let R be Ring; :: thesis: for V being LeftMod of R

for L being Linear_Combination of V

for v being Element of V holds

( L . v = 0. R iff not v in Carrier L )

let V be LeftMod of R; :: thesis: for L being Linear_Combination of V

for v being Element of V holds

( L . v = 0. R iff not v in Carrier L )

let L be Linear_Combination of V; :: thesis: for v being Element of V holds

( L . v = 0. R iff not v in Carrier L )

let v be Element of V; :: thesis: ( L . v = 0. R iff not v in Carrier L )

thus ( L . v = 0. R implies not v in Carrier L ) :: thesis: ( not v in Carrier L implies L . v = 0. R )

hence L . v = 0. R ; :: thesis: verum

for L being Linear_Combination of V

for v being Element of V holds

( L . v = 0. R iff not v in Carrier L )

let V be LeftMod of R; :: thesis: for L being Linear_Combination of V

for v being Element of V holds

( L . v = 0. R iff not v in Carrier L )

let L be Linear_Combination of V; :: thesis: for v being Element of V holds

( L . v = 0. R iff not v in Carrier L )

let v be Element of V; :: thesis: ( L . v = 0. R iff not v in Carrier L )

thus ( L . v = 0. R implies not v in Carrier L ) :: thesis: ( not v in Carrier L implies L . v = 0. R )

proof

assume
not v in Carrier L
; :: thesis: L . v = 0. R
assume A1:
L . v = 0. R
; :: thesis: not v in Carrier L

assume v in Carrier L ; :: thesis: contradiction

then ex u being Element of V st

( u = v & L . u <> 0. R ) ;

hence contradiction by A1; :: thesis: verum

end;assume v in Carrier L ; :: thesis: contradiction

then ex u being Element of V st

( u = v & L . u <> 0. R ) ;

hence contradiction by A1; :: thesis: verum

hence L . v = 0. R ; :: thesis: verum