let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for V being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for v being Element of V
for f being Function of the carrier of V,the carrier of GF holds f (#) <*v*> = <*((f . v) * v)*>
let V be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for v being Element of V
for f being Function of the carrier of V,the carrier of GF holds f (#) <*v*> = <*((f . v) * v)*>
let v be Element of V; :: thesis: for f being Function of the carrier of V,the carrier of GF holds f (#) <*v*> = <*((f . v) * v)*>
let f be Function of the carrier of V,the carrier of GF; :: thesis: f (#) <*v*> = <*((f . v) * v)*>
A1: len (f (#) <*v*>) = len <*v*> by Def8
.= 1 by FINSEQ_1:57 ;
then ( dom (f (#) <*v*>) = {1} & 1 in {1} ) by FINSEQ_1:4, FINSEQ_1:def 3, TARSKI:def 1;
then (f (#) <*v*>) . 1 = (f . (<*v*> /. 1)) * (<*v*> /. 1) by Def8
.= (f . (<*v*> /. 1)) * v by FINSEQ_4:25
.= (f . v) * v by FINSEQ_4:25 ;
hence f (#) <*v*> = <*((f . v) * v)*> by A1, FINSEQ_1:57; :: thesis: verum