let K be non empty right_complementable add-associative right_zeroed doubleLoopStr ; :: thesis: for L being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for J being Function of K,L holds
( J is linear iff opp J is antilinear )
let L be non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr ; :: thesis: for J being Function of K,L holds
( J is linear iff opp J is antilinear )
let J be Function of K,L; :: thesis: ( J is linear iff opp J is antilinear )
set J9 = opp J;
set L9 = opp L;

A1: now :: thesis: ( J is linear implies opp J is antilinear )

assume A2: J is linear ; :: thesis: opp J is antilinear
A3: for x, y being Scalar of K holds (opp J) . (x * y) = ((opp J) . y) * ((opp J) . x)

proof

let x, y be Scalar of K; :: thesis: (opp J) . (x * y) = ((opp J) . y) * ((opp J) . x)
thus (opp J) . (x * y) = (J . x) * (J . y) by A2, GROUP_6:def 6
.= ((opp J) . y) * ((opp J) . x) by Lm3 ; :: thesis: verum

end;

(opp J) . (1_ K) = 1_ L by A2, GROUP_1:def 13
.= 1_ (opp L) ;
then ( opp J is additive & opp J is antimultiplicative & opp J is unity-preserving ) by Lm12, A3, A2;
hence opp J is antilinear ; :: thesis: verum

end;

now :: thesis: ( opp J is antilinear implies J is linear )

assume A4: opp J is antilinear ; :: thesis: J is linear
A5: for x, y being Scalar of K holds J . (x + y) = (J . x) + (J . y)

proof

let x, y be Scalar of K; :: thesis: J . (x + y) = (J . x) + (J . y)
thus J . (x + y) = ((opp J) . x) + ((opp J) . y) by A4, VECTSP_1:def 20
.= (J . x) + (J . y) ; :: thesis: verum

end;

A6: for x, y being Scalar of K holds J . (x * y) = (J . x) * (J . y)

proof

let x, y be Scalar of K; :: thesis: J . (x * y) = (J . x) * (J . y)
thus J . (x * y) = ((opp J) . y) * ((opp J) . x) by A4, Def6
.= (J . x) * (J . y) by Lm3 ; :: thesis: verum

end;

J . (1_ K) = 1_ (opp L) by A4, GROUP_1:def 13
.= 1_ L ;
then ( J is additive & J is multiplicative & J is unity-preserving ) by A5, A6;
hence J is linear ; :: thesis: verum

end;

hence ( J is linear iff opp J is antilinear ) by A1; :: thesis: verum