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Theorem fvtransport 28075
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 6106 . 2  |-  ( <. A ,  B >.TransportTo <. C ,  D >. )  =  (TransportTo `  <. <. A ,  B >. ,  <. C ,  D >. >. )
2 opelxpi 4883 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
323ad2ant1 1009 . . . . . 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 4883 . . . . . . 7  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
543ad2ant2 1010 . . . . . 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 990 . . . . . . 7  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  C  =/=  D )
7 op1stg 6601 . . . . . . . 8  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 1st `  <. C ,  D >. )  =  C )
873ad2ant2 1010 . . . . . . 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 6602 . . . . . . . 8  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 2nd `  <. C ,  D >. )  =  D )
1093ad2ant2 1010 . . . . . . 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 2647 . . . . . 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 1168 . . . . 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 4077 . . . . . . . . 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 4317 . . . . . . . 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 4077 . . . . . . . . 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 4314 . . . . . . . 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 710 . . . . . . 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 6066 . . . . . 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 2448 . . . . 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 532 . . . 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 5703 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
2221, 21xpeq12d 4877 . . . . . . . 8  |-  ( n  =  N  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  N )  X.  ( EE `  N
) ) )
2322eleq2d 2510 . . . . . . 7  |-  ( n  =  N  ->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
2422eleq2d 2510 . . . . . . 7  |-  ( n  =  N  ->  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
2523, 243anbi12d 1290 . . . . . 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 6065 . . . . . . 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 2454 . . . . . 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 710 . . . . 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 3085 . . . 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 474 . . 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 4305 . . . . 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 28073 . . . . . 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 2507 . . . . 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 4568 . . . . . 6  |-  <. A ,  B >.  e.  _V
35 opex 4568 . . . . . 6  |-  <. C ,  D >.  e.  _V
36 riotaex 6068 . . . . . 6  |-  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  e.  _V
37 eleq1 2503 . . . . . . . . . 10  |-  ( p  =  <. A ,  B >.  ->  ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
38373anbi1d 1293 . . . . . . . . 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 4308 . . . . . . . . . . . 12  |-  ( p  =  <. A ,  B >.  ->  ( <. ( 2nd `  q ) ,  r >.Cgr p  <->  <. ( 2nd `  q ) ,  r
>.Cgr <. A ,  B >. ) )
4039anbi2d 703 . . . . . . . . . . 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 6066 . . . . . . . . . 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 2454 . . . . . . . . 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 710 . . . . . . . 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 2748 . . . . . . 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 2503 . . . . . . . . . 10  |-  ( q  =  <. C ,  D >.  ->  ( q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
46 fveq2 5703 . . . . . . . . . . 11  |-  ( q  =  <. C ,  D >.  ->  ( 1st `  q
)  =  ( 1st `  <. C ,  D >. ) )
47 fveq2 5703 . . . . . . . . . . 11  |-  ( q  =  <. C ,  D >.  ->  ( 2nd `  q
)  =  ( 2nd `  <. C ,  D >. ) )
4846, 47neeq12d 2635 . . . . . . . . . 10  |-  ( q  =  <. C ,  D >.  ->  ( ( 1st `  q )  =/=  ( 2nd `  q )  <->  ( 1st ` 
<. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
) )
4945, 483anbi23d 1292 . . . . . . . . 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 4077 . . . . . . . . . . . . 13  |-  ( q  =  <. C ,  D >.  ->  <. ( 1st `  q
) ,  r >.  =  <. ( 1st `  <. C ,  D >. ) ,  r >. )
5147, 50breq12d 4317 . . . . . . . . . . . 12  |-  ( q  =  <. C ,  D >.  ->  ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  <->  ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >. )
)
5247opeq1d 4077 . . . . . . . . . . . . 13  |-  ( q  =  <. C ,  D >.  ->  <. ( 2nd `  q
) ,  r >.  =  <. ( 2nd `  <. C ,  D >. ) ,  r >. )
5352breq1d 4314 . . . . . . . . . . . 12  |-  ( q  =  <. C ,  D >.  ->  ( <. ( 2nd `  q ) ,  r >.Cgr <. A ,  B >.  <->  <. ( 2nd `  <. C ,  D >. ) ,  r >.Cgr <. A ,  B >. ) )
5451, 53anbi12d 710 . . . . . . . . . . 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 6066 . . . . . . . . . 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 2454 . . . . . . . . 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 710 . . . . . . . 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 2748 . . . . . . 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 2449 . . . . . . . . 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 703 . . . . . . . 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 2748 . . . . . . 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 6190 . . . . . 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 1314 . . . . 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 271 . . . 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 28074 . . . . 5  |-  Fun TransportTo
66 funbrfv 5742 . . . . 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 213 . . 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 2487 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   E.wrex 2728   _Vcvv 2984   <.cop 3895   class class class wbr 4304    X. cxp 4850   Fun wfun 5424   ` cfv 5430   iota_crio 6063  (class class class)co 6103   {coprab 6104   1stc1st 6587   2ndc2nd 6588   NNcn 10334   EEcee 23146    Btwn cbtwn 23147  Cgrccgr 23148  TransportToctransport 28072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-z 10659  df-uz 10874  df-fz 11450  df-ee 23149  df-transport 28073
This theorem is referenced by:  transportcl  28076  transportprops  28077
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