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Theorem fvtransport 30799
Description: Calculate the value of the TransportTo function. This function takes four points,  A through  D, where  C and  D are distinct. It then returns the point that extends  C D by the length of  A B. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fvtransport  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  -> 
( <. A ,  B >.TransportTo <. C ,  D >. )  =  ( iota_ r  e.  ( EE `  N
) ( D  Btwn  <. C ,  r >.  /\ 
<. D ,  r >.Cgr <. A ,  B >. ) ) )
Distinct variable groups:    N, r    A, r    B, r    C, r    D, r

Proof of Theorem fvtransport
Dummy variables  n  p  q  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 6293 . 2  |-  ( <. A ,  B >.TransportTo <. C ,  D >. )  =  (TransportTo `  <. <. A ,  B >. ,  <. C ,  D >. >. )
2 opelxpi 4866 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
323ad2ant1 1029 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
4 opelxpi 4866 . . . . . . 7  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
543ad2ant2 1030 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
6 simp3 1010 . . . . . . 7  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  C  =/=  D )
7 op1stg 6805 . . . . . . . 8  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 1st `  <. C ,  D >. )  =  C )
873ad2ant2 1030 . . . . . . 7  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  ( 1st `  <. C ,  D >. )  =  C )
9 op2ndg 6806 . . . . . . . 8  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 2nd `  <. C ,  D >. )  =  D )
1093ad2ant2 1030 . . . . . . 7  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  ( 2nd `  <. C ,  D >. )  =  D )
116, 8, 103netr4d 2701 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. ) )
123, 5, 113jca 1188 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  ( <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) )  /\  <. C ,  D >.  e.  ( ( EE
`  N )  X.  ( EE `  N
) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. ) ) )
138opeq1d 4172 . . . . . . . . 9  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  <. ( 1st `  <. C ,  D >. ) ,  r >.  =  <. C ,  r
>. )
1410, 13breq12d 4415 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  (
( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  <->  D  Btwn  <. C ,  r >. ) )
1510opeq1d 4172 . . . . . . . . 9  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  <. ( 2nd `  <. C ,  D >. ) ,  r >.  =  <. D ,  r
>. )
1615breq1d 4412 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  ( <. ( 2nd `  <. C ,  D >. ) ,  r >.Cgr <. A ,  B >. 
<-> 
<. D ,  r >.Cgr <. A ,  B >. ) )
1714, 16anbi12d 717 . . . . . . 7  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  (
( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. )  <->  ( D  Btwn  <. C ,  r >.  /\ 
<. D ,  r >.Cgr <. A ,  B >. ) ) )
1817riotabidv 6254 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  ( iota_ r  e.  ( EE
`  N ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) ) )
1918eqcomd 2457 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  ( iota_ r  e.  ( EE
`  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  N ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) )
2012, 19jca 535 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  (
( <. A ,  B >.  e.  ( ( EE
`  N )  X.  ( EE `  N
) )  /\  <. C ,  D >.  e.  ( ( EE `  N
)  X.  ( EE
`  N ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  /\  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  N ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) )
21 fveq2 5865 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
2221sqxpeqd 4860 . . . . . . . 8  |-  ( n  =  N  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  N )  X.  ( EE `  N
) ) )
2322eleq2d 2514 . . . . . . 7  |-  ( n  =  N  ->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
2422eleq2d 2514 . . . . . . 7  |-  ( n  =  N  ->  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
2523, 243anbi12d 1340 . . . . . 6  |-  ( n  =  N  ->  (
( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  <->  ( <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) )  /\  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
) ) )
2621riotaeqdv 6253 . . . . . . 7  |-  ( n  =  N  ->  ( iota_ r  e.  ( EE
`  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  N ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) )
2726eqeq2d 2461 . . . . . 6  |-  ( n  =  N  ->  (
( iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) )  <->  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  N ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) )
2825, 27anbi12d 717 . . . . 5  |-  ( n  =  N  ->  (
( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  /\  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) )  <->  ( ( <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) )  /\  <. C ,  D >.  e.  ( ( EE
`  N )  X.  ( EE `  N
) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. ) )  /\  ( iota_ r  e.  ( EE
`  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  N ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) ) )
2928rspcev 3150 . . . 4  |-  ( ( N  e.  NN  /\  ( ( <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) )  /\  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  /\  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  N ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) )  ->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  /\  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) )
3020, 29sylan2 477 . . 3  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  ->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  /\  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) )
31 df-br 4403 . . . . 5  |-  ( <. <. A ,  B >. , 
<. C ,  D >. >.TransportTo (
iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  <->  <. <. <. A ,  B >. ,  <. C ,  D >. >. ,  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) ) >.  e. TransportTo )
32 df-transport 30797 . . . . . 6  |- TransportTo  =  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) }
3332eleq2i 2521 . . . . 5  |-  ( <. <. <. A ,  B >. ,  <. C ,  D >. >. ,  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) ) >.  e. TransportTo  <->  <. <. <. A ,  B >. ,  <. C ,  D >. >. ,  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) ) >.  e.  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) } )
34 opex 4664 . . . . . 6  |-  <. A ,  B >.  e.  _V
35 opex 4664 . . . . . 6  |-  <. C ,  D >.  e.  _V
36 riotaex 6256 . . . . . 6  |-  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  e.  _V
37 eleq1 2517 . . . . . . . . . 10  |-  ( p  =  <. A ,  B >.  ->  ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
38373anbi1d 1343 . . . . . . . . 9  |-  ( p  =  <. A ,  B >.  ->  ( ( p  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  <->  ( <. A ,  B >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) ) ) )
39 breq2 4406 . . . . . . . . . . . 12  |-  ( p  =  <. A ,  B >.  ->  ( <. ( 2nd `  q ) ,  r >.Cgr p  <->  <. ( 2nd `  q ) ,  r
>.Cgr <. A ,  B >. ) )
4039anbi2d 710 . . . . . . . . . . 11  |-  ( p  =  <. A ,  B >.  ->  ( ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p )  <->  ( ( 2nd `  q )  Btwn  <.
( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr <. A ,  B >. ) ) )
4140riotabidv 6254 . . . . . . . . . 10  |-  ( p  =  <. A ,  B >.  ->  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q ) ,  r >.Cgr p ) )  =  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q ) ,  r >.Cgr <. A ,  B >. ) ) )
4241eqeq2d 2461 . . . . . . . . 9  |-  ( p  =  <. A ,  B >.  ->  ( x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) )  <->  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr <. A ,  B >. ) ) ) )
4338, 42anbi12d 717 . . . . . . . 8  |-  ( p  =  <. A ,  B >.  ->  ( ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n
) )  /\  ( 1st `  q )  =/=  ( 2nd `  q
) )  /\  x  =  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q ) ,  r >.Cgr p ) ) )  <->  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr <. A ,  B >. ) ) ) ) )
4443rexbidv 2901 . . . . . . 7  |-  ( p  =  <. A ,  B >.  ->  ( E. n  e.  NN  ( ( p  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) )  <->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n )  X.  ( EE `  n
) )  /\  ( 1st `  q )  =/=  ( 2nd `  q
) )  /\  x  =  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q ) ,  r >.Cgr <. A ,  B >. ) ) ) ) )
45 eleq1 2517 . . . . . . . . . 10  |-  ( q  =  <. C ,  D >.  ->  ( q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
46 fveq2 5865 . . . . . . . . . . 11  |-  ( q  =  <. C ,  D >.  ->  ( 1st `  q
)  =  ( 1st `  <. C ,  D >. ) )
47 fveq2 5865 . . . . . . . . . . 11  |-  ( q  =  <. C ,  D >.  ->  ( 2nd `  q
)  =  ( 2nd `  <. C ,  D >. ) )
4846, 47neeq12d 2685 . . . . . . . . . 10  |-  ( q  =  <. C ,  D >.  ->  ( ( 1st `  q )  =/=  ( 2nd `  q )  <->  ( 1st ` 
<. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
) )
4945, 483anbi23d 1342 . . . . . . . . 9  |-  ( q  =  <. C ,  D >.  ->  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  <->  ( <. A ,  B >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. ) ) ) )
5046opeq1d 4172 . . . . . . . . . . . . 13  |-  ( q  =  <. C ,  D >.  ->  <. ( 1st `  q
) ,  r >.  =  <. ( 1st `  <. C ,  D >. ) ,  r >. )
5147, 50breq12d 4415 . . . . . . . . . . . 12  |-  ( q  =  <. C ,  D >.  ->  ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  <->  ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >. )
)
5247opeq1d 4172 . . . . . . . . . . . . 13  |-  ( q  =  <. C ,  D >.  ->  <. ( 2nd `  q
) ,  r >.  =  <. ( 2nd `  <. C ,  D >. ) ,  r >. )
5352breq1d 4412 . . . . . . . . . . . 12  |-  ( q  =  <. C ,  D >.  ->  ( <. ( 2nd `  q ) ,  r >.Cgr <. A ,  B >.  <->  <. ( 2nd `  <. C ,  D >. ) ,  r >.Cgr <. A ,  B >. ) )
5451, 53anbi12d 717 . . . . . . . . . . 11  |-  ( q  =  <. C ,  D >.  ->  ( ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr <. A ,  B >. )  <-> 
( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) )
5554riotabidv 6254 . . . . . . . . . 10  |-  ( q  =  <. C ,  D >.  ->  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  /\  <. ( 2nd `  q ) ,  r >.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) )
5655eqeq2d 2461 . . . . . . . . 9  |-  ( q  =  <. C ,  D >.  ->  ( x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q
)  Btwn  <. ( 1st `  q ) ,  r
>.  /\  <. ( 2nd `  q
) ,  r >.Cgr <. A ,  B >. ) )  <->  x  =  ( iota_ r  e.  ( EE
`  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) )
5749, 56anbi12d 717 . . . . . . . 8  |-  ( q  =  <. C ,  D >.  ->  ( ( (
<. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr <. A ,  B >. ) ) )  <->  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. ) )  /\  x  =  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r >.Cgr <. A ,  B >. ) ) ) ) )
5857rexbidv 2901 . . . . . . 7  |-  ( q  =  <. C ,  D >.  ->  ( E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr <. A ,  B >. ) ) )  <->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. ) )  /\  x  =  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r >.Cgr <. A ,  B >. ) ) ) ) )
59 eqeq1 2455 . . . . . . . . 9  |-  ( x  =  ( iota_ r  e.  ( EE `  N
) ( D  Btwn  <. C ,  r >.  /\ 
<. D ,  r >.Cgr <. A ,  B >. ) )  ->  ( x  =  ( iota_ r  e.  ( EE `  n
) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r >.Cgr <. A ,  B >. ) )  <->  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) )
