Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
|- ( T = 0 -> ( 1 - T ) = ( 1 - 0 ) ) |
2 |
|
1m0e1 |
|- ( 1 - 0 ) = 1 |
3 |
1 2
|
eqtrdi |
|- ( T = 0 -> ( 1 - T ) = 1 ) |
4 |
3
|
oveq1d |
|- ( T = 0 -> ( ( 1 - T ) x. ( A ` i ) ) = ( 1 x. ( A ` i ) ) ) |
5 |
|
oveq1 |
|- ( T = 0 -> ( T x. ( C ` i ) ) = ( 0 x. ( C ` i ) ) ) |
6 |
4 5
|
oveq12d |
|- ( T = 0 -> ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) = ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) ) |
7 |
6
|
eqeq2d |
|- ( T = 0 -> ( ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) <-> ( B ` i ) = ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) ) ) |
8 |
7
|
ralbidv |
|- ( T = 0 -> ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) <-> A. i e. ( 1 ... N ) ( B ` i ) = ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) ) ) |
9 |
8
|
biimpac |
|- ( ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) /\ T = 0 ) -> A. i e. ( 1 ... N ) ( B ` i ) = ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) ) |
10 |
|
eqeefv |
|- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) ) ) |
11 |
10
|
3adant1 |
|- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) ) ) |
12 |
11
|
3adant3r3 |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) ) ) |
13 |
|
simplr1 |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ i e. ( 1 ... N ) ) -> A e. ( EE ` N ) ) |
14 |
|
fveecn |
|- ( ( A e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. CC ) |
15 |
13 14
|
sylancom |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. CC ) |
16 |
|
simplr3 |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ i e. ( 1 ... N ) ) -> C e. ( EE ` N ) ) |
17 |
|
fveecn |
|- ( ( C e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( C ` i ) e. CC ) |
18 |
16 17
|
sylancom |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ i e. ( 1 ... N ) ) -> ( C ` i ) e. CC ) |
19 |
|
mulid2 |
|- ( ( A ` i ) e. CC -> ( 1 x. ( A ` i ) ) = ( A ` i ) ) |
20 |
|
mul02 |
|- ( ( C ` i ) e. CC -> ( 0 x. ( C ` i ) ) = 0 ) |
21 |
19 20
|
oveqan12d |
|- ( ( ( A ` i ) e. CC /\ ( C ` i ) e. CC ) -> ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) = ( ( A ` i ) + 0 ) ) |
22 |
|
addid1 |
|- ( ( A ` i ) e. CC -> ( ( A ` i ) + 0 ) = ( A ` i ) ) |
23 |
22
|
adantr |
|- ( ( ( A ` i ) e. CC /\ ( C ` i ) e. CC ) -> ( ( A ` i ) + 0 ) = ( A ` i ) ) |
24 |
21 23
|
eqtrd |
|- ( ( ( A ` i ) e. CC /\ ( C ` i ) e. CC ) -> ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) = ( A ` i ) ) |
25 |
15 18 24
|
syl2anc |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) = ( A ` i ) ) |
26 |
25
|
eqeq1d |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) = ( B ` i ) <-> ( A ` i ) = ( B ` i ) ) ) |
27 |
|
eqcom |
|- ( ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) = ( B ` i ) <-> ( B ` i ) = ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) ) |
28 |
26 27
|
bitr3di |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ i e. ( 1 ... N ) ) -> ( ( A ` i ) = ( B ` i ) <-> ( B ` i ) = ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) ) ) |
29 |
28
|
ralbidva |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) <-> A. i e. ( 1 ... N ) ( B ` i ) = ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) ) ) |
30 |
12 29
|
bitrd |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( B ` i ) = ( ( 1 x. ( A ` i ) ) + ( 0 x. ( C ` i ) ) ) ) ) |
31 |
9 30
|
syl5ibr |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) /\ T = 0 ) -> A = B ) ) |
32 |
31
|
expdimp |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) -> ( T = 0 -> A = B ) ) |
33 |
32
|
necon3d |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) -> ( A =/= B -> T =/= 0 ) ) |
34 |
33
|
3impia |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) /\ A =/= B ) -> T =/= 0 ) |