let R be Ring; :: thesis: for V being LeftMod of R
for v1, v2 being Vector of V st R = INT.Ring & V is Mult-cancelable holds
( v1 <> v2 & {v1,v2} is linearly-independent iff ( v2 <> 0. V & ( for a, b being Element of R st b <> 0. R holds
b * v1 <> a * v2 ) ) )
let V be LeftMod of R; :: thesis: for v1, v2 being Vector of V st R = INT.Ring & V is Mult-cancelable holds
( v1 <> v2 & {v1,v2} is linearly-independent iff ( v2 <> 0. V & ( for a, b being Element of R st b <> 0. R holds
b * v1 <> a * v2 ) ) )
let v1, v2 be Vector of V; :: thesis: ( R = INT.Ring & V is Mult-cancelable implies ( v1 <> v2 & {v1,v2} is linearly-independent iff ( v2 <> 0. V & ( for a, b being Element of R st b <> 0. R holds
b * v1 <> a * v2 ) ) ) )
assume A1: ( R = INT.Ring & V is Mult-cancelable ) ; :: thesis: ( v1 <> v2 & {v1,v2} is linearly-independent iff ( v2 <> 0. V & ( for a, b being Element of R st b <> 0. R holds
b * v1 <> a * v2 ) ) )
thus ( v1 <> v2 & {v1,v2} is linearly-independent implies ( v2 <> 0. V & ( for a, b being Element of R st b <> 0. R holds
b * v1 <> a * v2 ) ) ) :: thesis: ( v2 <> 0. V & ( for a, b being Element of R st b <> 0. R holds
b * v1 <> a * v2 ) implies ( v1 <> v2 & {v1,v2} is linearly-independent ) )

proof

set N0 = 0. R;
set N1 = - (1. R);
deffunc H₁( Element of V) -> Element of the carrier of R = 0. R;
assume that
A2: v1 <> v2 and
A3: {v1,v2} is linearly-independent ; :: thesis: ( v2 <> 0. V & ( for a, b being Element of R st b <> 0. R holds
b * v1 <> a * v2 ) )
thus v2 <> 0. V by A3, Th60, A1; :: thesis: for a, b being Element of R st b <> 0. R holds
b * v1 <> a * v2
let a, b be Element of R; :: thesis: ( b <> 0. R implies b * v1 <> a * v2 )
assume A4: b <> 0. R ; :: thesis: b * v1 <> a * v2
set Na = a;
set Nb = - b;
consider f being Function of V,R such that
A5: ( f . v1 = - b & f . v2 = a ) and
A6: for v being Element of V st v <> v1 & v <> v2 holds
f . v = H₁(v) from FUNCT_2:sch 7(A2);
reconsider f = f as Element of Funcs ( the carrier of V, the carrier of R) by FUNCT_2:8;

now :: thesis: for v being Vector of V st not v in {v1,v2} holds
f . v = 0. R

let v be Vector of V; :: thesis: ( not v in {v1,v2} implies f . v = 0. R )
assume not v in {v1,v2} ; :: thesis: f . v = 0. R
then ( v <> v1 & v <> v2 ) by TARSKI:def 2;
hence f . v = 0. R by A6; :: thesis: verum

end;

then reconsider f = f as Linear_Combination of V by VECTSP_6:def 1;
Carrier f c= {v1,v2}

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Carrier f or x in {v1,v2} )
assume x in Carrier f ; :: thesis: x in {v1,v2}
then A7: ex u being Vector of V st
( x = u & f . u <> 0. R ) ;
assume not x in {v1,v2} ; :: thesis: contradiction
then ( x <> v1 & x <> v2 ) by TARSKI:def 2;
hence contradiction by A6, A7; :: thesis: verum

end;

then reconsider f = f as Linear_Combination of {v1,v2} by VECTSP_6:def 4;
- b <> 0. R by A4, VECTSP_1:28;
then f . v1 <> 0. R by A5;
then A8: v1 in Carrier f ;
set w = a * v2;
assume A9: b * v1 = a * v2 ; :: thesis: contradiction
Sum f = ((- b) * v1) + (a * v2) by A2, A5, Th22
.= (b * (- v1)) + (a * v2) by ZMODUL01:5, A1
.= (- (a * v2)) + (a * v2) by A9, ZMODUL01:6, A1
.= - ((a * v2) - (a * v2)) by RLVECT_1:33
.= - (0. V) by RLVECT_1:15
.= 0. V by RLVECT_1:12 ;
then Carrier f = {} by VECTSP_7:def 1, A3;
hence contradiction by A8; :: thesis: verum

end;

assume A10: v2 <> 0. V ; :: thesis: ( ex a, b being Element of R st
( b <> 0. R & not b * v1 <> a * v2 ) or ( v1 <> v2 & {v1,v2} is linearly-independent ) )
assume A11: for a, b being Element of R st b <> 0. R holds
b * v1 <> a * v2 ; :: thesis: ( v1 <> v2 & {v1,v2} is linearly-independent )
A12: ( (1. R) * v2 = v2 & (1. R) * v1 = v1 ) ;
hence v1 <> v2 by A11, A1; :: thesis: {v1,v2} is linearly-independent
let l be Linear_Combination of {v1,v2}; :: according to VECTSP_7:def 1 :: thesis: ( not Sum l = 0. V or Carrier l = {} )
assume that
A13: Sum l = 0. V and
A14: Carrier l <> {} ; :: thesis: contradiction
A15: 0. V = ((l . v1) * v1) + ((l . v2) * v2) by A11, A12, A13, Th22, A1;
set x = the Element of Carrier l;
Carrier l c= {v1,v2} by VECTSP_6:def 4;
then A16: the Element of Carrier l in {v1,v2} by A14;
the Element of Carrier l in Carrier l by A14;
then A17: ex u being Vector of V st
( the Element of Carrier l = u & l . u <> 0. R ) ;

now :: thesis: contradiction

per cases ( l . v1 <> 0. R or ( l . v2 <> 0. R & l . v1 = 0. R ) ) by A17, A16, TARSKI:def 2;

suppose A18: l . v1 <> 0. R ; :: thesis: contradiction

(l . v1) * v1 = - ((l . v2) * v2) by A15, RLVECT_1:6
.= (- (1. R)) * ((l . v2) * v2) by ZMODUL01:2
.= ((- (1. R)) * (l . v2)) * v2 by VECTSP_1:def 16 ;
hence contradiction by A11, A18; :: thesis: verum

end;

suppose A19: ( l . v2 <> 0. R & l . v1 = 0. R ) ; :: thesis: contradiction

0. V = ((l . v1) * v1) + ((l . v2) * v2) by A11, A12, A13, Th22, A1
.= (0. V) + ((l . v2) * v2) by A19, ZMODUL01:1, A1
.= (l . v2) * v2 by RLVECT_1:4 ;
hence contradiction by A1, A10, A19; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum