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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvtransport Structured version   Unicode version
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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