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Theorem segleantisym 25992
Description: Antisymmetry law for segment comparison. Theorem 5.9 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)
Assertion
Ref Expression
segleantisym  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.  Seg<_  <. C ,  D >.  /\ 
<. C ,  D >.  Seg<_  <. A ,  B >. )  ->  <. A ,  B >.Cgr
<. C ,  D >. ) )

Proof of Theorem segleantisym
Dummy variables  y 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brsegle 25985 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.  Seg<_  <. C ,  D >.  <->  E. y  e.  ( EE `  N ) ( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. ) ) )
2 brsegle2 25986 . . . . 5  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( <. C ,  D >.  Seg<_  <. A ,  B >.  <->  E. t  e.  ( EE `  N ) ( D  Btwn  <. C , 
t >.  /\  <. C , 
t >.Cgr <. A ,  B >. ) ) )
323com23 1159 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. C ,  D >.  Seg<_  <. A ,  B >.  <->  E. t  e.  ( EE `  N ) ( D  Btwn  <. C , 
t >.  /\  <. C , 
t >.Cgr <. A ,  B >. ) ) )
41, 3anbi12d 692 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.  Seg<_  <. C ,  D >.  /\ 
<. C ,  D >.  Seg<_  <. A ,  B >. )  <-> 
( E. y  e.  ( EE `  N
) ( y  Btwn  <. C ,  D >.  /\ 
<. A ,  B >.Cgr <. C ,  y >. )  /\  E. t  e.  ( EE `  N
) ( D  Btwn  <. C ,  t >.  /\ 
<. C ,  t >.Cgr <. A ,  B >. ) ) ) )
5 reeanv 2862 . . 3  |-  ( E. y  e.  ( EE
`  N ) E. t  e.  ( EE
`  N ) ( ( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) )  <->  ( E. y  e.  ( EE `  N ) ( y 
Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr
<. C ,  y >.
)  /\  E. t  e.  ( EE `  N
) ( D  Btwn  <. C ,  t >.  /\ 
<. C ,  t >.Cgr <. A ,  B >. ) ) )
64, 5syl6bbr 255 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.  Seg<_  <. C ,  D >.  /\ 
<. C ,  D >.  Seg<_  <. A ,  B >. )  <->  E. y  e.  ( EE `  N ) E. t  e.  ( EE
`  N ) ( ( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) ) )
7 simpl1 960 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  ->  N  e.  NN )
8 simpl3l 1012 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
9 simprr 734 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  ->  t  e.  ( EE `  N
) )
10 simprl 733 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  ->  y  e.  ( EE `  N
) )
11 simpl3r 1013 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N
) )
12 simprll 739 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) )  -> 
y  Btwn  <. C ,  D >. )
13 simprrl 741 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) )  ->  D  Btwn  <. C ,  t
>. )
147, 8, 10, 11, 9, 12, 13btwnexchand 25903 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) )  -> 
y  Btwn  <. C , 
t >. )
15 simpl2l 1010 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
16 simpl2r 1011 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
17 simprrr 742 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) )  ->  <. C ,  t >.Cgr <. A ,  B >. )
18 simprlr 740 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) )  ->  <. A ,  B >.Cgr <. C ,  y >. )
197, 8, 9, 15, 16, 8, 10, 17, 18cgrtrand 25870 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) )  ->  <. C ,  t >.Cgr <. C ,  y >.
)
207, 8, 9, 10, 14, 19endofsegidand 25963 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) )  -> 
t  =  y )
21 opeq2 3972 . . . . . . . . . 10  |-  ( t  =  y  ->  <. C , 
t >.  =  <. C , 
y >. )
2221breq2d 4211 . . . . . . . . 9  |-  ( t  =  y  ->  ( D  Btwn  <. C ,  t
>. 
<->  D  Btwn  <. C , 
y >. ) )
2321breq1d 4209 . . . . . . . . 9  |-  ( t  =  y  ->  ( <. C ,  t >.Cgr <. A ,  B >.  <->  <. C ,  y >.Cgr <. A ,  B >. ) )
2422, 23anbi12d 692 . . . . . . . 8  |-  ( t  =  y  ->  (
( D  Btwn  <. C , 
t >.  /\  <. C , 
t >.Cgr <. A ,  B >. )  <->  ( D  Btwn  <. C ,  y >.  /\ 
<. C ,  y >.Cgr <. A ,  B >. ) ) )
2524anbi2d 685 . . . . . . 7  |-  ( t  =  y  ->  (
( ( y  Btwn  <. C ,  D >.  /\ 
<. A ,  B >.Cgr <. C ,  y >. )  /\  ( D  Btwn  <. C ,  t >.  /\ 
<. C ,  t >.Cgr <. A ,  B >. ) )  <->  ( ( y 
Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr
<. C ,  y >.
