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Theorem trisegint 25967
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
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 simpl1 961 . . . . 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 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 >. ) )  ->  C  e.  ( EE `  N ) )
3 simpl21 1036 . . . . . 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 1039 . . . . . 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 1135 . . . . 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 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 >. ) )  ->  E  e.  ( EE `  N ) )
7 simpl33 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 >. ) )  ->  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 1135 . . . 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 965 . . . . . 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 25953 . . . . . . 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 1187 . . . . . 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 966 . . . . 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 25885 . . . 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 59 . . 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 1051 . . . . . . 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 979 . . . . . . . 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 979 . . . . . . . 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 979 . . . . . . . 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 1135 . . . . . . 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 959 . . . . . . . 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 1037 . . . . . . . . 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 979 . . . . . . . 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 1135 . . . . . 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 986 . . . . . . 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 1054 . . . . . . . 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 25953 . . . . . . . . 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 1187 . . . . . . . 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 25885 . . . . . 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 59 . . . . 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 997 . . . . . . . . . . 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 16 . . . . . . . . . 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 16 . . . . . . . . . . 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 998 . . . . . . . . . . 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 733 . . . . . . . . . . 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 16 . . . . . . . . . . 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 1135 . . . . . . . . 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 1014 . . . . . . . . . 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 553 . . . . . . . . 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 25962 . . . . . . . . 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 59 . . . . . . . 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 2820 . . . . 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 15 . . . 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 2834 . . 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 15 . 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 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726   E.wrex 2708   <.cop 3819   class class class wbr 4215   ` cfv 5457   NNcn 10005   EEcee 25832    Btwn cbtwn 25833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-ee 25835  df-btwn 25836  df-cgr 25837  df-ofs 25922
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