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:23
.= F . (G . (the_unity_wrt F),d),(G . (the_unity_wrt F),d) by A4, BINOP_1:23 ;
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, Th63;
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, BINOP_1:def 3;
then the_unity_wrt F = F . (G . (the_unity_wrt F),d),(the_unity_wrt F) by A1, A2, A3, Th63;
then the_unity_wrt F = G . (the_unity_wrt F),d by A2, SETWISEO:23;
then G . ((id D) . (the_unity_wrt F)),d = the_unity_wrt F by FUNCT_1:35;
hence (G [:] (id D),d) . (the_unity_wrt F) = the_unity_wrt F by FUNCOP_1:60; :: thesis: verum