let D be non empty set ; 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; 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; ( 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
; (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; verum