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 |
|
fveq1 |
|- ( X = Y -> ( X ` 2 ) = ( Y ` 2 ) ) |
6 |
5
|
necon3i |
|- ( ( X ` 2 ) =/= ( Y ` 2 ) -> X =/= Y ) |
7 |
6
|
adantl |
|- ( ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) -> X =/= Y ) |
8 |
1 2 3 4
|
rrx2line |
|- ( ( 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 ) ) ) ) } ) |
9 |
7 8
|
syl3an3 |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( 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 ) ) ) ) } ) |
10 |
|
oveq2 |
|- ( ( Y ` 1 ) = ( X ` 1 ) -> ( t x. ( Y ` 1 ) ) = ( t x. ( X ` 1 ) ) ) |
11 |
10
|
oveq2d |
|- ( ( Y ` 1 ) = ( X ` 1 ) -> ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( X ` 1 ) ) ) ) |
12 |
11
|
eqcoms |
|- ( ( X ` 1 ) = ( Y ` 1 ) -> ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( X ` 1 ) ) ) ) |
13 |
12
|
adantr |
|- ( ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) -> ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( X ` 1 ) ) ) ) |
14 |
13
|
3ad2ant3 |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( X ` 1 ) ) ) ) |
15 |
14
|
adantr |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( X ` 1 ) ) ) ) |
16 |
15
|
adantr |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ t e. RR ) -> ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( X ` 1 ) ) ) ) |
17 |
1 3
|
rrx2pxel |
|- ( X e. P -> ( X ` 1 ) e. RR ) |
18 |
17
|
recnd |
|- ( X e. P -> ( X ` 1 ) e. CC ) |
19 |
18
|
3ad2ant1 |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( X ` 1 ) e. CC ) |
20 |
19
|
adantr |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( X ` 1 ) e. CC ) |
21 |
20
|
adantr |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ t e. RR ) -> ( X ` 1 ) e. CC ) |
22 |
|
recn |
|- ( t e. RR -> t e. CC ) |
23 |
22
|
adantl |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ t e. RR ) -> t e. CC ) |
24 |
21 23
|
affineid |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ t e. RR ) -> ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( X ` 1 ) ) ) = ( X ` 1 ) ) |
25 |
16 24
|
eqtrd |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ t e. RR ) -> ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) = ( X ` 1 ) ) |
26 |
25
|
eqeq2d |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ t e. RR ) -> ( ( p ` 1 ) = ( ( ( 1 - t ) x. ( X ` 1 ) ) + ( t x. ( Y ` 1 ) ) ) <-> ( p ` 1 ) = ( X ` 1 ) ) ) |
27 |
26
|
anbi1d |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ 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 ) ) ) ) <-> ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) ) ) |
28 |
27
|
rexbidva |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ 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 ) ) ) ) <-> E. t e. RR ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) ) ) |
29 |
|
simpl |
|- ( ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) -> ( p ` 1 ) = ( X ` 1 ) ) |
30 |
29
|
a1i |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ t e. RR ) -> ( ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) -> ( p ` 1 ) = ( X ` 1 ) ) ) |
31 |
30
|
rexlimdva |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( E. t e. RR ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) -> ( p ` 1 ) = ( X ` 1 ) ) ) |
32 |
1 3
|
rrx2pyel |
|- ( p e. P -> ( p ` 2 ) e. RR ) |
33 |
32
|
adantl |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( p ` 2 ) e. RR ) |
34 |
1 3
|
rrx2pyel |
|- ( X e. P -> ( X ` 2 ) e. RR ) |
35 |
34
|
3ad2ant1 |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( X ` 2 ) e. RR ) |
36 |
35
|
adantr |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( X ` 2 ) e. RR ) |
37 |
33 36
|
resubcld |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( p ` 2 ) - ( X ` 2 ) ) e. RR ) |
38 |
1 3
|
rrx2pyel |
|- ( Y e. P -> ( Y ` 2 ) e. RR ) |
39 |
38
|
3ad2ant2 |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( Y ` 2 ) e. RR ) |
40 |
39 35
|
resubcld |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( ( Y ` 2 ) - ( X ` 2 ) ) e. RR ) |
41 |
40
|
adantr |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( Y ` 2 ) - ( X ` 2 ) ) e. RR ) |
42 |
38
|
recnd |
|- ( Y e. P -> ( Y ` 2 ) e. CC ) |
43 |
42
|
3ad2ant2 |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( Y ` 2 ) e. CC ) |
44 |
34
|
recnd |
|- ( X e. P -> ( X ` 2 ) e. CC ) |
45 |
44
|
3ad2ant1 |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( X ` 2 ) e. CC ) |
46 |
|
simpr |
|- ( ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) -> ( X ` 2 ) =/= ( Y ` 2 ) ) |
47 |
46
|
necomd |
|- ( ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) -> ( Y ` 2 ) =/= ( X ` 2 ) ) |
48 |
47
|
3ad2ant3 |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( Y ` 2 ) =/= ( X ` 2 ) ) |
49 |
43 45 48
|
subne0d |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( ( Y ` 2 ) - ( X ` 2 ) ) =/= 0 ) |
50 |
49
|
adantr |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( Y ` 2 ) - ( X ` 2 ) ) =/= 0 ) |
51 |
37 41 50
|
redivcld |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) e. RR ) |
52 |
51
|
adantr |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ ( p ` 1 ) = ( X ` 1 ) ) -> ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) e. RR ) |
53 |
|
oveq2 |
|- ( t = ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) -> ( 1 - t ) = ( 1 - ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) ) |
54 |
53
|
oveq1d |
|- ( t = ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) -> ( ( 1 - t ) x. ( X ` 2 ) ) = ( ( 1 - ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) x. ( X ` 2 ) ) ) |
