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 antilinear iff opp J is linear )
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 antilinear iff opp J is linear )
let J be Function of K,L; :: thesis: ( J is antilinear iff opp J is linear )
set J9 = opp J;
set L9 = opp L;

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

assume A1: J is antilinear ; :: thesis: opp J is linear
A2: opp J is additive

proof

let x, y be Scalar of K; :: according to VECTSP_1:def 19 :: thesis: (opp J) . (x + y) = ((opp J) . x) + ((opp J) . y)
thus (opp J) . (x + y) = (J . x) + (J . y) by A1, VECTSP_1:def 20
.= ((opp J) . x) + ((opp J) . y) ; :: thesis: verum

end;

A3: opp J is multiplicative

proof

let x, y be Scalar of K; :: according to GROUP_6:def 6 :: thesis: (opp J) . (x * y) = ((opp J) . x) * ((opp J) . y)
thus (opp J) . (x * y) = (J . y) * (J . x) by A1, Def6
.= ((opp J) . x) * ((opp J) . y) by Lm3 ; :: thesis: verum

end;

(opp J) . (1_ K) = 1_ L by A1, GROUP_1:def 13
.= 1_ (opp L) ;
then opp J is unity-preserving ;
hence opp J is linear by A2, A3; :: thesis: verum

end;

assume A4: ( opp J is additive & opp J is multiplicative & opp J is unity-preserving ) ; :: according to RINGCAT1:def 1 :: thesis: J is antilinear

hereby :: according to VECTSP_1:def 19,MOD_4:def 7 :: thesis: ( J is antimultiplicative & J is unity-preserving )

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
.= (J . x) + (J . y) ; :: thesis: verum

end;

hereby :: according to MOD_4:def 6 :: thesis: J is unity-preserving

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

end;

thus J . (1_ K) = 1_ L by A4; :: according to GROUP_1:def 13 :: thesis: verum