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Theorem fvtransport 32139
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 6653 . 2 |- ( <. A , B >.TransportTo <. C , D >. ) = (TransportTo ` <. <. A , B >. , <. C , D >. >. )
2 opelxpi 5148 . . . . . . 7 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) )
323ad2ant1 1082 . . . . . 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 5148 . . . . . . 7 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) )
543ad2ant2 1083 . . . . . 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 1063 . . . . . . 7 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ C =/= D ) -> C =/= D )
7 op1stg 7180 . . . . . . . 8 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( 1st ` <. C , D >. ) = C )
873ad2ant2 1083 . . . . . . 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 7181 . . . . . . . 8 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( 2nd ` <. C , D >. ) = D )
1093ad2ant2 1083 . . . . . . 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 2871 . . . . . 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 1242 . . . . 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 4408 . . . . . . . . 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 4666 . . . . . . . 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 4408 . . . . . . . . 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 4663 . . . . . . . 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 747 . . . . . . 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 6613 . . . . . 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 2628 . . . . 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 554 . . . 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 6191 . . . . . . . . 9 |- ( n = N -> ( EE ` n ) = ( EE ` N ) )
2221sqxpeqd 5141 . . . . . . . 8 |- ( n = N -> ( ( EE ` n ) X. ( EE ` n ) ) = ( ( EE ` N ) X. ( EE ` N ) ) )
2322eleq2d 2687 . . . . . . 7 |- ( n = N -> ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) )
2422eleq2d 2687 . . . . . . 7 |- ( n = N -> ( <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) )
2523, 243anbi12d 1400 . . . . . 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 6612 . . . . . . 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 2632 . . . . . 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 747 . . . . 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 3309 . . . 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 491 . . 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 4654 . . . . 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 32137 . . . . . 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 2693 . . . . 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 4932 . . . . . 6 |- <. A , B >. e. _V
35 opex 4932 . . . . . 6 |- <. C , D >. e. _V
36 riotaex 6615 . . . . . 6 |- ( iota_ r e. ( EE ` N ) ( D Btwn <. C , r >. /\ <. D , r >.Cgr <. A , B >. ) ) e. _V
37 eleq1 2689 . . . . . . . . . 10 |- ( p = <. A , B >. -> ( p e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) )
38373anbi1d 1403 . . . . . . . . 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 4657 . . . . . . . . . . . 12 |- ( p = <. A , B >. -> ( <. ( 2nd ` q ) , r >.Cgr p <-> <. ( 2nd ` q ) , r >.Cgr <. A , B >. ) )
4039anbi2d 740 . . . . . . . . . . 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 6613 . . . . . . . . . 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 2632 . . . . . . . . 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 747 . . . . . . . 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 3052 . . . . . . 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 2689 . . . . . . . . . 10 |- ( q = <. C , D >. -> ( q e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) )
46 fveq2 6191 . . . . . . . . . . 11 |- ( q = <. C , D >. -> ( 1st ` q ) = ( 1st ` <. C , D >. ) )
47 fveq2 6191 . . . . . . . . . . 11 |- ( q = <. C , D >. -> ( 2nd ` q ) = ( 2nd ` <. C , D >. ) )
4846, 47neeq12d 2855 . . . . . . . . . 10 |- ( q = <. C , D >. -> ( ( 1st ` q ) =/= ( 2nd ` q ) <-> ( 1st ` <. C , D >. ) =/= ( 2nd ` <. C , D >. ) ) )
4945, 483anbi23d 1402 . . . . . . . . 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 4408 . . . . . . . . . . . . 13 |- ( q = <. C , D >. -> <. ( 1st ` q ) , r >. = <. ( 1st ` <. C , D >. ) , r >. )
5147, 50breq12d 4666 . . . . . . . . . . . 12 |- ( q = <. C , D >. -> ( ( 2nd ` q ) Btwn <. ( 1st ` q ) , r >. <-> ( 2nd ` <. C , D >. ) Btwn <. ( 1st ` <. C , D >. ) , r >. ) )
5247opeq1d 4408 . . . . . . . . . . . . 13 |- ( q = <. C , D >. -> <. ( 2nd ` q ) , r >. = <. ( 2nd ` <. C , D >. ) , r >. )
5352breq1d 4663 . . . . . . . . . . . 12 |- ( q = <. C , D >. -> ( <. ( 2nd ` q ) , r >.Cgr <. A , B >. <-> <. ( 2nd ` <. C , D >. ) , r >.Cgr <. A , B >. ) )
5451, 53anbi12d 747 . . . . . . . . . . 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 6613 . . . . . . . . . 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 2632 . . . . . . . . 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 747 . . . . . . . 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 3052 . . . . . . 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 2626 . . . . . . . . 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 740 . . . . . . . 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 3052 . . . . . . 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 6748 . . . . . 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 1424 . . . . 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 286 . . . 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 32138 . . . . 5 |- Fun TransportTo
66 funbrfv 6234 . . . . 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 225 . . 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 2668 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   <.cop 4183   class class class wbr 4653   X. cxp 5112   Fun wfun 5882   ` cfv 5888   iota_crio 6610  (class class class)co 6650   {coprab 6651   1stc1st 7166   2ndc2nd 7167   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770  TransportToctransport 32136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-z 11378  df-uz 11688  df-fz 12327  df-ee 25771  df-transport 32137
This theorem is referenced by:  transportcl  32140  transportprops  32141
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