let G1, G2 be Group; for f being Homomorphism of G1,G2 st f is one-to-one holds
FuncLatt f is one-to-one
let f be Homomorphism of G1,G2; ( f is one-to-one implies FuncLatt f is one-to-one )
assume A1:
f is one-to-one
; FuncLatt f is one-to-one
for x1, x2 being object st x1 in dom (FuncLatt f) & x2 in dom (FuncLatt f) & (FuncLatt f) . x1 = (FuncLatt f) . x2 holds
x1 = x2
proof
reconsider y =
f . (1_ G1) as
Element of
G2 ;
let x1,
x2 be
object ;
( x1 in dom (FuncLatt f) & x2 in dom (FuncLatt f) & (FuncLatt f) . x1 = (FuncLatt f) . x2 implies x1 = x2 )
assume that A2:
(
x1 in dom (FuncLatt f) &
x2 in dom (FuncLatt f) )
and A3:
(FuncLatt f) . x1 = (FuncLatt f) . x2
;
x1 = x2
reconsider x1 =
x1,
x2 =
x2 as
strict Subgroup of
G1 by A2, GROUP_3:def 1;
reconsider A1 =
f .: the
carrier of
x1,
A2 =
f .: the
carrier of
x2 as
Subset of
G2 ;
A4:
for
g being
Element of
G2 st
g in f .: the
carrier of
x1 holds
g " in f .: the
carrier of
x1
A8:
for
g1,
g2 being
Element of
G2 st
g1 in f .: the
carrier of
x1 &
g2 in f .: the
carrier of
x1 holds
g1 * g2 in f .: the
carrier of
x1
A17:
for
g being
Element of
G2 st
g in f .: the
carrier of
x2 holds
g " in f .: the
carrier of
x2
A21:
dom f = the
carrier of
G1
by FUNCT_2:def 1;
A22:
for
g1,
g2 being
Element of
G2 st
g1 in f .: the
carrier of
x2 &
g2 in f .: the
carrier of
x2 holds
g1 * g2 in f .: the
carrier of
x2
1_ G1 in x2
by GROUP_2:46;
then A31:
1_ G1 in the
carrier of
x2
by STRUCT_0:def 5;
A32:
(
(FuncLatt f) . x1 = gr A1 &
(FuncLatt f) . x2 = gr A2 )
by Def3;
ex
y being
Element of
G2 st
y = f . (1_ G1)
;
then
f .: the
carrier of
x2 <> {}
by A21, A31, FUNCT_1:def 6;
then consider B2 being
strict Subgroup of
G2 such that A33:
the
carrier of
B2 = f .: the
carrier of
x2
by A17, A22, GROUP_2:52;
1_ G1 in x1
by GROUP_2:46;
then
1_ G1 in the
carrier of
x1
by STRUCT_0:def 5;
then
y in f .: the
carrier of
x1
by A21, FUNCT_1:def 6;
then A34:
ex
B1 being
strict Subgroup of
G2 st the
carrier of
B1 = f .: the
carrier of
x1
by A4, A8, GROUP_2:52;
gr (f .: the carrier of x2) = B2
by A33, Th3;
then A35:
f .: the
carrier of
x1 = f .: the
carrier of
x2
by A3, A32, A34, A33, Th3;
the
carrier of
x2 c= dom f
by A21, GROUP_2:def 5;
then A36:
the
carrier of
x2 c= the
carrier of
x1
by A1, A35, FUNCT_1:87;
the
carrier of
x1 c= dom f
by A21, GROUP_2:def 5;
then
the
carrier of
x1 c= the
carrier of
x2
by A1, A35, FUNCT_1:87;
then
the
carrier of
x1 = the
carrier of
x2
by A36;
hence
x1 = x2
by GROUP_2:59;
verum
end;
hence
FuncLatt f is one-to-one
by FUNCT_1:def 4; verum