Metamath Proof Explorer

Theorem lcfrlem8

Description: Lemma for lcf1o and lcfr . (Contributed by NM, 21-Feb-2015)

Ref Expression
Hypotheses lcf1o.h 
|- H = ( LHyp ` K )
lcf1o.o 
|- ._|_ = ( ( ocH ` K ) ` W )
lcf1o.u 
|- U = ( ( DVecH ` K ) ` W )
lcf1o.v 
|- V = ( Base ` U )
lcf1o.a 
|- .+ = ( +g ` U )
lcf1o.t 
|- .x. = ( .s ` U )
lcf1o.s 
|- S = ( Scalar ` U )
lcf1o.r 
|- R = ( Base ` S )
lcf1o.z 
|- .0. = ( 0g ` U )
lcf1o.f 
|- F = ( LFnl ` U )
lcf1o.l 
|- L = ( LKer ` U )
lcf1o.d 
|- D = ( LDual ` U )
lcf1o.q 
|- Q = ( 0g ` D )
lcf1o.c 
|- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
lcf1o.j 
|- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
lcflo.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
lcfrlem8.x 
|- ( ph -> X e. ( V \ { .0. } ) )
Assertion lcfrlem8 
|- ( ph -> ( J ` X ) = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) )

Proof

Step Hyp Ref Expression
1 lcf1o.h 
 |-  H = ( LHyp ` K )
2 lcf1o.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
3 lcf1o.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 lcf1o.v 
 |-  V = ( Base ` U )
5 lcf1o.a 
 |-  .+ = ( +g ` U )
6 lcf1o.t 
 |-  .x. = ( .s ` U )
7 lcf1o.s 
 |-  S = ( Scalar ` U )
8 lcf1o.r 
 |-  R = ( Base ` S )
9 lcf1o.z 
 |-  .0. = ( 0g ` U )
10 lcf1o.f 
 |-  F = ( LFnl ` U )
11 lcf1o.l 
 |-  L = ( LKer ` U )
12 lcf1o.d 
 |-  D = ( LDual ` U )
13 lcf1o.q 
 |-  Q = ( 0g ` D )
14 lcf1o.c 
 |-  C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
15 lcf1o.j 
 |-  J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
16 lcflo.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
17 lcfrlem8.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
18 sneq 
 |-  ( x = X -> { x } = { X } )
19 18 fveq2d 
 |-  ( x = X -> ( ._|_ ` { x } ) = ( ._|_ ` { X } ) )
20 oveq2 
 |-  ( x = X -> ( k .x. x ) = ( k .x. X ) )
21 20 oveq2d 
 |-  ( x = X -> ( w .+ ( k .x. x ) ) = ( w .+ ( k .x. X ) ) )
22 21 eqeq2d 
 |-  ( x = X -> ( v = ( w .+ ( k .x. x ) ) <-> v = ( w .+ ( k .x. X ) ) ) )
23 19 22 rexeqbidv 
 |-  ( x = X -> ( E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) <-> E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) )
24 23 riotabidv 
 |-  ( x = X -> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) = ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) )
25 24 mpteq2dv 
 |-  ( x = X -> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) )
26 25 15 4 mptfvmpt 
 |-  ( X e. ( V \ { .0. } ) -> ( J ` X ) = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) )
27 17 26 syl 
 |-  ( ph -> ( J ` X ) = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) )