let V be RealLinearSpace; :: thesis: for v being VECTOR of V holds

( {v} is linearly-independent iff v <> 0. V )

let v be VECTOR of V; :: 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 )

let l be Linear_Combination of {v}; :: according to RLVECT_3:def 1 :: thesis: ( Sum l = 0. V implies Carrier l = {} )

A2: Carrier l c= {v} by RLVECT_2:def 6;

assume A3: Sum l = 0. V ; :: thesis: Carrier l = {}

( {v} is linearly-independent iff v <> 0. V )

let v be VECTOR of V; :: 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 A1:
v <> 0. V
; :: thesis: {v} is linearly-independent
assume
{v} is linearly-independent
; :: thesis: v <> 0. V

then not 0. V in {v} by Th6;

hence v <> 0. V by TARSKI:def 1; :: thesis: verum

end;then not 0. V in {v} by Th6;

hence v <> 0. V by TARSKI:def 1; :: thesis: verum

let l be Linear_Combination of {v}; :: according to RLVECT_3:def 1 :: thesis: ( Sum l = 0. V implies Carrier l = {} )

A2: Carrier l c= {v} by RLVECT_2:def 6;

assume A3: Sum l = 0. V ; :: thesis: Carrier l = {}

now :: thesis: Carrier l = {} end;

hence
Carrier l = {}
; :: thesis: verumper cases
( Carrier l = {} or Carrier l = {v} )
by A2, ZFMISC_1:33;

end;

suppose A4:
Carrier l = {v}
; :: thesis: Carrier l = {}

A5:
0. V = (l . v) * v
by A3, RLVECT_2:32;

end;now :: thesis: not v in Carrier l

hence
Carrier l = {}
by A4, TARSKI:def 1; :: thesis: verumassume
v in Carrier l
; :: thesis: contradiction

then ex u being VECTOR of V st

( v = u & l . u <> 0 ) ;

hence contradiction by A1, A5, RLVECT_1:11; :: thesis: verum

end;then ex u being VECTOR of V st

( v = u & l . u <> 0 ) ;

hence contradiction by A1, A5, RLVECT_1:11; :: thesis: verum