Step |
Hyp |
Ref |
Expression |
1 |
|
rrx2pnecoorneor.i |
|- I = { 1 , 2 } |
2 |
|
rrx2pnecoorneor.b |
|- P = ( RR ^m I ) |
3 |
|
rrx2pnedifcoorneor.a |
|- A = ( ( Y ` 1 ) - ( X ` 1 ) ) |
4 |
|
rrx2pnedifcoorneor.b |
|- B = ( ( Y ` 2 ) - ( X ` 2 ) ) |
5 |
1 2
|
rrx2pnecoorneor |
|- ( ( X e. P /\ Y e. P /\ X =/= Y ) -> ( ( X ` 1 ) =/= ( Y ` 1 ) \/ ( X ` 2 ) =/= ( Y ` 2 ) ) ) |
6 |
3
|
neeq1i |
|- ( A =/= 0 <-> ( ( Y ` 1 ) - ( X ` 1 ) ) =/= 0 ) |
7 |
4
|
neeq1i |
|- ( B =/= 0 <-> ( ( Y ` 2 ) - ( X ` 2 ) ) =/= 0 ) |
8 |
6 7
|
orbi12i |
|- ( ( A =/= 0 \/ B =/= 0 ) <-> ( ( ( Y ` 1 ) - ( X ` 1 ) ) =/= 0 \/ ( ( Y ` 2 ) - ( X ` 2 ) ) =/= 0 ) ) |
9 |
1 2
|
rrx2pxel |
|- ( Y e. P -> ( Y ` 1 ) e. RR ) |
10 |
9
|
recnd |
|- ( Y e. P -> ( Y ` 1 ) e. CC ) |
11 |
1 2
|
rrx2pxel |
|- ( X e. P -> ( X ` 1 ) e. RR ) |
12 |
11
|
recnd |
|- ( X e. P -> ( X ` 1 ) e. CC ) |
13 |
|
subeq0 |
|- ( ( ( Y ` 1 ) e. CC /\ ( X ` 1 ) e. CC ) -> ( ( ( Y ` 1 ) - ( X ` 1 ) ) = 0 <-> ( Y ` 1 ) = ( X ` 1 ) ) ) |
14 |
10 12 13
|
syl2anr |
|- ( ( X e. P /\ Y e. P ) -> ( ( ( Y ` 1 ) - ( X ` 1 ) ) = 0 <-> ( Y ` 1 ) = ( X ` 1 ) ) ) |
15 |
14
|
necon3bid |
|- ( ( X e. P /\ Y e. P ) -> ( ( ( Y ` 1 ) - ( X ` 1 ) ) =/= 0 <-> ( Y ` 1 ) =/= ( X ` 1 ) ) ) |
16 |
1 2
|
rrx2pyel |
|- ( Y e. P -> ( Y ` 2 ) e. RR ) |
17 |
16
|
recnd |
|- ( Y e. P -> ( Y ` 2 ) e. CC ) |
18 |
1 2
|
rrx2pyel |
|- ( X e. P -> ( X ` 2 ) e. RR ) |
19 |
18
|
recnd |
|- ( X e. P -> ( X ` 2 ) e. CC ) |
20 |
|
subeq0 |
|- ( ( ( Y ` 2 ) e. CC /\ ( X ` 2 ) e. CC ) -> ( ( ( Y ` 2 ) - ( X ` 2 ) ) = 0 <-> ( Y ` 2 ) = ( X ` 2 ) ) ) |
21 |
17 19 20
|
syl2anr |
|- ( ( X e. P /\ Y e. P ) -> ( ( ( Y ` 2 ) - ( X ` 2 ) ) = 0 <-> ( Y ` 2 ) = ( X ` 2 ) ) ) |
22 |
21
|
necon3bid |
|- ( ( X e. P /\ Y e. P ) -> ( ( ( Y ` 2 ) - ( X ` 2 ) ) =/= 0 <-> ( Y ` 2 ) =/= ( X ` 2 ) ) ) |
23 |
15 22
|
orbi12d |
|- ( ( X e. P /\ Y e. P ) -> ( ( ( ( Y ` 1 ) - ( X ` 1 ) ) =/= 0 \/ ( ( Y ` 2 ) - ( X ` 2 ) ) =/= 0 ) <-> ( ( Y ` 1 ) =/= ( X ` 1 ) \/ ( Y ` 2 ) =/= ( X ` 2 ) ) ) ) |
24 |
|
necom |
|- ( ( Y ` 1 ) =/= ( X ` 1 ) <-> ( X ` 1 ) =/= ( Y ` 1 ) ) |
25 |
|
necom |
|- ( ( Y ` 2 ) =/= ( X ` 2 ) <-> ( X ` 2 ) =/= ( Y ` 2 ) ) |
26 |
24 25
|
orbi12i |
|- ( ( ( Y ` 1 ) =/= ( X ` 1 ) \/ ( Y ` 2 ) =/= ( X ` 2 ) ) <-> ( ( X ` 1 ) =/= ( Y ` 1 ) \/ ( X ` 2 ) =/= ( Y ` 2 ) ) ) |
27 |
23 26
|
bitrdi |
|- ( ( X e. P /\ Y e. P ) -> ( ( ( ( Y ` 1 ) - ( X ` 1 ) ) =/= 0 \/ ( ( Y ` 2 ) - ( X ` 2 ) ) =/= 0 ) <-> ( ( X ` 1 ) =/= ( Y ` 1 ) \/ ( X ` 2 ) =/= ( Y ` 2 ) ) ) ) |
28 |
8 27
|
syl5bb |
|- ( ( X e. P /\ Y e. P ) -> ( ( A =/= 0 \/ B =/= 0 ) <-> ( ( X ` 1 ) =/= ( Y ` 1 ) \/ ( X ` 2 ) =/= ( Y ` 2 ) ) ) ) |
29 |
28
|
3adant3 |
|- ( ( X e. P /\ Y e. P /\ X =/= Y ) -> ( ( A =/= 0 \/ B =/= 0 ) <-> ( ( X ` 1 ) =/= ( Y ` 1 ) \/ ( X ` 2 ) =/= ( Y ` 2 ) ) ) ) |
30 |
5 29
|
mpbird |
|- ( ( X e. P /\ Y e. P /\ X =/= Y ) -> ( A =/= 0 \/ B =/= 0 ) ) |