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Theorem outsidele 25970
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
2 simpr1 963 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
3 simpr2 964 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
4 simpr3 965 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
5 brsegle2 25947 . . . . . 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 1193 . . . . 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 452 . . . 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 733 . . . . . . . . . 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
<. A ,  B >. )
9 outsideofcom 25966 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  <-> 
POutsideOf <. B ,  A >. ) )
109ad2antrr 707 . . . . . . . . . 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 <. A ,  B >.  <-> 
POutsideOf <. B ,  A >. ) )
118, 10mpbid 202 . . . . . . . . 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 >. )
12 simpll 731 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  y  e.  ( EE `  N
) )  ->  N  e.  NN )
13 simplr1 999 . . . . . . . . . . 11  |-  ( ( ( 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 1001 . . . . . . . . . . 11  |-  ( ( ( 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 25826 . . . . . . . . . 10  |-  ( ( ( 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 452 . . . . . . . . 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 >. ) ) )  ->  <. P ,  B >.Cgr <. P ,  B >. )
1711, 16jca 519 . . . . . . . 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 >. ) ) )  -> 
( POutsideOf <. B ,  A >.  /\  <. P ,  B >.Cgr
<. P ,  B >. ) )
18 simprrl 741 . . . . . . . . . . 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 >. ) ) )  ->  A  Btwn  <. P ,  y
>. )
19 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( 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 1000 . . . . . . . . . . . . 13  |-  ( ( ( 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 25907 . . . . . . . . . . . . 13  |-  ( ( 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 1186 . . . . . . . . . . . 12  |-  ( ( ( 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 452 . . . . . . . . . . 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 >. ) ) )  -> 
( A  Btwn  <. P , 
y >.  ->  P  Colinear  <. y ,  A >. ) )
2418, 23mpd 15 . . . . . . . . . 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  Colinear  <. y ,  A >. )
25 outsidene1 25961 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  ->  A  =/=  P
) )
2625ad2antrr 707 . . . . . . . . . . . . 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 >. ) ) )  -> 
( POutsideOf <. A ,  B >.  ->  A  =/=  P
) )
278, 26mpd 15 . . . . . . . . . . . 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 )
2827neneqd 2583 . . . . . . . . . . 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 >. ) ) )  ->  -.  A  =  P
)
29 df-3an 938 . . . . . . . . . . . . 13  |-  ( ( 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 1016 . . . . . . . . . . . . . . 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  <. P ,  y >. )
3112, 20, 13, 19, 30btwncomand 25853 . . . . . . . . . . . . . 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  Btwn  <.
y ,  P >. )
32 simpr3 965 . . . . . . . . . . . . . 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 >. ) )  ->  P  Btwn  <.
y ,  A >. )
33 btwnswapid2 25856 . . . . . . . . . . . . . . . 16  |-  ( ( 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 1186 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 452 . . . . . . . . . . . . . 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  Btwn  <. y ,  P >.  /\  P  Btwn  <. y ,  A >. )  ->  A  =  P ) )
3631, 32, 35mp2and 661 . . . . . . . . . . . . 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 )
3729, 36sylan2br 463 . . . . . . . . . . . 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 )
3837expr 599 . . . . . . . . . . 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 >.  ->  A  =  P ) )
3928, 38mtod 170 . . . . . . . . . 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  Btwn  <. y ,  A >. )
40 broutsideof 25959 . . . . . . . . . 10  |-  ( POutsideOf <. y ,  A >.  <->  ( P  Colinear  <. y ,  A >.  /\  -.  P  Btwn  <.
y ,  A >. ) )
4124, 39, 40sylanbrc 646 . . . . . . . . 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 >. )
42 simprrr 742 . . . . . . . . 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 >. ) ) )  ->  <. P ,  y >.Cgr <. P ,  B >. )
4341, 42jca 519 . . . . . . . 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 >. ) ) )  -> 
( POutsideOf <. y ,  A >.  /\  <. P ,  y
>.Cgr <. P ,  B >. ) )
44 outsideofeq 25968 . . . . . . . . . 10  |-  ( ( 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 1207 . . . . . . . . 9  |-  ( ( ( 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 452 . . . . . . . 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 >. ) ) )  -> 
( ( ( POutsideOf <. B ,  A >.  /\ 
<. P ,  B >.Cgr <. P ,  B >. )  /\  ( POutsideOf <. y ,  A >.  /\  <. P , 
y >.Cgr <. P ,  B >. ) )  ->  B  =  y ) )
4717, 43, 46mp2and 661 . . . . . . 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 >. ) ) )  ->  B  =  y )
48 opeq2 3945 . . . . . . . . 9  |-  ( B  =  y  ->  <. P ,  B >.  =  <. P , 
y >. )
4948breq2d 4184 . . . . . . . 8  |-  ( B  =  y  ->  ( A  Btwn  <. P ,  B >.  <-> 
A  Btwn  <. P , 
y >. ) )
5018, 49syl5ibrcom 214 . . . . . . 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 >. ) ) )  -> 
( B  =  y  ->  A  Btwn  <. P ,  B >. ) )
5147, 50mpd 15 . . . . . 6  |-  ( ( ( ( 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 800 . . . . 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 >. )
5352rexlimdvaa 2791 . . . 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 >. ) )
547, 53sylbid 207 . . 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 >. ) )
55 btwnsegle 25955 . . . 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 >. ) )
5655adantr 452 . . 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 >. ) )
5754, 56impbid 184 . 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 >. ) )
5857ex 424 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 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   <.cop 3777   class class class wbr 4172   ` cfv 5413   NNcn 9956   EEcee 25731    Btwn cbtwn 25732  Cgrccgr 25733    Colinear ccolin 25875    Seg<_ csegle 25944  OutsideOfcoutsideof 25957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-ee 25734  df-btwn 25735  df-cgr 25736  df-ofs 25821  df-ifs 25877  df-cgr3 25878  df-colinear 25879  df-fs 25880  df-segle 25945  df-outsideof 25958
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