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Theorem fvtransport 30799
Description: Calculate the value of the TransportTo function. This function takes four points, A through D, where C and D are distinct. It then returns the point that extends C D by the length of A B. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fvtransport |- ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) ) -> ( <. A , B >.TransportTo <. C , D >. ) = ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) )
Distinct variable groups:   N, r   A, r   B, r   C, r   D, r
Proof of Theorem fvtransport
Dummy variables n p q x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 6293 . 2 |- ( <. A , B >.TransportTo <. C , D >. ) = (TransportTo ` <. <. A , B >. , <. C , D >. >. )
2 opelxpi 4866 . . . . . . 7 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) )
323ad2ant1 1029 . . . . . 6 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) )
4 opelxpi 4866 . . . . . . 7 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) )
543ad2ant2 1030 . . . . . 6 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) )
6 simp3 1010 . . . . . . 7 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> C =/= D )
7 op1stg 6805 . . . . . . . 8 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( 1st ` <. C , D >. ) = C )
873ad2ant2 1030 . . . . . . 7 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> ( 1st ` <. C , D >. ) = C )
9 op2ndg 6806 . . . . . . . 8 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( 2nd ` <. C , D >. ) = D )
1093ad2ant2 1030 . . . . . . 7 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> ( 2nd ` <. C , D >. ) = D )
116, 8, 103netr4d 2701 . . . . . 6 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) )
123, 5, 113jca 1188 . . . . 5 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) )
138opeq1d 4172 . . . . . . . . 9 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> <. ( 1st ` <. C , D >. ) , r >. = <. C , r >. )
1410, 13breq12d 4415 . . . . . . . 8 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. <-> D Btwn <. C , r >. ) )
1510opeq1d 4172 . . . . . . . . 9 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> <. ( 2nd ` <. C , D >. ) , r >. = <. D , r >. )
1615breq1d 4412 . . . . . . . 8 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> ( <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. <-> <. D , r >.Cgr <. A , B >. ) )
1714, 16anbi12d 717 . . . . . . 7 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> ( ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) <-> ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) )
1817riotabidv 6254 . . . . . 6 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> ( iota_ r e. ( EE ` N ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) )
1918eqcomd 2457 . . . . 5 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` N ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) )
2012, 19jca 535 . . . 4 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> ( ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` N ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) )
21 fveq2 5865 . . . . . . . . 9 |- ( n = N -> ( EE ` n ) = ( EE ` N ) )
2221sqxpeqd 4860 . . . . . . . 8 |- ( n = N -> ( ( EE ` n ) X. ( EE ` n ) ) = ( ( EE ` N ) X. ( EE ` N ) ) )
2322eleq2d 2514 . . . . . . 7 |- ( n = N -> ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) )
2422eleq2d 2514 . . . . . . 7 |- ( n = N -> ( <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) )
2523, 243anbi12d 1340 . . . . . 6 |- ( n = N -> ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) <-> ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) ) )
2621riotaeqdv 6253 . . . . . . 7 |- ( n = N -> ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` N ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) )
2726eqeq2d 2461 . . . . . 6 |- ( n = N -> ( ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) <-> ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` N ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) )
2825, 27anbi12d 717 . . . . 5 |- ( n = N -> ( ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) <-> ( ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` N ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) ) )
2928rspcev 3150 . . . 4 |- ( ( N e. NN /\ ( ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` N ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) ) -> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) )
3020, 29sylan2 477 . . 3 |- ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) ) -> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) )
31 df-br 4403 . . . . 5 |- ( <. <. A , B >. , <. C , D >. >.TransportTo ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) <-> <. <. <. A , B >. , <. C , D >. >. , ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) >. e. TransportTo )
32 df-transport 30797 . . . . . 6 |- TransportTo = { <. <. p , q >. , x >. | E. n e. NN ( ( p e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr p ) ) ) }
3332eleq2i 2521 . . . . 5 |- ( <. <. <. A , B >. , <. C , D >. >. , ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) >. e. TransportTo <-> <. <. <. A , B >. , <. C , D >. >. , ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) >. e. { <. <. p , q >. , x >. | E. n e. NN ( ( p e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr p ) ) ) } )
34 opex 4664 . . . . . 6 |- <. A , B >. e. _V
35 opex 4664 . . . . . 6 |- <. C , D >. e. _V
36 riotaex 6256 . . . . . 6 |- ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) e. _V
37 eleq1 2517 . . . . . . . . . 10 |- ( p = <. A , B >. -> ( p e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) )
38373anbi1d 1343 . . . . . . . . 9 |- ( p = <. A , B >. -> ( ( p e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) <-> ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) ) )
39 breq2 4406 . . . . . . . . . . . 12 |- ( p = <. A , B >. -> ( <. ( 2nd ` q ) , r >.Cgr p <-> <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) )
4039anbi2d 710 . . . . . . . . . . 11 |- ( p = <. A , B >. -> ( ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr p ) <-> ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) ) )
4140riotabidv 6254 . . . . . . . . . 10 |- ( p = <. A , B >. -> ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr p ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) ) )