55 |
|
oveq1 |
|- ( t = ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) -> ( t x. ( Y ` 2 ) ) = ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( Y ` 2 ) ) ) |
56 |
54 55
|
oveq12d |
|- ( t = ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) -> ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) = ( ( ( 1 - ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( Y ` 2 ) ) ) ) |
57 |
56
|
eqeq2d |
|- ( t = ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) -> ( ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) <-> ( p ` 2 ) = ( ( ( 1 - ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( Y ` 2 ) ) ) ) ) |
58 |
57
|
anbi2d |
|- ( t = ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) -> ( ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) <-> ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( Y ` 2 ) ) ) ) ) ) |
59 |
58
|
adantl |
|- ( ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ ( p ` 1 ) = ( X ` 1 ) ) /\ t = ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) -> ( ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) <-> ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( Y ` 2 ) ) ) ) ) ) |
60 |
|
simpr |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ ( p ` 1 ) = ( X ` 1 ) ) -> ( p ` 1 ) = ( X ` 1 ) ) |
61 |
44
|
mulid2d |
|- ( X e. P -> ( 1 x. ( X ` 2 ) ) = ( X ` 2 ) ) |
62 |
61
|
3ad2ant1 |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( 1 x. ( X ` 2 ) ) = ( X ` 2 ) ) |
63 |
62
|
adantr |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( 1 x. ( X ` 2 ) ) = ( X ` 2 ) ) |
64 |
37
|
recnd |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( p ` 2 ) - ( X ` 2 ) ) e. CC ) |
65 |
42
|
adantl |
|- ( ( X e. P /\ Y e. P ) -> ( Y ` 2 ) e. CC ) |
66 |
44
|
adantr |
|- ( ( X e. P /\ Y e. P ) -> ( X ` 2 ) e. CC ) |
67 |
65 66
|
subcld |
|- ( ( X e. P /\ Y e. P ) -> ( ( Y ` 2 ) - ( X ` 2 ) ) e. CC ) |
68 |
67
|
3adant3 |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( ( Y ` 2 ) - ( X ` 2 ) ) e. CC ) |
69 |
68
|
adantr |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( Y ` 2 ) - ( X ` 2 ) ) e. CC ) |
70 |
64 69 50
|
divcan1d |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( ( Y ` 2 ) - ( X ` 2 ) ) ) = ( ( p ` 2 ) - ( X ` 2 ) ) ) |
71 |
63 70
|
oveq12d |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( 1 x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) = ( ( X ` 2 ) + ( ( p ` 2 ) - ( X ` 2 ) ) ) ) |
72 |
45
|
adantr |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( X ` 2 ) e. CC ) |
73 |
32
|
recnd |
|- ( p e. P -> ( p ` 2 ) e. CC ) |
74 |
73
|
adantl |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( p ` 2 ) e. CC ) |
75 |
72 74
|
pncan3d |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( X ` 2 ) + ( ( p ` 2 ) - ( X ` 2 ) ) ) = ( p ` 2 ) ) |
76 |
71 75
|
eqtr2d |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( p ` 2 ) = ( ( 1 x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) ) |
77 |
76
|
adantr |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ ( p ` 1 ) = ( X ` 1 ) ) -> ( p ` 2 ) = ( ( 1 x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) ) |
78 |
|
1cnd |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> 1 e. CC ) |
79 |
51
|
recnd |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) e. CC ) |
80 |
43
|
adantr |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( Y ` 2 ) e. CC ) |
81 |
78 79 72 80
|
submuladdmuld |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( ( 1 - ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( Y ` 2 ) ) ) = ( ( 1 x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) ) |
82 |
81
|
adantr |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ ( p ` 1 ) = ( X ` 1 ) ) -> ( ( ( 1 - ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( Y ` 2 ) ) ) = ( ( 1 x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) ) |
83 |
77 82
|
eqtr4d |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ ( p ` 1 ) = ( X ` 1 ) ) -> ( p ` 2 ) = ( ( ( 1 - ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( Y ` 2 ) ) ) ) |
84 |
60 83
|
jca |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ ( p ` 1 ) = ( X ` 1 ) ) -> ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) ) x. ( X ` 2 ) ) + ( ( ( ( p ` 2 ) - ( X ` 2 ) ) / ( ( Y ` 2 ) - ( X ` 2 ) ) ) x. ( Y ` 2 ) ) ) ) ) |
85 |
52 59 84
|
rspcedvd |
|- ( ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) /\ ( p ` 1 ) = ( X ` 1 ) ) -> E. t e. RR ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) ) |
86 |
85
|
ex |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( ( p ` 1 ) = ( X ` 1 ) -> E. t e. RR ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) ) ) |
87 |
31 86
|
impbid |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ p e. P ) -> ( E. t e. RR ( ( p ` 1 ) = ( X ` 1 ) /\ ( p ` 2 ) = ( ( ( 1 - t ) x. ( X ` 2 ) ) + ( t x. ( Y ` 2 ) ) ) ) <-> ( p ` 1 ) = ( X ` 1 ) ) ) |
88 |
28 87
|
bitrd |
|- ( ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) /\ 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 ) ) ) ) <-> ( p ` 1 ) = ( X ` 1 ) ) ) |
89 |
88
|
rabbidva |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> { 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 ) ) ) ) } = { p e. P | ( p ` 1 ) = ( X ` 1 ) } ) |
90 |
9 89
|
eqtrd |
|- ( ( X e. P /\ Y e. P /\ ( ( X ` 1 ) = ( Y ` 1 ) /\ ( X ` 2 ) =/= ( Y ` 2 ) ) ) -> ( X L Y ) = { p e. P | ( p ` 1 ) = ( X ` 1 ) } ) |