let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for v being Element of V
for f being Function of V,GF holds f (#) <*v*> = <*((f . v) * v)*>
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for v being Element of V
for f being Function of V,GF holds f (#) <*v*> = <*((f . v) * v)*>
let v be Element of V; :: thesis: for f being Function of V,GF holds f (#) <*v*> = <*((f . v) * v)*>
let f be Function of V,GF; :: thesis: f (#) <*v*> = <*((f . v) * v)*>
A1: 1 in {1} by TARSKI:def 1;
A2: len (f (#) <*v*>) = len <*v*> by Def5
.= 1 by FINSEQ_1:40 ;
then dom (f (#) <*v*>) = {1} by FINSEQ_1:2, FINSEQ_1:def 3;
then (f (#) <*v*>) . 1 = (f . (<*v*> /. 1)) * (<*v*> /. 1) by A1, Def5
.= (f . (<*v*> /. 1)) * v by FINSEQ_4:16
.= (f . v) * v by FINSEQ_4:16 ;
hence f (#) <*v*> = <*((f . v) * v)*> by A2, FINSEQ_1:40; :: thesis: verum