let L be Semilattice; :: thesis: for x being Element of L holds downarrow x is meet-closed

let x be Element of L; :: thesis: downarrow x is meet-closed

reconsider x1 = x as Element of L ;

hence downarrow x is meet-closed ; :: thesis: verum

let x be Element of L; :: thesis: downarrow x is meet-closed

reconsider x1 = x as Element of L ;

now :: thesis: for y, z being Element of L st y in the carrier of (subrelstr (downarrow x)) & z in the carrier of (subrelstr (downarrow x)) & ex_inf_of {y,z},L holds

inf {y,z} in the carrier of (subrelstr (downarrow x))

then
subrelstr (downarrow x) is meet-inheriting
;inf {y,z} in the carrier of (subrelstr (downarrow x))

let y, z be Element of L; :: thesis: ( y in the carrier of (subrelstr (downarrow x)) & z in the carrier of (subrelstr (downarrow x)) & ex_inf_of {y,z},L implies inf {y,z} in the carrier of (subrelstr (downarrow x)) )

assume that

A1: y in the carrier of (subrelstr (downarrow x)) and

z in the carrier of (subrelstr (downarrow x)) and

ex_inf_of {y,z},L ; :: thesis: inf {y,z} in the carrier of (subrelstr (downarrow x))

y in downarrow x by A1, YELLOW_0:def 15;

then A2: y <= x1 by WAYBEL_0:17;

y "/\" z <= y by YELLOW_0:23;

then y "/\" z <= x1 by A2, YELLOW_0:def 2;

then y "/\" z in downarrow x by WAYBEL_0:17;

then inf {y,z} in downarrow x by YELLOW_0:40;

hence inf {y,z} in the carrier of (subrelstr (downarrow x)) by YELLOW_0:def 15; :: thesis: verum

end;assume that

A1: y in the carrier of (subrelstr (downarrow x)) and

z in the carrier of (subrelstr (downarrow x)) and

ex_inf_of {y,z},L ; :: thesis: inf {y,z} in the carrier of (subrelstr (downarrow x))

y in downarrow x by A1, YELLOW_0:def 15;

then A2: y <= x1 by WAYBEL_0:17;

y "/\" z <= y by YELLOW_0:23;

then y "/\" z <= x1 by A2, YELLOW_0:def 2;

then y "/\" z in downarrow x by WAYBEL_0:17;

then inf {y,z} in downarrow x by YELLOW_0:40;

hence inf {y,z} in the carrier of (subrelstr (downarrow x)) by YELLOW_0:def 15; :: thesis: verum

hence downarrow x is meet-closed ; :: thesis: verum