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Metamath Proof Explorer

Theorem lshpkrlem2

Description: Lemma for lshpkrex . The value of tentative functional G is a scalar. (Contributed by NM, 16-Jul-2014)

Ref Expression
Hypotheses lshpkrlem.v 
|- V = ( Base ` W )
lshpkrlem.a 
|- .+ = ( +g ` W )
lshpkrlem.n 
|- N = ( LSpan ` W )
lshpkrlem.p 
|- .(+) = ( LSSum ` W )
lshpkrlem.h 
|- H = ( LSHyp ` W )
lshpkrlem.w 
|- ( ph -> W e. LVec )
lshpkrlem.u 
|- ( ph -> U e. H )
lshpkrlem.z 
|- ( ph -> Z e. V )
lshpkrlem.x 
|- ( ph -> X e. V )
lshpkrlem.e 
|- ( ph -> ( U .(+) ( N ` { Z } ) ) = V )
lshpkrlem.d 
|- D = ( Scalar ` W )
lshpkrlem.k 
|- K = ( Base ` D )
lshpkrlem.t 
|- .x. = ( .s ` W )
lshpkrlem.o 
|- .0. = ( 0g ` D )
lshpkrlem.g 
|- G = ( x e. V |-> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) )
Assertion lshpkrlem2 
|- ( ph -> ( G ` X ) e. K )

Proof

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 )