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

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

reconsider x1 = x as Element of L ;

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

let x be Element of L; :: thesis: downarrow x is join-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_sup_of {y,z},L holds

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

then
subrelstr (downarrow x) is join-inheriting
;sup {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_sup_of {y,z},L implies sup {y,z} in the carrier of (subrelstr (downarrow x)) )

assume that

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

A2: z in the carrier of (subrelstr (downarrow x)) and

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

z in downarrow x by A2, YELLOW_0:def 15;

then A3: z <= x1 by WAYBEL_0:17;

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

then y <= x1 by WAYBEL_0:17;

then y "\/" z <= x1 by A3, YELLOW_5:9;

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

then sup {y,z} in downarrow x by YELLOW_0:41;

hence sup {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

A2: z in the carrier of (subrelstr (downarrow x)) and

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

z in downarrow x by A2, YELLOW_0:def 15;

then A3: z <= x1 by WAYBEL_0:17;

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

then y <= x1 by WAYBEL_0:17;

then y "\/" z <= x1 by A3, YELLOW_5:9;

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

then sup {y,z} in downarrow x by YELLOW_0:41;

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

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