let BL be Boolean Lattice; :: thesis: for B being Element of Fin the carrier of BL holds (FinMeet B) ` = FinJoin (B,(comp BL))

let B be Element of Fin the carrier of BL; :: thesis: (FinMeet B) ` = FinJoin (B,(comp BL))

set M = the L_meet of BL;

set J = the L_join of BL;

A1: for a, b being Element of BL holds (comp BL) . ( the L_meet of BL . (a,b)) = the L_join of BL . (((comp BL) . a),((comp BL) . b))

.= (Top BL) ` by LATTICE2:57

.= Bottom BL by Th29

.= the_unity_wrt the L_join of BL by LATTICE2:52 ;

thus (FinMeet B) ` = ( the L_meet of BL $$ (B,(id BL))) ` by LATTICE2:def 4

.= (comp BL) . ( the L_meet of BL $$ (B,(id BL))) by Def12

.= the L_join of BL $$ (B,((comp BL) * (id BL))) by A2, A1, SETWISEO:36

.= the L_join of BL $$ (B,(comp BL)) by FUNCT_2:17

.= FinJoin (B,(comp BL)) by LATTICE2:def 3 ; :: thesis: verum

let B be Element of Fin the carrier of BL; :: thesis: (FinMeet B) ` = FinJoin (B,(comp BL))

set M = the L_meet of BL;

set J = the L_join of BL;

A1: for a, b being Element of BL holds (comp BL) . ( the L_meet of BL . (a,b)) = the L_join of BL . (((comp BL) . a),((comp BL) . b))

proof

A2: (comp BL) . (the_unity_wrt the L_meet of BL) =
(the_unity_wrt the L_meet of BL) `
by Def12
let a, b be Element of BL; :: thesis: (comp BL) . ( the L_meet of BL . (a,b)) = the L_join of BL . (((comp BL) . a),((comp BL) . b))

thus (comp BL) . ( the L_meet of BL . (a,b)) = (a "/\" b) ` by Def12

.= (a `) "\/" (b `) by LATTICES:23

.= the L_join of BL . (((comp BL) . a),(b `)) by Def12

.= the L_join of BL . (((comp BL) . a),((comp BL) . b)) by Def12 ; :: thesis: verum

end;thus (comp BL) . ( the L_meet of BL . (a,b)) = (a "/\" b) ` by Def12

.= (a `) "\/" (b `) by LATTICES:23

.= the L_join of BL . (((comp BL) . a),(b `)) by Def12

.= the L_join of BL . (((comp BL) . a),((comp BL) . b)) by Def12 ; :: thesis: verum

.= (Top BL) ` by LATTICE2:57

.= Bottom BL by Th29

.= the_unity_wrt the L_join of BL by LATTICE2:52 ;

thus (FinMeet B) ` = ( the L_meet of BL $$ (B,(id BL))) ` by LATTICE2:def 4

.= (comp BL) . ( the L_meet of BL $$ (B,(id BL))) by Def12

.= the L_join of BL $$ (B,((comp BL) * (id BL))) by A2, A1, SETWISEO:36

.= the L_join of BL $$ (B,(comp BL)) by FUNCT_2:17

.= FinJoin (B,(comp BL)) by LATTICE2:def 3 ; :: thesis: verum