let F be Field; :: thesis: for x being Element of F
for V being VectSp of F
for v being Element of V holds
( x * v = 0. V iff ( x = 0. F or v = 0. V ) )
let x be Element of F; :: thesis: for V being VectSp of F
for v being Element of V holds
( x * v = 0. V iff ( x = 0. F or v = 0. V ) )
let V be VectSp of F; :: thesis: for v being Element of V holds
( x * v = 0. V iff ( x = 0. F or v = 0. V ) )
let v be Element of V; :: thesis: ( x * v = 0. V iff ( x = 0. F or v = 0. V ) )
( not x * v = 0. V or x = 0. F or v = 0. V )
proof
assume x * v = 0. V ; :: thesis: ( x = 0. F or v = 0. V )
then A1: ((x ") * x) * v = (x ") * (0. V) by Def15
.= 0. V by Th10 ;
assume x <> 0. F ; :: thesis: v = 0. V
then 0. V = (1. F) * v by A1, Def10;
hence v = 0. V by Def16; :: thesis: verum
end;
hence ( x * v = 0. V iff ( x = 0. F or v = 0. V ) ) by Th10; :: thesis: verum