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Theorem fvtransport 25971
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 6087 . 2  |-  ( <. A ,  B >.TransportTo <. C ,  D >. )  =  (TransportTo `  <. <. A ,  B >. ,  <. C ,  D >. >. )
2 opelxpi 4913 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
323ad2ant1 979 . . . . . 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 4913 . . . . . . 7  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
543ad2ant2 980 . . . . . 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 960 . . . . . . 7  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  C  =/=  D )
7 op1stg 6362 . . . . . . . 8  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 1st `  <. C ,  D >. )  =  C )
873ad2ant2 980 . . . . . . 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 6363 . . . . . . . 8  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 2nd `  <. C ,  D >. )  =  D )
1093ad2ant2 980 . . . . . . 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 2630 . . . . . 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 1135 . . . . 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 3992 . . . . . . . . 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 4228 . . . . . . . 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 3992 . . . . . . . . 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 4225 . . . . . . . 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 693 . . . . . . 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 6554 . . . . . 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 2443 . . . . 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 520 . . . 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 5731 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
2221, 21xpeq12d 4906 . . . . . . . 8  |-  ( n  =  N  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  N )  X.  ( EE `  N
) ) )
2322eleq2d 2505 . . . . . . 7  |-  ( n  =  N  ->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
2422eleq2d 2505 . . . . . . 7  |-  ( n  =  N  ->  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
2523, 243anbi12d 1256 . . . . . 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 6553 . . . . . . 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 2449 . . . . . 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 693 . . . . 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 3054 . . . 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 462 . . 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 4216 . . . . 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 25969 . . . . . 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 2502 . . . . 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 4430 . . . . . 6  |-  <. A ,  B >.  e.  _V
35 opex 4430 . . . . . 6  |-  <. C ,  D >.  e.  _V
36 riotaex 6556 . . . . . 6  |-  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  e.  _V
37 eleq1 2498 . . . . . . . . . 10  |-  ( p  =  <. A ,  B >.  ->  ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
38373anbi1d 1259 . . . . . . . . 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 4219 . . . . . . . . . . . 12  |-  ( p  =  <. A ,  B >.  ->  ( <. ( 2nd `  q ) ,  r >.Cgr p  <->  <. ( 2nd `  q ) ,  r
>.Cgr <. A ,  B >. ) )
4039anbi2d 686 . . . . . . . . . . 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 6554 . . . . . . . . . 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 2449 . . . . . . . . 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 693 . . . . . . . 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 2728 . . . . . . 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 2498 . . . . . . . . . 10  |-  ( q  =  <. C ,  D >.  ->  ( q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
46 fveq2 5731 . . . . . . . . . . 11  |-  ( q  =  <. C ,  D >.  ->  ( 1st `  q
)  =  ( 1st `  <. C ,  D >. ) )
47 fveq2 5731 . . . . . . . . . . 11  |-  ( q  =  <. C ,  D >.  ->  ( 2nd `  q
)  =  ( 2nd `  <. C ,  D >. ) )
4846, 47neeq12d 2618 . . . . . . . . . 10  |-  ( q  =  <. C ,  D >.  ->  ( ( 1st `  q )  =/=  ( 2nd `  q )  <->  ( 1st ` 
<. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
) )
4945, 483anbi23d 1258 . . . . . . . . 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 3992 . . . . . . . . . . . . 13  |-  ( q  =  <. C ,  D >.  ->  <. ( 1st `  q
) ,  r >.  =  <. ( 1st `  <. C ,  D >. ) ,  r >. )
5147, 50breq12d 4228 . . . . . . . . . . . 12  |-  ( q  =  <. C ,  D >.  ->  ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  <->  ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >. )
)
5247opeq1d 3992 . . . . . . . . . . . . 13  |-  ( q  =  <. C ,  D >.  ->  <. ( 2nd `  q
) ,  r >.  =  <. ( 2nd `  <. C ,  D >. ) ,  r >. )
5352breq1d 4225 . . . . . . . . . . . 12  |-  ( q  =  <. C ,  D >.  ->  ( <. ( 2nd `  q ) ,  r >.Cgr <. A ,  B >.  <->  <. ( 2nd `  <. C ,  D >. ) ,  r >.Cgr <. A ,  B >. ) )
5451, 53anbi12d 693 . . . . . . . . . . 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 6554 . . . . . . . . . 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 2449 . . . . . . . . 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 693 . . . . . . . 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 2728 . . . . . . 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 2444 . . . . . . . . 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 686 . . . . . . . 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 2728 . . . . . . 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 6164 . . . . . 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 1280 . . . . 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 264 . . . 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 25970 . . . . 5  |-  Fun TransportTo
66 funbrfv 5768 . . . . 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 206 . . 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 16 . 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 2482 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   _Vcvv 2958   <.cop 3819   class class class wbr 4215    X. cxp 4879   Fun wfun 5451   ` cfv 5457  (class class class)co 6084   {coprab 6085   1stc1st 6350   2ndc2nd 6351   iota_crio 6545   NNcn 10005   EEcee 25832    Btwn cbtwn 25833  Cgrccgr 25834  TransportToctransport 25968
This theorem is referenced by:  transportcl  25972  transportprops  25973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-z 10288  df-uz 10494  df-fz 11049  df-ee 25835  df-transport 25969
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