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Theorem trisegint 23825
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 23811 . . . . . . 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 23743 . . . 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 23811 . . . . . . . . 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 23743 . . . . . 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 23820 . . . . . . . . 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 2617 . . . . 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 2631 . . 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 2510   <.cop 3547   class class class wbr 3920   ` cfv 4592   NNcn 9626   EEcee 23690    Btwn cbtwn 23691
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-ee 23693  df-btwn 23694  df-cgr 23695  df-ofs 23780
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