Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvtransport Structured version   Unicode version
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
  Copyright terms: Public domain W3C validator