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Theorem outsidele 24095
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 24072 . . . . . 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 24091 . . . . . . . . . . . 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 23951 . . . . . . . . . . 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 24032 . . . . . . . . . . . . . 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 24086 . . . . . . . . . . . . . . 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 2435 . . . . . . . . . . . 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 23978 . . . . . . . . . . . . . . 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 23981 . . . . . . . . . . . . . . . . 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 24084 . . . . . . . . . . 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 24093 . . . . . . . . . . 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 3738 . . . . . . . . . 10  |-  ( B  =  y  ->  <. P ,  B >.  =  <. P , 
y >. )
4948breq2d 3975 . . . . . . . . 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 2637 . . . 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 24080 . . . 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 2419   E.wrex 2517   <.cop 3584   class class class wbr 3963   ` cfv 4638   NNcn 9679   EEcee 23856    Btwn cbtwn 23857  Cgrccgr 23858    Colinear ccolin 24000    Seg<_ csegle 24069  OutsideOfcoutsideof 24082
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-ee 23859  df-btwn 23860  df-cgr 23861  df-ofs 23946  df-ifs 24002  df-cgr3 24003  df-colinear 24004  df-fs 24005  df-segle 24070  df-outsideof 24083
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