Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  outsidele Unicode version

Theorem outsidele 24130
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 24107 . . . . . 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 24126 . . . . . . . . . . . 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 23986 . . . . . . . . . . 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 24067 . . . . . . . . . . . . . 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 24121 . . . . . . . . . . . . . . 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 2437 . . . . . . . . . . . 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 24013 . . . . . . . . . . . . . . 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 24016 . . . . . . . . . . . . . . . . 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 24119 . . . . . . . . . . 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 24128 . . . . . . . . . . 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 3771 . . . . . . . . . 10  |-  ( B  =  y  ->  <. P ,  B >.  =  <. P , 
y >. )
4948breq2d 4009 . . . . . . . . 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 2641 . . . 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 24115 . . . 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 2421   E.wrex 2519   <.cop 3617   class class class wbr 3997   ` cfv 4673   NNcn 9714   EEcee 23891    Btwn cbtwn 23892  Cgrccgr 23893    Colinear ccolin 24035    Seg<_ csegle 24104  OutsideOfcoutsideof 24117
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-ico 10628  df-icc 10629  df-fz 10749  df-fzo 10837  df-seq 11013  df-exp 11071  df-hash 11304  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-clim 11927  df-sum 12124  df-ee 23894  df-btwn 23895  df-cgr 23896  df-ofs 23981  df-ifs 24037  df-cgr3 24038  df-colinear 24039  df-fs 24040  df-segle 24105  df-outsideof 24118
  Copyright terms: Public domain W3C validator