let L be with_infima Poset; :: thesis: for I being Ideal of L holds
( I is prime iff for A being non empty finite Subset of L st inf A in I holds
ex a being Element of L st
( a in A & a in I ) )
let I be Ideal of L; :: thesis: ( I is prime iff for A being non empty finite Subset of L st inf A in I holds
ex a being Element of L st
( a in A & a in I ) )
thus ( I is prime implies for A being non empty finite Subset of L st inf A in I holds
ex a being Element of L st
( a in A & a in I ) ) :: thesis: ( ( for A being non empty finite Subset of L st inf A in I holds
ex a being Element of L st
( a in A & a in I ) ) implies I is prime )

proof

assume A1: for x, y being Element of L holds
( not x "/\" y in I or x in I or y in I ) ; :: according to WAYBEL_7:def 1 :: thesis: for A being non empty finite Subset of L st inf A in I holds
ex a being Element of L st
( a in A & a in I )
let A be non empty finite Subset of L; :: thesis: ( inf A in I implies ex a being Element of L st
( a in A & a in I ) )
A2: A is finite ;
defpred S₁[ set ] means ( not $1 is empty & "/\" $1,L in I implies ex a being Element of L st
( a in $1 & a in I ) );
A3: S₁[ {} ] ;

A4: now

let x, B be set ; :: thesis: ( x in A & B c= A & S₁[B] implies S₁[B \/ {x}] )
assume that
A5: ( x in A & B c= A ) and
A6: S₁[B] ; :: thesis: S₁[B \/ {x}]
thus S₁[B \/ {x}] :: thesis: verum

proof

assume A7: ( not B \/ {x} is empty & "/\" (B \/ {x}),L in I ) ; :: thesis: ex a being Element of L st
( a in B \/ {x} & a in I )
reconsider C = B as finite Subset of L by A5, XBOOLE_1:1;
reconsider a = x as Element of L by A5;

per cases ( B = {} or B <> {} ) ;

suppose B = {} ; :: thesis: ex a being Element of L st
( a in B \/ {x} & a in I )

then ( "/\" (B \/ {a}),L = a & a in B \/ {a} ) by TARSKI:def 1, YELLOW_0:39;
hence ex a being Element of L st
( a in B \/ {x} & a in I ) by A7; :: thesis: verum

end;

suppose A8: B <> {} ; :: thesis: ex a being Element of L st
( a in B \/ {x} & a in I )

then ( ex_inf_of C,L & ex_inf_of {a},L ) by YELLOW_0:55;
then A9: ( "/\" (B \/ {x}),L = (inf C) "/\" (inf {a}) & inf {a} = a ) by YELLOW_0:39, YELLOW_2:4;

hereby :: thesis: verum

per cases ( inf C in I or a in I ) by A1, A7, A9;

suppose inf C in I ; :: thesis: ex b being Element of L st
( b in B \/ {x} & b in I )

then consider b being Element of L such that
A10: ( b in B & b in I ) by A6, A8;
take b = b; :: thesis: ( b in B \/ {x} & b in I )
thus ( b in B \/ {x} & b in I ) by A10, XBOOLE_0:def 3; :: thesis: verum

end;

suppose A11: a in I ; :: thesis: ex a being Element of L st
( a in B \/ {x} & a in I )

take a = a; :: thesis: ( a in B \/ {x} & a in I )
a in {a} by TARSKI:def 1;
hence ( a in B \/ {x} & a in I ) by A11, XBOOLE_0:def 3; :: thesis: verum

end;

end;

end;

end;

end;

end;

end;

S₁[A] from FINSET_1:sch 2(A2, A3, A4);
hence ( inf A in I implies ex a being Element of L st
( a in A & a in I ) ) ; :: thesis: verum

end;

assume A12: for A being non empty finite Subset of L st inf A in I holds
ex a being Element of L st
( a in A & a in I ) ; :: thesis: I is prime
let a, b be Element of L; :: according to WAYBEL_7:def 1 :: thesis: ( not a "/\" b in I or a in I or b in I )
assume a "/\" b in I ; :: thesis: ( a in I or b in I )
then inf {a,b} in I by YELLOW_0:40;
then consider x being Element of L such that
A13: ( x in {a,b} & x in I ) by A12;
thus ( a in I or b in I ) by A13, TARSKI:def 2; :: thesis: verum