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 [:] ((id D),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 [:] ((id D),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 [:] ((id D),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 [:] ((id D),d)) . (the_unity_wrt F) = the_unity_wrt F
set e = the_unity_wrt F;
set i = the_inverseOp_wrt F;
G . ((the_unity_wrt F),d) = G . ((F . ((the_unity_wrt F),(the_unity_wrt F))),d) by A2, SETWISEO:15
.= F . ((G . ((the_unity_wrt F),d)),(G . ((the_unity_wrt F),d))) by A4, BINOP_1:11 ;
then the_unity_wrt F = F . ((F . ((G . ((the_unity_wrt F),d)),(G . ((the_unity_wrt F),d)))),((the_inverseOp_wrt F) . (G . ((the_unity_wrt F),d)))) by A1, A2, A3, Th59;
then the_unity_wrt F = F . ((G . ((the_unity_wrt F),d)),(F . ((G . ((the_unity_wrt F),d)),((the_inverseOp_wrt F) . (G . ((the_unity_wrt F),d)))))) by A1;
then the_unity_wrt F = F . ((G . ((the_unity_wrt F),d)),(the_unity_wrt F)) by A1, A2, A3, Th59;
then the_unity_wrt F = G . ((the_unity_wrt F),d) by A2, SETWISEO:15;
then G . (((id D) . (the_unity_wrt F)),d) = the_unity_wrt F ;
hence (G [:] ((id D),d)) . (the_unity_wrt F) = the_unity_wrt F by FUNCOP_1:48; :: thesis: verum