4241eqeq2d 2461 . . . . . . . . 9 |- ( p = <. A , B >. -> ( x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr p ) ) <-> x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) ) ) )
4338, 42anbi12d 717 . . . . . . . 8 |- ( p = <. A , B >. -> ( ( ( p e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr p ) ) ) <-> ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) ) ) ) )
4443rexbidv 2901 . . . . . . 7 |- ( p = <. A , B >. -> ( E. n e. NN ( ( p e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr p ) ) ) <-> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) ) ) ) )
45 eleq1 2517 . . . . . . . . . 10 |- ( q = <. C , D >. -> ( q e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) )
46 fveq2 5865 . . . . . . . . . . 11 |- ( q = <. C , D >. -> ( 1st ` q ) = ( 1st ` <. C , D >. ) )
47 fveq2 5865 . . . . . . . . . . 11 |- ( q = <. C , D >. -> ( 2nd ` q ) = ( 2nd ` <. C , D >. ) )
4846, 47neeq12d 2685 . . . . . . . . . 10 |- ( q = <. C , D >. -> ( ( 1st ` q ) =/= ( 2nd ` q ) <-> ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) )
4945, 483anbi23d 1342 . . . . . . . . 9 |- ( q = <. C , D >. -> ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) <-> ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) ) )
5046opeq1d 4172 . . . . . . . . . . . . 13 |- ( q = <. C , D >. -> <. ( 1st ` q ) , r >. = <. ( 1st ` <. C , D >. ) , r >. )
5147, 50breq12d 4415 . . . . . . . . . . . 12 |- ( q = <. C , D >. -> ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. <-> ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. ) )
5247opeq1d 4172 . . . . . . . . . . . . 13 |- ( q = <. C , D >. -> <. ( 2nd ` q ) , r >. = <. ( 2nd ` <. C , D >. ) , r >. )
5352breq1d 4412 . . . . . . . . . . . 12 |- ( q = <. C , D >. -> ( <. ( 2nd ` q ) , r >.Cgr <. A , B >. <-> <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) )
5451, 53anbi12d 717 . . . . . . . . . . 11 |- ( q = <. C , D >. -> ( ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) <-> ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) )
5554riotabidv 6254 . . . . . . . . . 10 |- ( q = <. C , D >. -> ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) )
5655eqeq2d 2461 . . . . . . . . 9 |- ( q = <. C , D >. -> ( x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) ) <-> x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) )
5749, 56anbi12d 717 . . . . . . . 8 |- ( q = <. C , D >. -> ( ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) ) ) <-> ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) ) )
5857rexbidv 2901 . . . . . . 7 |- ( q = <. C , D >. -> ( E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) ) ) <-> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) ) )
59 eqeq1 2455 . . . . . . . . 9 |- ( x = ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) -> ( x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) <-> ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) )
6059anbi2d 710 . . . . . . . 8 |- ( x = ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) -> ( ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) <-> ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) ) )
6160rexbidv 2901 . . . . . . 7 |- ( x = ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) -> ( E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) <-> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) ) )
6244, 58, 61eloprabg 6384 . . . . . 6 |- ( ( <. A , B >. e. _V /\ <. C , D >. e. _V /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) e. _V ) -> ( <. <. <. A , B >. , <. C , D >. >. , ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) >. e. { <. <. p , q >. , x >. | E. n e. NN ( ( p e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr p ) ) ) } <-> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) ) )
6334, 35, 36, 62mp3an 1364 . . . . 5 |- ( <. <. <. A , B >. , <. C , D >. >. , ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) >. e. { <. <. p , q >. , x >. | E. n e. NN ( ( p e. ( ( EE ` n ) X. ( EE ` n ) ) /\ q e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` q ) =/= ( 2nd ` q ) ) /\ x = ( iota_ r e. ( EE ` n ) ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. /\ <. ( 2nd ` q ) , r >.Cgr p ) ) ) } <-> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) )
6431, 33, 633bitri 275 . . . 4 |- ( <. <. A , B >. , <. C , D >. >.TransportTo ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) <-> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) )
65 funtransport 30798 . . . . 5 |- Fun TransportTo
66 funbrfv 5903 . . . . 5 |- ( Fun TransportTo -> ( <. <. A , B >. , <. C , D >. >.TransportTo ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) -> (TransportTo ` <. <. A , B >. , <. C , D >. >. ) = ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) ) )
6765, 66ax-mp 5 . . . 4 |- ( <. <. A , B >. , <. C , D >. >.TransportTo ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) -> (TransportTo ` <. <. A , B >. , <. C , D >. >. ) = ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) )
6864, 67sylbir 217 . . 3 |- ( E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) /\ ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) = ( iota_ r e. ( EE ` n ) ( ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. /\ <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) ) ) -> (TransportTo ` <. <. A , B >. , <. C , D >. >. ) = ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) )
6930, 68syl 17 . 2 |- ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) ) -> (TransportTo ` <. <. A , B >. , <. C , D >. >. ) = ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) )
701, 69syl5eq 2497 1 |- ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) ) -> ( <. A , B >.TransportTo <. C , D >. ) = ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   E.wrex 2738   _Vcvv 3045   <.cop 3974   class class class wbr 4402   X. cxp 4832   Fun wfun 5576   ` cfv 5582   iota_crio 6251  (class class class)co 6290   {coprab 6291   1stc1st 6791   2ndc2nd 6792   NNcn 10609   EEcee 24918   Btwn cbtwn 24919  Cgrccgr 24920  TransportToctransport 30796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-z 10938  df-uz 11160  df-fz 11785  df-ee 24921  df-transport 30797
This theorem is referenced by:  transportcl  30800  transportprops  30801
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