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Theorem trisegint 27905
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 984 . . . . 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 1061 . . . . . 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 1059 . . . . . 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 1062 . . . . . 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 1161 . . . . 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 1063 . . . . . 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 1064 . . . . . 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 529 . . . . 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 1161 . . . 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 988 . . . . . 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 27891 . . . . . . 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 1213 . . . . . 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 210 . . . . 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 989 . . . . 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 529 . . . 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 23009 . . . 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 60 . . 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 1074 . . . . . . 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 1002 . . . . . . . 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 1002 . . . . . . . 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 1002 . . . . . . . 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 1161 . . . . . . 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 982 . . . . . . . 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 1060 . . . . . . . . 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 1002 . . . . . . . 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 529 . . . . . . 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 1161 . . . . . 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 1009 . . . . . . 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 1077 . . . . . . . 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 27891 . . . . . . . . 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 1213 . . . . . . . 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 210 . . . . . . 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 529 . . . . . 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 23009 . . . . . 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 60 . . . . 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 1020 . . . . . . . . . . 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 1021 . . . . . . . . . . 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 529 . . . . . . . . . 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 747 . . . . . . . . . . 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 529 . . . . . . . . . 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 1161 . . . . . . . . 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 1037 . . . . . . . . . 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 563 . . . . . . . . 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 27900 . . . . . . . . 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 60 . . . . . . . 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 434 . . . . . . 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 559 . . . . . 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 2818 . . . . 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 2833 . . 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 434 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    e. wcel 1755   E.wrex 2706   <.cop 3871   class class class wbr 4280   ` cfv 5406   NNcn 10309   EEcee 22956    Btwn cbtwn 22957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-n0 10567  df-z 10634  df-uz 10849  df-rp 10979  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-seq 11790  df-exp 11849  df-hash 12087  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-clim 12949  df-sum 13147  df-ee 22959  df-btwn 22960  df-cgr 22961  df-ofs 27860
This theorem is referenced by: (None)
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