  ( EE `  N ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  C >.  /\  P  Btwn  <. A ,  D >. )  ->  E. q  e.  ( EE `  N
) ( q  Btwn  <. P ,  C >.  /\  q  Btwn  <. B ,  E >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    e. wcel 1621   E.wrex 2519   <.cop 3617   class class class wbr 3997   ` cfv 4673   NNcn 9714   EEcee 23891    Btwn cbtwn 23892
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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