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Theorem segleantisym 24738
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 24731 . . . 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 24732 . . . . 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 1157 . . . 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 691 . . 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 2707 . . 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 254 . 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 958 . . . . . 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 1010 . . . . . 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 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 ) ) )  ->  t  e.  ( EE `  N
) )
10 simprl 732 . . . . . 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 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 ) ) )  ->  D  e.  ( EE `  N
) )
12 simprll 738 . . . . . . 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 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 >. ) ) )  ->  D  Btwn  <. C ,  t
>. )
147, 8, 10, 11, 9, 12, 13btwnexchand 24649 . . . . . 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 1008 . . . . . . 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 1009 . . . . . . 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 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 >. ) ) )  ->  <. C ,  t >.Cgr <. A ,  B >. )
18 simprlr 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 >. ) ) )  ->  <. A ,  B >.Cgr <. C ,  y >. )
197, 8, 9, 15, 16, 8, 10, 17, 18cgrtrand 24616 . . . . . 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 24709 . . . . 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 3797 . . . . . . . . . 10  |-  ( t  =  y  ->  <. C , 
t >.  =  <. C , 
y >. )
2221breq2d 4035 . . . . . . . . 9  |-  ( t  =  y  ->  ( D  Btwn  <. C ,  t
>. 
<->  D  Btwn  <. C , 
y >. ) )
2321breq1d 4033 . . . . . . . . 9  |-  ( t  =  y  ->  ( <. C ,  t >.Cgr <. A ,  B >.  <->  <. C ,  y >.Cgr <. A ,  B >. ) )
2422, 23anbi12d 691 . . . . . . . 8  |-  ( t  =  y  ->  (
( D  Btwn  <. C , 
t >.  /\  <. C , 
t >.Cgr <. A ,  B >. )  <->  ( D  Btwn  <. C ,  y >.  /\ 
<. C ,  y >.Cgr <. A ,  B >. ) ) )
2524anbi2d 684 . . . . . . 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 684 . . . . . 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 740 . . . . . . . . 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 24638 . . . . . . . 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 738 . . . . . . . . 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 24638 . . . . . . . 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 24640 . . . . . . . . . 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 1184 . . . . . . . . 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 451 . . . . . . . 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 660 . . . . . . 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 739 . . . . . . . 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 3797 . . . . . . . . 9  |-  ( D  =  y  ->  <. C ,  D >.  =  <. C , 
y >. )
3736breq2d 4035 . . . . . . . 8  |-  ( D  =  y  ->  ( <. A ,  B >.Cgr <. C ,  D >.  <->  <. A ,  B >.Cgr <. C , 
y >. ) )
3835, 37syl5ibrcom 213 . . . . . . 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 14 . . . . . 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 219 . . . . 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 32 . . . 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 587 . . 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 2673 . 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 206 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   <.cop 3643   class class class wbr 4023   ` cfv 5255   NNcn 9746   EEcee 24516    Btwn cbtwn 24517  Cgrccgr 24518    Seg<_ csegle 24729
This theorem is referenced by:  colinbtwnle  24741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-ee 24519  df-btwn 24520  df-cgr 24521  df-ofs 24606  df-ifs 24662  df-cgr3 24663  df-colinear 24664  df-segle 24730
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