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Theorem segleantisym 24146
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 >. ) )
Dummy variables  y 
t are mutually distinct and distinct from all other variables.

Proof of Theorem segleantisym
StepHypRef Expression
1 brsegle 24139 . . . 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 24140 . . . . 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 693 . . 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 2709 . . 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 256 . 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 735 . . . . . 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 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 ) ) )  ->  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 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 >. ) ) )  -> 
y  Btwn  <. C ,  D >. )
13 simprrl 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 >. ) ) )  ->  D  Btwn  <. C ,  t
>. )
147, 8, 10, 11, 9, 12, 13btwnexchand 24057 . . . . . 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 743 . . . . . . 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 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 >. ) ) )  ->  <. A ,  B >.Cgr <. C ,  y >. )
197, 8, 9, 15, 16, 8, 10, 17, 18cgrtrand 24024 . . . . . 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 24117 . . . . 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 3799 . . . . . . . . . 10  |-  ( t  =  y  ->  <. C , 
t >.  =  <. C , 
y >. )
2221breq2d 4037 . . . . . . . . 9  |-  ( t  =  y  ->  ( D  Btwn  <. C ,  t
>. 
<->  D  Btwn  <. C , 
y >. ) )
2321breq1d 4035 . . . . . . . . 9  |-  ( t  =  y  ->  ( <. C ,  t >.Cgr <. A ,  B >.  <->  <. C ,  y >.Cgr <. A ,  B >. ) )
2422, 23anbi12d 693 . . . . . . . 8  |-  ( t  =  y  ->  (
( D  Btwn  <. C , 
t >.  /\  <. C , 
t >.Cgr <. A ,  B >. )  <->  ( D  Btwn  <. C ,  y >.  /\ 
<. C ,  y >.Cgr <. A ,  B >. ) ) )
2524anbi2d 686 . . . . . . 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 686 . . . . . 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 742 . . . . . . . . 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 24046 . . . . . . . 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 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 >. ) ) )  -> 
y  Btwn  <. C ,  D >. )
307, 10, 8, 11, 29btwncomand 24046 . . . . . . . 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 24048 . . . . . . . . . 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 453 . . . . . . . 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 662 . . . . . . 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 741 . . . . . . . 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 3799 . . . . . . . . 9  |-  ( D  =  y  ->  <. C ,  D >.  =  <. C , 
y >. )
3736breq2d 4037 . . . . . . . 8  |-  ( D  =  y  ->  ( <. A ,  B >.Cgr <. C ,  D >.  <->  <. A ,  B >.Cgr <. C , 
y >. ) )
3835, 37syl5ibrcom 215 . . . . . . 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 16 . . . . . 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 221 . . . . 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 589 . . 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 2675 . 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 208 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 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   E.wrex 2546   <.cop 3645   class class class wbr 4025   ` cfv 5222   NNcn 9742   EEcee 23924    Btwn cbtwn 23925  Cgrccgr 23926    Seg<_ csegle 24137
This theorem is referenced by:  colinbtwnle  24149
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-ico 10657  df-icc 10658  df-fz 10778  df-fzo 10866  df-seq 11042  df-exp 11100  df-hash 11333  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-clim 11957  df-sum 12154  df-ee 23927  df-btwn 23928  df-cgr 23929  df-ofs 24014  df-ifs 24070  df-cgr3 24071  df-colinear 24072  df-segle 24138
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