let C, D be non empty set ; :: thesis: for B being Element of Fin C
for d being Element of D
for F, G being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))
let B be Element of Fin C; :: thesis: for d being Element of D
for F, G being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))
let d be Element of D; :: thesis: for F, G being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))
let F, G be BinOp of D; :: thesis: for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
(G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))
let f be Function of C,D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F implies (G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f)) )
assume that
A1: ( F is commutative & F is associative & F is having_a_unity ) and
A2: F is having_an_inverseOp and
A3: G is_distributive_wrt F ; :: thesis: (G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f))
set e = the_unity_wrt F;
set u = G [;] (d,(id D));
G [;] (d,(id D)) is_distributive_wrt F by A3, FINSEQOP:54;
then A4: for d1, d2 being Element of D holds (G [;] (d,(id D))) . (F . (d1,d2)) = F . (((G [;] (d,(id D))) . d1),((G [;] (d,(id D))) . d2)) ;
(G [;] (d,(id D))) . (the_unity_wrt F) = the_unity_wrt F by A1, A2, A3, FINSEQOP:69;
hence (G [;] (d,(id D))) . (F $$ (B,f)) = F $$ (B,((G [;] (d,(id D))) * f)) by A1, A4, Th16; :: thesis: verum