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Theorem outsidele 23929
Description: Relate OutsideOf to  Seg<_. Theorem 6.13 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsidele  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  ->  ( <. P ,  A >.  Seg<_  <. P ,  B >.  <-> 
A  Btwn  <. P ,  B >. ) ) )

Proof of Theorem outsidele
StepHypRef Expression
1 simpl 445 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
2 simpr1 966 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
3 simpr2 967 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
4 simpr3 968 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
5 brsegle2 23906 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( P  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( <. P ,  A >.  Seg<_  <. P ,  B >.  <->  E. y  e.  ( EE `  N ) ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )
61, 2, 3, 2, 4, 5syl122anc 1196 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( <. P ,  A >. 
Seg<_ 
<. P ,  B >.  <->  E. y  e.  ( EE `  N ) ( A 
Btwn  <. P ,  y
>.  /\  <. P ,  y
>.Cgr <. P ,  B >. ) ) )
76adantr 453 . . . 4  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  POutsideOf <. A ,  B >. )  ->  ( <. P ,  A >.  Seg<_  <. P ,  B >.  <->  E. y  e.  ( EE `  N ) ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )
8 simprl 735 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  POutsideOf
<. A ,  B >. )
9 outsideofcom 23925 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  <-> 
POutsideOf <. B ,  A >. ) )
109ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  -> 
( POutsideOf <. A ,  B >.  <-> 
POutsideOf <. B ,  A >. ) )
118, 10mpbid 203 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  POutsideOf
<. B ,  A >. )
12 simpll 733 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  ->  N  e.  NN )
13 simplr1 1002 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  ->  P  e.  ( EE `  N
) )
14 simplr3 1004 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
1512, 13, 14cgrrflxd 23785 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  ->  <. P ,  B >.Cgr <. P ,  B >. )
1615adantr 453 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  <. P ,  B >.Cgr <. P ,  B >. )
1711, 16jca 520 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  -> 
( POutsideOf <. B ,  A >.  /\  <. P ,  B >.Cgr
<. P ,  B >. ) )
18 simprrl 743 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  A  Btwn  <. P ,  y
>. )
19 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  ->  y  e.  ( EE `  N
) )
20 simplr2 1003 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
21 btwncolinear1 23866 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  y  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. P , 
y >.  ->  P  Colinear  <. y ,  A >. ) )
2212, 13, 19, 20, 21syl13anc 1189 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  ->  ( A  Btwn  <. P ,  y
>.  ->  P  Colinear  <. y ,  A >. ) )
2322adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  -> 
( A  Btwn  <. P , 
y >.  ->  P  Colinear  <. y ,  A >. ) )
2418, 23mpd 16 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  P  Colinear  <. y ,  A >. )
25 outsidene1 23920 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  ->  A  =/=  P
) )
2625ad2antrr 709 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  -> 
( POutsideOf <. A ,  B >.  ->  A  =/=  P
) )
278, 26mpd 16 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  A  =/=  P )
2827neneqd 2428 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  -.  A  =  P
)
29 df-3an 941 . . . . . . . . . . . . . 14  |-  ( ( POutsideOf <. A ,  B >.  /\  ( A  Btwn  <. P ,  y >.  /\ 
<. P ,  y >.Cgr <. P ,  B >. )  /\  P  Btwn  <. y ,  A >. )  <->  ( ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) )  /\  P  Btwn  <. y ,  A >. ) )
30 simpr2l 1019 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. )  /\  P  Btwn  <.
y ,  A >. ) )  ->  A  Btwn  <. P ,  y >. )
3112, 20, 13, 19, 30btwncomand 23812 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. )  /\  P  Btwn  <.
y ,  A >. ) )  ->  A  Btwn  <.
y ,  P >. )
32 simpr3 968 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. )  /\  P  Btwn  <.
y ,  A >. ) )  ->  P  Btwn  <.
y ,  A >. )
33 btwnswapid2 23815 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  y  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <.
y ,  P >.  /\  P  Btwn  <. y ,  A >. )  ->  A  =  P ) )
3412, 20, 19, 13, 33syl13anc 1189 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  ->  (
( A  Btwn  <. y ,  P >.  /\  P  Btwn  <.
y ,  A >. )  ->  A  =  P ) )
3534adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. )  /\  P  Btwn  <.
y ,  A >. ) )  ->  ( ( A  Btwn  <. y ,  P >.  /\  P  Btwn  <. y ,  A >. )  ->  A  =  P ) )
3631, 32, 35mp2and 663 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. )  /\  P  Btwn  <.
y ,  A >. ) )  ->  A  =  P )
3729, 36sylan2br 464 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  (
( POutsideOf <. A ,  B >.  /\  ( A  Btwn  <. P ,  y >.  /\ 
<. P ,  y >.Cgr <. P ,  B >. ) )  /\  P  Btwn  <.