6059anbi2d 710 . . . . . . . 8  |-  ( x  =  ( iota_ r  e.  ( EE `  N
) ( D  Btwn  <. C ,  r >.  /\ 
<. D ,  r >.Cgr <. A ,  B >. ) )  ->  ( (
( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) )  <->  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. ) )  /\  ( iota_ r  e.  ( EE
`  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) ) )
6160rexbidv 2901 . . . . . . 7  |-  ( x  =  ( iota_ r  e.  ( EE `  N
) ( D  Btwn  <. C ,  r >.  /\ 
<. D ,  r >.Cgr <. A ,  B >. ) )  ->  ( E. n  e.  NN  (
( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) )  <->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. ) )  /\  ( iota_ r  e.  ( EE
`  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) ) )
6244, 58, 61eloprabg 6384 . . . . . 6  |-  ( (
<. A ,  B >.  e. 
_V  /\  <. C ,  D >.  e.  _V  /\  ( iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  e.  _V )  ->  ( <. <. <. A ,  B >. ,  <. C ,  D >. >. ,  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) ) >.  e.  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) }  <->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  /\  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) ) )
6334, 35, 36, 62mp3an 1364 . . . . 5  |-  ( <. <. <. A ,  B >. ,  <. C ,  D >. >. ,  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) ) >.  e.  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  q  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  q
)  =/=  ( 2nd `  q ) )  /\  x  =  ( iota_ r  e.  ( EE `  n ) ( ( 2nd `  q ) 
Btwn  <. ( 1st `  q
) ,  r >.  /\  <. ( 2nd `  q
) ,  r >.Cgr p ) ) ) }  <->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  /\  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) )
6431, 33, 633bitri 275 . . . 4  |-  ( <. <. A ,  B >. , 
<. C ,  D >. >.TransportTo (
iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  <->  E. n  e.  NN  ( ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  <. C ,  D >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. ) )  /\  ( iota_ r  e.  ( EE
`  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) ) )
65 funtransport 30798 . . . . 5  |-  Fun TransportTo
66 funbrfv 5903 . . . . 5  |-  ( Fun TransportTo  -> 
( <. <. A ,  B >. ,  <. C ,  D >. >.TransportTo ( iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  ->  (TransportTo ` 
<. <. A ,  B >. ,  <. C ,  D >. >. )  =  (
iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) ) ) )
6765, 66ax-mp 5 . . . 4  |-  ( <. <. A ,  B >. , 
<. C ,  D >. >.TransportTo (
iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )  ->  (TransportTo ` 
<. <. A ,  B >. ,  <. C ,  D >. >. )  =  (
iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) ) )
6864, 67sylbir 217 . . 3  |-  ( E. n  e.  NN  (
( <. A ,  B >.  e.  ( ( EE
`  n )  X.  ( EE `  n
) )  /\  <. C ,  D >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) )  /\  ( 1st `  <. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
)  /\  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  =  (
iota_ r  e.  ( EE `  n ) ( ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >.  /\  <. ( 2nd `  <. C ,  D >. ) ,  r
>.Cgr <. A ,  B >. ) ) )  -> 
(TransportTo `  <. <. A ,  B >. ,  <. C ,  D >. >. )  =  (
iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) ) )
6930, 68syl 17 . 2  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  -> 
(TransportTo `  <. <. A ,  B >. ,  <. C ,  D >. >. )  =  (
iota_ r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) ) )
701, 69syl5eq 2497 1  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  -> 
( <. A ,  B >.TransportTo <. C ,  D >. )  =  ( iota_ r  e.  ( EE `  N
) ( D  Btwn  <. C ,  r >.  /\ 
<. D ,  r >.Cgr <. A ,  B >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   _Vcvv 3045   <.cop 3974   class class class wbr 4402    X. cxp 4832   Fun wfun 5576   ` cfv 5582   iota_crio 6251  (class class class)co 6290   {coprab 6291   1stc1st 6791   2ndc2nd 6792   NNcn 10609   EEcee 24918    Btwn cbtwn 24919  Cgrccgr 24920  TransportToctransport 30796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-z 10938  df-uz 11160  df-fz 11785  df-ee 24921  df-transport 30797
This theorem is referenced by:  transportcl  30800  transportprops  30801
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