let G be Group; :: thesis: for H being Subgroup of G holds Left_Cosets H, Right_Cosets H are_equipotent
let H be Subgroup of G; :: thesis:
defpred S1[ object , object ] means ex g being Element of G st
( \$1 = g * H & \$2 = H * (g ") );
A1: for x being object st x in Left_Cosets H holds
ex y being object st S1[x,y]
proof
let x be object ; :: thesis: ( x in Left_Cosets H implies ex y being object st S1[x,y] )
assume x in Left_Cosets H ; :: thesis: ex y being object st S1[x,y]
then consider g being Element of G such that
A2: x = g * H by Def15;
reconsider y = H * (g ") as set ;
take y ; :: thesis: S1[x,y]
take g ; :: thesis: ( x = g * H & y = H * (g ") )
thus ( x = g * H & y = H * (g ") ) by A2; :: thesis: verum
end;
consider f being Function such that
A3: dom f = Left_Cosets H and
A4: for x being object st x in Left_Cosets H holds
S1[x,f . x] from A5: rng f = Right_Cosets H
proof
thus rng f c= Right_Cosets H :: according to XBOOLE_0:def 10 :: thesis:
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng f or x in Right_Cosets H )
assume x in rng f ; :: thesis:
then consider y being object such that
A6: y in dom f and
A7: f . y = x by FUNCT_1:def 3;
ex g being Element of G st
( y = g * H & f . y = H * (g ") ) by A3, A4, A6;
hence x in Right_Cosets H by ; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Right_Cosets H or x in rng f )
assume A8: x in Right_Cosets H ; :: thesis: x in rng f
then reconsider A = x as Subset of G ;
consider g being Element of G such that
A9: A = H * g by ;
A10: (g ") * H in Left_Cosets H by Def15;
then A11: f . ((g ") * H) in rng f by ;
consider a being Element of G such that
A12: (g ") * H = a * H and
A13: f . ((g ") * H) = H * (a ") by ;
(a ") * (g ") in H by ;
hence x in rng f by ; :: thesis: verum
end;
f is one-to-one
proof
let x, y be object ; :: according to FUNCT_1:def 4 :: thesis: ( not x in dom f or not y in dom f or not f . x = f . y or x = y )
assume that
A14: x in dom f and
A15: y in dom f and
A16: f . x = f . y ; :: thesis: x = y
consider b being Element of G such that
A17: y = b * H and
A18: f . y = H * (b ") by A3, A4, A15;
consider a being Element of G such that
A19: x = a * H and
A20: f . x = H * (a ") by A3, A4, A14;
(b ") * ((a ") ") in H by ;
hence x = y by ; :: thesis: verum
end;
hence Left_Cosets H, Right_Cosets H are_equipotent by ; :: thesis: verum