let V be RealLinearSpace; :: thesis: for u, v, w being Element of V st ( u = 0. V or v = 0. V or w = 0. V ) holds
u,v,w are_LinDep
let u, v, w be Element of V; :: thesis: ( ( u = 0. V or v = 0. V or w = 0. V ) implies u,v,w are_LinDep )
A1: for u, v, w being Element of V st u = 0. V holds
u,v,w are_LinDep
proof
let u, v, w be Element of V; :: thesis: ( u = 0. V implies u,v,w are_LinDep )
assume A2: u = 0. V ; :: thesis: u,v,w are_LinDep
0. V = (0. V) + (0. V)
.= (1 * u) + (0. V) by A2
.= (1 * u) + (0 * v) by RLVECT_1:10
.= ((1 * u) + (0 * v)) + (0. V)
.= ((1 * u) + (0 * v)) + (0 * w) by RLVECT_1:10 ;
hence u,v,w are_LinDep ; :: thesis: verum
end;
A3: now :: thesis: ( v = 0. V & ( u = 0. V or v = 0. V or w = 0. V ) implies u,v,w are_LinDep )
assume v = 0. V ; :: thesis: ( ( u = 0. V or v = 0. V or w = 0. V ) implies u,v,w are_LinDep )
then v,w,u are_LinDep by A1;
hence ( ( u = 0. V or v = 0. V or w = 0. V ) implies u,v,w are_LinDep ) by Th5; :: thesis: verum
end;
A4: now :: thesis: ( w = 0. V & ( u = 0. V or v = 0. V or w = 0. V ) implies u,v,w are_LinDep )
assume w = 0. V ; :: thesis: ( ( u = 0. V or v = 0. V or w = 0. V ) implies u,v,w are_LinDep )
then w,u,v are_LinDep by A1;
hence ( ( u = 0. V or v = 0. V or w = 0. V ) implies u,v,w are_LinDep ) by Th5; :: thesis: verum
end;
assume ( u = 0. V or v = 0. V or w = 0. V ) ; :: thesis: u,v,w are_LinDep
hence u,v,w are_LinDep by A1, A3, A4; :: thesis: verum