let V be free finite-rank Z_Module; :: thesis: ( rank V = 0 iff (Omega). V = (0). V )
consider I being finite Subset of V such that
A1: I is Basis of V by ZMODUL03:def 3;
hereby :: thesis: ( (Omega). V = (0). V implies rank V = 0 )
consider I being finite Subset of V such that
A2: I is Basis of V by ZMODUL03:def 3;
assume rank V = 0 ; :: thesis: (Omega). V = (0). V
then card I = 0 by A2, ZMODUL03:def 5;
then A3: I = {} the carrier of V ;
(Omega). V = Lin I by A2, VECTSP_7:def 3
.= (0). V by A3, ZMODUL02:67 ;
hence (Omega). V = (0). V ; :: thesis: verum
end;
assume (Omega). V = (0). V ; :: thesis: rank V = 0
then Lin I = (0). V by A1, VECTSP_7:def 3;
then ( I = {} or I = {(0. V)} ) by ZMODUL02:68;
hence rank V = 0 by A1, VECTSP_7:def 3, ZMODUL03:def 5, CARD_1:27; :: thesis: verum