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:55;
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) by BINOP_1:def 12;
(G [;] d,(id D)) . (the_unity_wrt F) = the_unity_wrt F by A1, A2, A3, FINSEQOP:73;
hence (G [;] d,(id D)) . (F $$ B,f) = F $$ B,((G [;] d,(id D)) * f) by A1, A4, Th18; :: thesis: verum