let G be addGroup; :: thesis: for a being Element of G
for H1, H2 being Subgroup of G holds (H1 /\ H2) * a = (H1 * a) /\ (H2 * a)
let a be Element of G; :: thesis: for H1, H2 being Subgroup of G holds (H1 /\ H2) * a = (H1 * a) /\ (H2 * a)
let H1, H2 be Subgroup of G; :: thesis: (H1 /\ H2) * a = (H1 * a) /\ (H2 * a)
let g be Element of G; :: according to GROUP_1A:def 16 :: thesis: ( g in (H1 /\ H2) * a iff g in (H1 * a) /\ (H2 * a) )
thus ( g in (H1 /\ H2) * a implies g in (H1 * a) /\ (H2 * a) ) :: thesis: ( g in (H1 * a) /\ (H2 * a) implies g in (H1 /\ H2) * a )
proof
assume g in (H1 /\ H2) * a ; :: thesis: g in (H1 * a) /\ (H2 * a)
then consider h being Element of G such that
A1: g = h * a and
A2: h in H1 /\ H2 by Th58;
h in H2 by A2, Th82;
then A3: g in H2 * a by A1, Th58;
h in H1 by A2, Th82;
then g in H1 * a by A1, Th58;
hence g in (H1 * a) /\ (H2 * a) by A3, Th82; :: thesis: verum
end;
assume A4: g in (H1 * a) /\ (H2 * a) ; :: thesis: g in (H1 /\ H2) * a
then g in H1 * a by Th82;
then consider b being Element of G such that
A5: g = b * a and
A6: b in H1 by Th58;
g in H2 * a by A4, Th82;
then consider c being Element of G such that
A7: g = c * a and
A8: c in H2 by Th58;
b = c by A5, A7, ThB16;
then b in H1 /\ H2 by A6, A8, Th82;
hence g in (H1 /\ H2) * a by A5, Th58; :: thesis: verum