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Theorem trisegint 29909
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 997 . . . . 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 1074 . . . . . 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 1072 . . . . . 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 1075 . . . . . 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 1174 . . . . 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 1076 . . . . . 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 1077 . . . . . 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 530 . . . . 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 1174 . . . 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 1001 . . . . . 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 29895 . . . . . . 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 1228 . . . . . 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 1002 . . . . 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 530 . . . 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 24449 . . . 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 1087 . . . . . . 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 1015 . . . . . . . 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 1015 . . . . . . . 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 1015 . . . . . . . 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 1174 . . . . . . 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 995 . . . . . . . 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 1073 . . . . . . . . 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 1015 . . . . . . . 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 530 . . . . . . 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 1174 . . . . . 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 1022 . . . . . . 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 1090 . . . . . . . 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 29895 . . . . . . . . 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 1228 . . . . . . . 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 530 . . . . . 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 24449 . . . . . 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 1033 . . . . . . . . . . 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 1034 . . . . . . . . . . 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 530 . . . . . . . . . 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 753 . . . . . . . . . . 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 530 . . . . . . . . . 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 1174 . . . . . . . . 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 1050 . . . . . . . . . 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 566 . . . . . . . . 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 29904 . . . . . . . . 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 432 . . . . . . 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 562 . . . . . 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 2929 . . . . 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 2948 . . 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 432 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 367    /\ w3a 971    e. wcel 1823   E.wrex 2805   <.cop 4022   class class class wbr 4439   ` cfv 5570   NNcn 10531   EEcee 24396    Btwn cbtwn 24397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-sum 13594  df-ee 24399  df-btwn 24400  df-cgr 24401  df-ofs 29864
This theorem is referenced by: (None)
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