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

Theorem outsidele 29677
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 457 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
2 simpr1 1002 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
3 simpr2 1003 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
4 simpr3 1004 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
5 brsegle2 29654 . . . . . 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 1237 . . . . 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 465 . . . 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 755 . . . . . . . . . 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 29673 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  <-> 
POutsideOf <. B ,  A >. ) )
109ad2antrr 725 . . . . . . . . . 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 210 . . . . . . . . 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 753 . . . . . . . . . . 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 1038 . . . . . . . . . . 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 1040 . . . . . . . . . . 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 29533 . . . . . . . . . 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 465 . . . . . . . . 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 532 . . . . . . . 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 763 . . . . . . . . . . 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 461 . . . . . . . . . . . . 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 1039 . . . . . . . . . . . . 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 29614 . . . . . . . . . . . . 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 1230 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 29668 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  ->  A  =/=  P
) )
2625ad2antrr 725 . . . . . . . . . . . . 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 2669 . . . . . . . . . . 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 975 . . . . . . . . . . . . 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 1055 . . . . . . . . . . . . . . 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 29560 . . . . . . . . . . . . . 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 1004 . . . . . . . . . . . . . 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 29563 . . . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 679 . . . . . . . . . . . . 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 476 . . . . . . . . . . . 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 615 . . . . . . . . . . 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 177 . . . . . . . . . 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 29666 . . . . . . . . . 10  |-  ( POutsideOf <. y ,  A >.  <->  ( P  Colinear  <. y ,  A >.  /\  -.  P  Btwn  <.
y ,  A >. ) )
4124, 39, 40sylanbrc 664 . . . . . . . . 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 764 . . . . . . . . 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 532 . . . . . . . 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 29675 . . . . . . . . . 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 1251 . . . . . . . . 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 465 . . . . . . . 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 679 . . . . . . 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 4219 . . . . . . . . 9  |-  ( B  =  y  ->  <. P ,  B >.  =  <. P , 
y >. )
4948breq2d 4464 . . . . . . . 8  |-  ( B  =  y  ->  ( A  Btwn  <. P ,  B >.  <-> 
A  Btwn  <. P , 
y >. ) )
5018, 49syl5ibrcom 222 . . . . . . 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 824 . . . . 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 2960 . . . 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 215 . . 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 29662 . . . 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 465 . . 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 191 . 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 434 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   <.cop 4038   class class class wbr 4452   ` cfv 5593   NNcn 10546   EEcee 23982    Btwn cbtwn 23983  Cgrccgr 23984    Colinear ccolin 29582    Seg<_ csegle 29651  OutsideOfcoutsideof 29664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-sup 7911  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-ico 11545  df-icc 11546  df-fz 11683  df-fzo 11803  df-seq 12086  df-exp 12145  df-hash 12384  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-clim 13286  df-sum 13484  df-ee 23985  df-btwn 23986  df-cgr 23987  df-ofs 29528  df-colinear 29584  df-ifs 29585  df-cgr3 29586  df-fs 29587  df-segle 29652  df-outsideof 29665
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator