Step |
Hyp |
Ref |
Expression |
1 |
|
rrx2line.i |
|- I = { 1 , 2 } |
2 |
|
rrx2line.e |
|- E = ( RR^ ` I ) |
3 |
|
rrx2line.b |
|- P = ( RR ^m I ) |
4 |
|
rrx2line.l |
|- L = ( LineM ` E ) |
5 |
|
prfi |
|- { 1 , 2 } e. Fin |
6 |
1 5
|
eqeltri |
|- I e. Fin |
7 |
2 3 4
|
rrxlinec |
|- ( ( I e. Fin /\ ( X e. P /\ Y e. P /\ X =/= Y ) ) -> ( X L Y ) = { p e. P | E. t e. RR A. i e. I ( p ` i ) = ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) } ) |
8 |
6 7
|
mpan |
|- ( ( X e. P /\ Y e. P /\ X =/= Y ) -> ( X L Y ) = { p e. P | E. t e. RR A. i e. I ( p ` i ) = ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) } ) |
9 |
1
|
a1i |
|- ( ( ( ( X e. P /\ Y e. P /\ X =/= Y ) /\ p e. P ) /\ t e. RR ) -> I = { 1 , 2 } ) |
10 |
9
|
raleqdv |
|- ( ( ( ( X e. P /\ Y e. P /\ X =/= Y ) /\ p e. P ) /\ t e. RR ) -> ( A. i e. I ( p ` i ) = ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) <-> A. i e. { 1 , 2 } ( p ` i ) = ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) ) ) |
11 |
|
1ex |
|- 1 e. _V |
12 |
|
2ex |
|- 2 e. _V |
13 |
|
fveq2 |
|- ( i = 1 -> ( p ` i ) = ( p ` 1 ) ) |
14 |
|
fveq2 |
|- ( i = 1 -> ( X ` i ) = ( X ` 1 ) ) |
15 |
14
|
oveq2d |
|- ( i = 1 -> ( ( 1 - t ) x. ( X ` i ) ) = ( ( 1 - t ) x. ( X ` 1 ) ) ) |
16 |
|
fveq2 |
|- ( i = 1 -> ( Y ` i ) = ( Y ` 1 ) ) |
17 |
16
|
oveq2d |
|- ( i = 1 -> ( t x. ( Y ` i ) ) = ( t x. ( Y ` 1 ) ) ) |
18 |
15 17
|
oveq12d |
|- ( i = 1 -> ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) ) |
19 |
13 18
|
eqeq12d |
|- ( i = 1 -> ( ( p ` i ) = ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) <-> ( p ` 1 ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) ) ) |
20 |
|
fveq2 |
|- ( i = 2 -> ( p ` i ) = ( p ` 2 ) ) |
21 |
|
fveq2 |
|- ( i = 2 -> ( X ` i ) = ( X ` 2 ) ) |
22 |
21
|
oveq2d |
|- ( i = 2 -> ( ( 1 - t ) x. ( X ` i ) ) = ( ( 1 - t ) x. ( X ` 2 ) ) ) |
23 |
|
fveq2 |
|- ( i = 2 -> ( Y ` i ) = ( Y ` 2 ) ) |
24 |
23
|
oveq2d |
|- ( i = 2 -> ( t x. ( Y ` i ) ) = ( t x. ( Y ` 2 ) ) ) |
25 |
22 24
|
oveq12d |
|- ( i = 2 -> ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) |
26 |
20 25
|
eqeq12d |
|- ( i = 2 -> ( ( p ` i ) = ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) <-> ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) ) |
27 |
11 12 19 26
|
ralpr |
|- ( A. i e. { 1 , 2 } ( p ` i ) = ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) <-> ( ( p ` 1 ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) ) |
28 |
10 27
|
bitrdi |
|- ( ( ( ( X e. P /\ Y e. P /\ X =/= Y ) /\ p e. P ) /\ t e. RR ) -> ( A. i e. I ( p ` i ) = ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) <-> ( ( p ` 1 ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) ) ) |
29 |
28
|
rexbidva |
|- ( ( ( X e. P /\ Y e. P /\ X =/= Y ) /\ p e. P ) -> ( E. t e. RR A. i e. I ( p ` i ) = ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) <-> E. t e. RR ( ( p ` 1 ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) ) ) |
30 |
29
|
rabbidva |
|- ( ( X e. P /\ Y e. P /\ X =/= Y ) -> { p e. P | E. t e. RR A. i e. I ( p ` i ) = ( ( ( 1 - t ) x. ( X ` i ) ) + ( t x. ( Y ` i ) ) ) } = { p e. P | E. t e. RR ( ( p ` 1 ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) } ) |
31 |
8 30
|
eqtrd |
|- ( ( X e. P /\ Y e. P /\ X =/= Y ) -> ( X L Y ) = { p e. P | E. t e. RR ( ( p ` 1 ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) } ) |