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Theorem segleantisym 28283
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 28276 . . . 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 28277 . . . . 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 1194 . . . 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 710 . . 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 2987 . . 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 263 . 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 991 . . . . . 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 1043 . . . . . 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 756 . . . . . 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 755 . . . . . 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 1044 . . . . . . 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 761 . . . . . . 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 763 . . . . . . 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 28194 . . . . . 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 1041 . . . . . . 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 1042 . . . . . . 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 764 . . . . . . 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 762 . . . . . . 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 28161 . . . . . 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 28254 . . . . 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 4161 . . . . . . . . . 10  |-  ( t  =  y  ->  <. C , 
t >.  =  <. C , 
y >. )
2221breq2d 4405 . . . . . . . . 9  |-  ( t  =  y  ->  ( D  Btwn  <. C ,  t
>. 
<->  D  Btwn  <. C , 
y >. ) )
2321breq1d 4403 . . . . . . . . 9  |-  ( t  =  y  ->  ( <. C ,  t >.Cgr <. A ,  B >.  <->  <. C ,  y >.Cgr <. A ,  B >. ) )
2422, 23anbi12d 710 . . . . . . . 8  |-  ( t  =  y  ->  (
( D  Btwn  <. C , 
t >.  /\  <. C , 
t >.Cgr <. A ,  B >. )  <->  ( D  Btwn  <. C ,  y >.  /\ 
<. C ,  y >.Cgr <. A ,  B >. ) ) )
2524anbi2d 703 . . . . . . 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 703 . . . . . 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 763 . . . . . . . . 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 28183 . . . . . . . 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 761 . . . . . . . . 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 28183 . . . . . . . 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 28185 . . . . . . . . . 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 1221 . . . . . . . . 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 465 . . . . . . . 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 679 . . . . . . 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 762 . . . . . . . 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 4161 . . . . . . . . 9  |-  ( D  =  y  ->  <. C ,  D >.  =  <. C , 
y >. )
3736breq2d 4405 . . . . . . . 8  |-  ( D  =  y  ->  ( <. A ,  B >.Cgr <. C ,  D >.  <->  <. A ,  B >.Cgr <. C , 
y >. ) )
3835, 37syl5ibrcom 222 . . . . . . 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 228 . . . . 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 36 . . . 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 604 . . 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 2946 . 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 215 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2796   <.cop 3984   class class class wbr 4393   ` cfv 5519   NNcn 10426   EEcee 23279    Btwn cbtwn 23280  Cgrccgr 23281    Seg<_ csegle 28274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-sum 13275  df-ee 23282  df-btwn 23283  df-cgr 23284  df-ofs 28151  df-colinear 28207  df-ifs 28208  df-cgr3 28209  df-segle 28275
This theorem is referenced by:  colinbtwnle  28286
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