Step |
Hyp |
Ref |
Expression |
1 |
|
lshpkrlem.v |
|- V = ( Base ` W ) |
2 |
|
lshpkrlem.a |
|- .+ = ( +g ` W ) |
3 |
|
lshpkrlem.n |
|- N = ( LSpan ` W ) |
4 |
|
lshpkrlem.p |
|- .(+) = ( LSSum ` W ) |
5 |
|
lshpkrlem.h |
|- H = ( LSHyp ` W ) |
6 |
|
lshpkrlem.w |
|- ( ph -> W e. LVec ) |
7 |
|
lshpkrlem.u |
|- ( ph -> U e. H ) |
8 |
|
lshpkrlem.z |
|- ( ph -> Z e. V ) |
9 |
|
lshpkrlem.x |
|- ( ph -> X e. V ) |
10 |
|
lshpkrlem.e |
|- ( ph -> ( U .(+) ( N ` { Z } ) ) = V ) |
11 |
|
lshpkrlem.d |
|- D = ( Scalar ` W ) |
12 |
|
lshpkrlem.k |
|- K = ( Base ` D ) |
13 |
|
lshpkrlem.t |
|- .x. = ( .s ` W ) |
14 |
|
lshpkrlem.o |
|- .0. = ( 0g ` D ) |
15 |
|
lshpkrlem.g |
|- G = ( x e. V |-> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) ) |
16 |
|
eqeq1 |
|- ( x = X -> ( x = ( y .+ ( k .x. Z ) ) <-> X = ( y .+ ( k .x. Z ) ) ) ) |
17 |
16
|
rexbidv |
|- ( x = X -> ( E. y e. U x = ( y .+ ( k .x. Z ) ) <-> E. y e. U X = ( y .+ ( k .x. Z ) ) ) ) |
18 |
17
|
riotabidv |
|- ( x = X -> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) = ( iota_ k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) ) |
19 |
|
riotaex |
|- ( iota_ k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) e. _V |
20 |
18 15 19
|
fvmpt |
|- ( X e. V -> ( G ` X ) = ( iota_ k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) ) |
21 |
9 20
|
syl |
|- ( ph -> ( G ` X ) = ( iota_ k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) ) |
22 |
1 2 3 4 5 6 7 8 9 10 11 12 13
|
lshpsmreu |
|- ( ph -> E! k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) |
23 |
|
riotacl |
|- ( E! k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) -> ( iota_ k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) e. K ) |
24 |
22 23
|
syl |
|- ( ph -> ( iota_ k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) e. K ) |
25 |
21 24
|
eqeltrd |
|- ( ph -> ( G ` X ) e. K ) |