y ,  A >. ) )  ->  A  =  P )
3837expr 601 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  -> 
( P  Btwn  <. y ,  A >.  ->  A  =  P ) )
3928, 38mtod 170 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  -.  P  Btwn  <. y ,  A >. )
40 broutsideof 23918 . . . . . . . . . . 11  |-  ( POutsideOf <. y ,  A >.  <->  ( P  Colinear  <. y ,  A >.  /\  -.  P  Btwn  <.
y ,  A >. ) )
4124, 39, 40sylanbrc 648 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  POutsideOf
<. y ,  A >. )
42 simprrr 744 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  <. P ,  y >.Cgr <. P ,  B >. )
4341, 42jca 520 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  -> 
( POutsideOf <. y ,  A >.  /\  <. P ,  y
>.Cgr <. P ,  B >. ) )
44 outsideofeq 23927 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  y  e.  ( EE `  N ) ) )  ->  (
( ( POutsideOf <. B ,  A >.  /\  <. P ,  B >.Cgr <. P ,  B >. )  /\  ( POutsideOf <. y ,  A >.  /\ 
<. P ,  y >.Cgr <. P ,  B >. ) )  ->  B  =  y ) )
4512, 13, 20, 13, 14, 14, 19, 44syl133anc 1210 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  ->  (
( ( POutsideOf <. B ,  A >.  /\  <. P ,  B >.Cgr <. P ,  B >. )  /\  ( POutsideOf <. y ,  A >.  /\ 
<. P ,  y >.Cgr <. P ,  B >. ) )  ->  B  =  y ) )
4645adantr 453 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  -> 
( ( ( POutsideOf <. B ,  A >.  /\ 
<. P ,  B >.Cgr <. P ,  B >. )  /\  ( POutsideOf <. y ,  A >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) )  ->  B  =  y ) )
4717, 43, 46mp2and 663 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  B  =  y )
48 opeq2 3697 . . . . . . . . . 10  |-  ( B  =  y  ->  <. P ,  B >.  =  <. P , 
y >. )
4948breq2d 3932 . . . . . . . . 9  |-  ( B  =  y  ->  ( A  Btwn  <. P ,  B >.  <-> 
A  Btwn  <. P , 
y >. ) )
5018, 49syl5ibrcom 215 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  -> 
( B  =  y  ->  A  Btwn  <. P ,  B >. ) )
5147, 50mpd 16 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  /\  ( POutsideOf
<. A ,  B >.  /\  ( A  Btwn  <. P , 
y >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) ) )  ->  A  Btwn  <. P ,  B >. )
5251an4s 802 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  POutsideOf <. A ,  B >. )  /\  ( y  e.  ( EE `  N
)  /\  ( A  Btwn  <. P ,  y
>.  /\  <. P ,  y
>.Cgr <. P ,  B >. ) ) )  ->  A  Btwn  <. P ,  B >. )
5352exp32 591 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  POutsideOf <. A ,  B >. )  ->  ( y  e.  ( EE `  N
)  ->  ( ( A  Btwn  <. P ,  y
>.  /\  <. P ,  y
>.Cgr <. P ,  B >. )  ->  A  Btwn  <. P ,  B >. ) ) )
5453rexlimdv 2628 . . . 4  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  POutsideOf <. A ,  B >. )  ->  ( E. y  e.  ( EE `  N
) ( A  Btwn  <. P ,  y >.  /\ 
<. P ,  y >.Cgr <. P ,  B >. )  ->  A  Btwn  <. P ,  B >. ) )
557, 54sylbid 208 . . 3  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  POutsideOf <. A ,  B >. )  ->  ( <. P ,  A >.  Seg<_  <. P ,  B >.  ->  A  Btwn  <. P ,  B >. ) )
56 btwnsegle 23914 . . . 4  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. P ,  B >.  ->  <. P ,  A >.  Seg<_  <. P ,  B >. ) )
5756adantr 453 . . 3  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  POutsideOf <. A ,  B >. )  ->  ( A  Btwn  <. P ,  B >.  ->  <. P ,  A >.  Seg<_  <. P ,  B >. ) )
5855, 57impbid 185 . 2  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  POutsideOf <. A ,  B >. )  ->  ( <. P ,  A >.  Seg<_  <. P ,  B >.  <-> 
A  Btwn  <. P ,  B >. ) )
5958ex 425 1  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  ->  ( <. P ,  A >.  Seg<_  <. P ,  B >.  <-> 
A  Btwn  <. P ,  B >. ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510   <.cop 3547   class class class wbr 3920   ` cfv 4592   NNcn 9626   EEcee 23690    Btwn cbtwn 23691  Cgrccgr 23692    Colinear ccolin 23834    Seg<_ csegle 23903  OutsideOfcoutsideof 23916
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  df-ifs 23836  df-cgr3 23837  df-colinear 23838  df-fs 23839  df-segle 23904  df-outsideof 23917
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