let D be non empty set ; :: thesis: for d being Element of D
for F, G being BinOp of D st F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
(G [;] d,(id D)) . (the_unity_wrt F) = the_unity_wrt F
let d be Element of D; :: thesis: for F, G being BinOp of D st F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
(G [;] d,(id D)) . (the_unity_wrt F) = the_unity_wrt F
let F, G be BinOp of D; :: thesis: ( F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F implies (G [;] d,(id D)) . (the_unity_wrt F) = the_unity_wrt F )
assume that
A1: F is associative and
A2: F is having_a_unity and
A3: F is having_an_inverseOp and
A4: G is_distributive_wrt F ; :: thesis: (G [;] d,(id D)) . (the_unity_wrt F) = the_unity_wrt F
set e = the_unity_wrt F;
set i = the_inverseOp_wrt F;
G . d,(the_unity_wrt F) = G . d,(F . (the_unity_wrt F),(the_unity_wrt F)) by A2, SETWISEO:23
.= F . (G . d,(the_unity_wrt F)),(G . d,(the_unity_wrt F)) by A4, BINOP_1:23 ;
then the_unity_wrt F = F . (F . (G . d,(the_unity_wrt F)),(G . d,(the_unity_wrt F))),((the_inverseOp_wrt F) . (G . d,(the_unity_wrt F))) by A1, A2, A3, Th63;
then the_unity_wrt F = F . (G . d,(the_unity_wrt F)),(F . (G . d,(the_unity_wrt F)),((the_inverseOp_wrt F) . (G . d,(the_unity_wrt F)))) by A1, BINOP_1:def 3;
then the_unity_wrt F = F . (G . d,(the_unity_wrt F)),(the_unity_wrt F) by A1, A2, A3, Th63;
then the_unity_wrt F = G . d,(the_unity_wrt F) by A2, SETWISEO:23;
then G . d,((id D) . (the_unity_wrt F)) = the_unity_wrt F by FUNCT_1:35;
hence (G [;] d,(id D)) . (the_unity_wrt F) = the_unity_wrt F by FUNCOP_1:66; :: thesis: verum