)  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) ) )
2625anbi2d 685 . . . . . 6  |-  ( t  =  y  ->  (
( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) )  <->  ( (
( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) ) ) )
27 simprrl 741 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) )  ->  D  Btwn  <. C ,  y
>. )
287, 11, 8, 10, 27btwncomand 25892 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) )  ->  D  Btwn  <. y ,  C >. )
29 simprll 739 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) )  -> 
y  Btwn  <. C ,  D >. )
307, 10, 8, 11, 29btwncomand 25892 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) )  -> 
y  Btwn  <. D ,  C >. )
31 btwnswapid 25894 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  y  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( D  Btwn  <.
y ,  C >.  /\  y  Btwn  <. D ,  C >. )  ->  D  =  y ) )
327, 11, 10, 8, 31syl13anc 1186 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  ->  (
( D  Btwn  <. y ,  C >.  /\  y  Btwn  <. D ,  C >. )  ->  D  =  y ) )
3332adantr 452 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) )  -> 
( ( D  Btwn  <.
y ,  C >.  /\  y  Btwn  <. D ,  C >. )  ->  D  =  y ) )
3428, 30, 33mp2and 661 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) )  ->  D  =  y )
35 simprlr 740 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) )  ->  <. A ,  B >.Cgr <. C ,  y >. )
36 opeq2 3972 . . . . . . . . 9  |-  ( D  =  y  ->  <. C ,  D >.  =  <. C , 
y >. )
3736breq2d 4211 . . . . . . . 8  |-  ( D  =  y  ->  ( <. A ,  B >.Cgr <. C ,  D >.  <->  <. A ,  B >.Cgr <. C , 
y >. ) )
3835, 37syl5ibrcom 214 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) )  -> 
( D  =  y  ->  <. A ,  B >.Cgr
<. C ,  D >. ) )
3934, 38mpd 15 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  y
>.  /\  <. C ,  y
>.Cgr <. A ,  B >. ) ) )  ->  <. A ,  B >.Cgr <. C ,  D >. )
4026, 39syl6bi 220 . . . . 5  |-  ( t  =  y  ->  (
( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) )  ->  <. A ,  B >.Cgr <. C ,  D >. ) )
4120, 40mpcom 34 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N ) ) )  /\  (
( y  Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr <. C ,  y
>. )  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) ) )  ->  <. A ,  B >.Cgr <. C ,  D >. )
4241exp31 588 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( y  e.  ( EE `  N )  /\  t  e.  ( EE `  N
) )  ->  (
( ( y  Btwn  <. C ,  D >.  /\ 
<. A ,  B >.Cgr <. C ,  y >. )  /\  ( D  Btwn  <. C ,  t >.  /\ 
<. C ,  t >.Cgr <. A ,  B >. ) )  ->  <. A ,  B >.Cgr <. C ,  D >. ) ) )
4342rexlimdvv 2823 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( E. y  e.  ( EE `  N
) E. t  e.  ( EE `  N
) ( ( y 
Btwn  <. C ,  D >.  /\  <. A ,  B >.Cgr
<. C ,  y >.
)  /\  ( D  Btwn  <. C ,  t
>.  /\  <. C ,  t
>.Cgr <. A ,  B >. ) )  ->  <. A ,  B >.Cgr <. C ,  D >. ) )
446, 43sylbid 207 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.  Seg<_  <. C ,  D >.  /\ 
<. C ,  D >.  Seg<_  <. A ,  B >. )  ->  <. A ,  B >.Cgr
<. C ,  D >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2693   <.cop 3804   class class class wbr 4199   ` cfv 5440   NNcn 9984   EEcee 25770    Btwn cbtwn 25771  Cgrccgr 25772    Seg<_ csegle 25983
This theorem is referenced by:  colinbtwnle  25995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-oadd 6714  df-er 6891  df-map 7006  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-sup 7432  df-oi 7463  df-card 7810  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-n0 10206  df-z 10267  df-uz 10473  df-rp 10597  df-ico 10906  df-icc 10907  df-fz 11028  df-fzo 11119  df-seq 11307  df-exp 11366  df-hash 11602  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-clim 12265  df-sum 12463  df-ee 25773  df-btwn 25774  df-cgr 25775  df-ofs 25860  df-ifs 25916  df-cgr3 25917  df-colinear 25918  df-segle 25984
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