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 )

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 ex x being Element of L st

( x in {a,b} & x in I ) by A15;

hence ( a in I or b in I ) by TARSKI:def 2; :: thesis: verum

( 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 A15:
for A being non empty finite Subset of L st inf A in I holds
defpred S_{1}[ set ] means ( not $1 is empty & "/\" ($1,L) in I implies ex a being Element of L st

( a in $1 & a in I ) );

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 ) )

_{1}[ {} ]
;

A14: A is finite ;

S_{1}[A]
from FINSET_1:sch 2(A14, A13, A2);

hence ( inf A in I implies ex a being Element of L st

( a in A & a in I ) ) ; :: thesis: verum

end;( a in $1 & a in I ) );

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: now :: thesis: for x, B being set st x in A & B c= A & S_{1}[B] holds

S_{1}[B \/ {x}]

A13:
SS

let x, B be set ; :: thesis: ( x in A & B c= A & S_{1}[B] implies S_{1}[B \/ {x}] )

assume that

A3: x in A and

A4: B c= A and

A5: S_{1}[B]
; :: thesis: S_{1}[B \/ {x}]

thus S_{1}[B \/ {x}]
:: thesis: verum

end;assume that

A3: x in A and

A4: B c= A and

A5: S

thus S

proof

reconsider a = x as Element of L by A3;

reconsider C = B as finite Subset of L by A4, XBOOLE_1:1;

assume that

not B \/ {x} is empty and

A6: "/\" ((B \/ {x}),L) in I ; :: thesis: ex a being Element of L st

( a in B \/ {x} & a in I )

end;reconsider C = B as finite Subset of L by A4, XBOOLE_1:1;

assume that

not B \/ {x} is empty and

A6: "/\" ((B \/ {x}),L) in I ; :: thesis: ex a being Element of L st

( a in B \/ {x} & a in I )

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

end;

suppose
B = {}
; :: thesis: ex a being Element of L st

( a in B \/ {x} & a in I )

( 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 A6; :: thesis: verum

end;hence ex a being Element of L st

( a in B \/ {x} & a in I ) by A6; :: thesis: verum

suppose A7:
B <> {}
; :: thesis: ex a being Element of L st

( a in B \/ {x} & a in I )

( a in B \/ {x} & a in I )

A8:
inf {a} = a
by YELLOW_0:39;

A9: ex_inf_of {a},L by YELLOW_0:55;

ex_inf_of C,L by A7, YELLOW_0:55;

then A10: "/\" ((B \/ {x}),L) = (inf C) "/\" (inf {a}) by A9, YELLOW_2:4;

end;A9: ex_inf_of {a},L by YELLOW_0:55;

ex_inf_of C,L by A7, YELLOW_0:55;

then A10: "/\" ((B \/ {x}),L) = (inf C) "/\" (inf {a}) by A9, YELLOW_2:4;

hereby :: thesis: verum
end;

per cases
( inf C in I or a in I )
by A1, A6, A10, A8;

end;

A14: A is finite ;

S

hence ( inf A in I implies ex a being Element of L st

( a in A & a in I ) ) ; :: thesis: verum

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 ex x being Element of L st

( x in {a,b} & x in I ) by A15;

hence ( a in I or b in I ) by TARSKI:def 2; :: thesis: verum