Step |
Hyp |
Ref |
Expression |
1 |
|
axlowdimlem10.1 |
|- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) |
2 |
1
|
fveq1i |
|- ( Q ` K ) = ( ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) ` K ) |
3 |
|
eldifsn |
|- ( K e. ( ( 1 ... N ) \ { ( I + 1 ) } ) <-> ( K e. ( 1 ... N ) /\ K =/= ( I + 1 ) ) ) |
4 |
|
disjdif |
|- ( { ( I + 1 ) } i^i ( ( 1 ... N ) \ { ( I + 1 ) } ) ) = (/) |
5 |
|
ovex |
|- ( I + 1 ) e. _V |
6 |
|
1ex |
|- 1 e. _V |
7 |
5 6
|
fnsn |
|- { <. ( I + 1 ) , 1 >. } Fn { ( I + 1 ) } |
8 |
|
c0ex |
|- 0 e. _V |
9 |
8
|
fconst |
|- ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) : ( ( 1 ... N ) \ { ( I + 1 ) } ) --> { 0 } |
10 |
|
ffn |
|- ( ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) : ( ( 1 ... N ) \ { ( I + 1 ) } ) --> { 0 } -> ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) Fn ( ( 1 ... N ) \ { ( I + 1 ) } ) ) |
11 |
9 10
|
ax-mp |
|- ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) Fn ( ( 1 ... N ) \ { ( I + 1 ) } ) |
12 |
|
fvun2 |
|- ( ( { <. ( I + 1 ) , 1 >. } Fn { ( I + 1 ) } /\ ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) Fn ( ( 1 ... N ) \ { ( I + 1 ) } ) /\ ( ( { ( I + 1 ) } i^i ( ( 1 ... N ) \ { ( I + 1 ) } ) ) = (/) /\ K e. ( ( 1 ... N ) \ { ( I + 1 ) } ) ) ) -> ( ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) ` K ) = ( ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ` K ) ) |
13 |
7 11 12
|
mp3an12 |
|- ( ( ( { ( I + 1 ) } i^i ( ( 1 ... N ) \ { ( I + 1 ) } ) ) = (/) /\ K e. ( ( 1 ... N ) \ { ( I + 1 ) } ) ) -> ( ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) ` K ) = ( ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ` K ) ) |
14 |
4 13
|
mpan |
|- ( K e. ( ( 1 ... N ) \ { ( I + 1 ) } ) -> ( ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) ` K ) = ( ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ` K ) ) |
15 |
8
|
fvconst2 |
|- ( K e. ( ( 1 ... N ) \ { ( I + 1 ) } ) -> ( ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ` K ) = 0 ) |
16 |
14 15
|
eqtrd |
|- ( K e. ( ( 1 ... N ) \ { ( I + 1 ) } ) -> ( ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) ` K ) = 0 ) |
17 |
3 16
|
sylbir |
|- ( ( K e. ( 1 ... N ) /\ K =/= ( I + 1 ) ) -> ( ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) ` K ) = 0 ) |
18 |
2 17
|
syl5eq |
|- ( ( K e. ( 1 ... N ) /\ K =/= ( I + 1 ) ) -> ( Q ` K ) = 0 ) |