let X be BCI-algebra; :: thesis: for I being Ideal of X holds

( I is p-ideal of X iff for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds

x \ y in I )

let I be Ideal of X; :: thesis: ( I is p-ideal of X iff for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds

x \ y in I )

thus ( I is p-ideal of X implies for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds

x \ y in I ) :: thesis: ( ( for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds

x \ y in I ) implies I is p-ideal of X )

x \ y in I ; :: thesis: I is p-ideal of X

A4: for x, y, z being Element of X st (x \ z) \ (y \ z) in I & y in I holds

x in I

hence I is p-ideal of X by A4, Def5; :: thesis: verum

( I is p-ideal of X iff for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds

x \ y in I )

let I be Ideal of X; :: thesis: ( I is p-ideal of X iff for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds

x \ y in I )

thus ( I is p-ideal of X implies for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds

x \ y in I ) :: thesis: ( ( for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds

x \ y in I ) implies I is p-ideal of X )

proof

assume A3:
for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds
assume A1:
I is p-ideal of X
; :: thesis: for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds

x \ y in I

let x, y, z be Element of X; :: thesis: ( (x \ z) \ (y \ z) in I implies x \ y in I )

assume A2: (x \ z) \ (y \ z) in I ; :: thesis: x \ y in I

((x \ z) \ (y \ z)) \ (x \ y) = 0. X by BCIALG_1:def 3;

then (x \ z) \ (y \ z) <= x \ y ;

hence x \ y in I by A1, A2, Th21; :: thesis: verum

end;x \ y in I

let x, y, z be Element of X; :: thesis: ( (x \ z) \ (y \ z) in I implies x \ y in I )

assume A2: (x \ z) \ (y \ z) in I ; :: thesis: x \ y in I

((x \ z) \ (y \ z)) \ (x \ y) = 0. X by BCIALG_1:def 3;

then (x \ z) \ (y \ z) <= x \ y ;

hence x \ y in I by A1, A2, Th21; :: thesis: verum

x \ y in I ; :: thesis: I is p-ideal of X

A4: for x, y, z being Element of X st (x \ z) \ (y \ z) in I & y in I holds

x in I

proof

0. X in I
by BCIALG_1:def 18;
let x, y, z be Element of X; :: thesis: ( (x \ z) \ (y \ z) in I & y in I implies x in I )

assume that

A5: (x \ z) \ (y \ z) in I and

A6: y in I ; :: thesis: x in I

x \ y in I by A3, A5;

hence x in I by A6, BCIALG_1:def 18; :: thesis: verum

end;assume that

A5: (x \ z) \ (y \ z) in I and

A6: y in I ; :: thesis: x in I

x \ y in I by A3, A5;

hence x in I by A6, BCIALG_1:def 18; :: thesis: verum

hence I is p-ideal of X by A4, Def5; :: thesis: verum