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Theorem fvtransport 29887
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 6299 . 2  |-  ( <. A ,  B >.TransportTo <. C ,  D >. )  =  (TransportTo `  <. <. A ,  B >. ,  <. C ,  D >. >. )
2 opelxpi 5040 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
323ad2ant1 1017 . . . . . 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 5040 . . . . . . 7  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  ->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) )
543ad2ant2 1018 . . . . . 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 998 . . . . . . 7  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  ->  C  =/=  D )
7 op1stg 6811 . . . . . . . 8  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 1st `  <. C ,  D >. )  =  C )
873ad2ant2 1018 . . . . . . 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 6812 . . . . . . . 8  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( 2nd `  <. C ,  D >. )  =  D )
1093ad2ant2 1018 . . . . . . 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 2762 . . . . . 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 1176 . . . . 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 4225 . . . . . . . . 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 4469 . . . . . . . 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 4225 . . . . . . . . 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 4466 . . . . . . . 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 6260 . . . . . 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 2465 . . . . 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 5872 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
2221sqxpeqd 5034 . . . . . . . 8  |-  ( n  =  N  ->  (
( EE `  n
)  X.  ( EE
`  n ) )  =  ( ( EE
`  N )  X.  ( EE `  N
) ) )
2322eleq2d 2527 . . . . . . 7  |-  ( n  =  N  ->  ( <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
2422eleq2d 2527 . . . . . . 7  |-  ( n  =  N  ->  ( <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  N )  X.  ( EE `  N ) ) ) )
2523, 243anbi12d 1300 . . . . . 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 6259 . . . . . . 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 2471 . . . . . 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 3210 . . . 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 4457 . . . . 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 29885 . . . . . 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 2535 . . . . 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 4720 . . . . . 6  |-  <. A ,  B >.  e.  _V
35 opex 4720 . . . . . 6  |-  <. C ,  D >.  e.  _V
36 riotaex 6262 . . . . . 6  |-  ( iota_ r  e.  ( EE `  N ) ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. ) )  e.  _V
37 eleq1 2529 . . . . . . . . . 10  |-  ( p  =  <. A ,  B >.  ->  ( p  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. A ,  B >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
38373anbi1d 1303 . . . . . . . . 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 4460 . . . . . . . . . . . 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 6260 . . . . . . . . . 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 2471 . . . . . . . . 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 2968 . . . . . . 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 2529 . . . . . . . . . 10  |-  ( q  =  <. C ,  D >.  ->  ( q  e.  ( ( EE `  n )  X.  ( EE `  n ) )  <->  <. C ,  D >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) ) )
46 fveq2 5872 . . . . . . . . . . 11  |-  ( q  =  <. C ,  D >.  ->  ( 1st `  q
)  =  ( 1st `  <. C ,  D >. ) )
47 fveq2 5872 . . . . . . . . . . 11  |-  ( q  =  <. C ,  D >.  ->  ( 2nd `  q
)  =  ( 2nd `  <. C ,  D >. ) )
4846, 47neeq12d 2736 . . . . . . . . . 10  |-  ( q  =  <. C ,  D >.  ->  ( ( 1st `  q )  =/=  ( 2nd `  q )  <->  ( 1st ` 
<. C ,  D >. )  =/=  ( 2nd `  <. C ,  D >. )
) )
4945, 483anbi23d 1302 . . . . . . . . 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 4225 . . . . . . . . . . . . 13  |-  ( q  =  <. C ,  D >.  ->  <. ( 1st `  q
) ,  r >.  =  <. ( 1st `  <. C ,  D >. ) ,  r >. )
5147, 50breq12d 4469 . . . . . . . . . . . 12  |-  ( q  =  <. C ,  D >.  ->  ( ( 2nd `  q )  Btwn  <. ( 1st `  q ) ,  r >.  <->  ( 2nd `  <. C ,  D >. )  Btwn  <. ( 1st `  <. C ,  D >. ) ,  r >. )
)
5247opeq1d 4225 . . . . . . . . . . . . 13  |-  ( q  =  <. C ,  D >.  ->  <. ( 2nd `  q
) ,  r >.  =  <. ( 2nd `  <. C ,  D >. ) ,  r >. )
5352breq1d 4466 . . . . . . . . . . . 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 6260 . . . . . . . . . 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 2471 . . . . . . . . 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 2968 . . . . . . 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 2461 . . . . . . . . 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 2968 . . . . . . 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 6389 . . . . . 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 1324 . . . . 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 29886 . . . . 5  |-  Fun TransportTo
66 funbrfv 5911 . . . . 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 2510 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   _Vcvv 3109   <.cop 4038   class class class wbr 4456    X. cxp 5006   Fun wfun 5588   ` cfv 5594   iota_crio 6257  (class class class)co 6296   {coprab 6297   1stc1st 6797   2ndc2nd 6798   NNcn 10556   EEcee 24318    Btwn cbtwn 24319  Cgrccgr 24320  TransportToctransport 29884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-z 10886  df-uz 11107  df-fz 11698  df-ee 24321  df-transport 29885
This theorem is referenced by:  transportcl  29888  transportprops  29889
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