let R be Ring; :: thesis: for V being LeftMod of R
for v being Vector of V st not R is degenerated & V is Mult-cancelable holds
( {v} is linearly-independent iff v <> 0. V )
let V be LeftMod of R; :: thesis: for v being Vector of V st not R is degenerated & V is Mult-cancelable holds
( {v} is linearly-independent iff v <> 0. V )
let v be Vector of V; :: thesis: ( not R is degenerated & V is Mult-cancelable implies ( {v} is linearly-independent iff v <> 0. V ) )
assume A1: ( not R is degenerated & V is Mult-cancelable ) ; :: thesis: ( {v} is linearly-independent iff v <> 0. V )
thus ( {v} is linearly-independent implies v <> 0. V ) :: thesis: ( v <> 0. V implies {v} is linearly-independent )

proof

assume {v} is linearly-independent ; :: thesis: v <> 0. V
then not 0. V in {v} by Th57, A1;
hence v <> 0. V by TARSKI:def 1; :: thesis: verum

end;

assume A2: v <> 0. V ; :: thesis: {v} is linearly-independent
let l be Linear_Combination of {v}; :: according to VECTSP_7:def 1 :: thesis: ( not Sum l = 0. V or Carrier l = {} )
A3: Carrier l c= {v} by VECTSP_6:def 4;
assume A4: Sum l = 0. V ; :: thesis: Carrier l = {}

now :: thesis: Carrier l = {}

per cases ( Carrier l = {} or Carrier l = {v} ) by A3, ZFMISC_1:33;

suppose Carrier l = {} ; :: thesis: Carrier l = {}

hence Carrier l = {} ; :: thesis: verum

end;

suppose A5: Carrier l = {v} ; :: thesis: Carrier l = {}

then A6: 0. V = (l . v) * v by A4, Th24;

now :: thesis: not v in Carrier l

assume v in Carrier l ; :: thesis: contradiction
then ex u being Vector of V st
( v = u & l . u <> 0. R ) ;
hence contradiction by A2, A6, A1; :: thesis: verum

end;

hence Carrier l = {} by A5, TARSKI:def 1; :: thesis: verum

end;

end;

end;

hence Carrier l = {} ; :: thesis: verum