:: LATSUBGR semantic presentation

for G being Group
for H1, H2 being Subgroup of G holds the carrier of (H1 /\ H2) = the carrier of H1 /\ the carrier of H2

proof end;

theorem Th2: :: LATSUBGR:2

for G being Group
for h being set holds
( h in Subgroups G iff ex H being strict Subgroup of G st h = H )

proof end;

theorem Th3: :: LATSUBGR:3

for G being Group
for A being Subset of G
for H being strict Subgroup of G st A = the carrier of H holds
gr A = H

proof end;

theorem Th4: :: LATSUBGR:4

for G being Group
for H1, H2 being Subgroup of G
for A being Subset of G st A = the carrier of H1 \/ the carrier of H2 holds
H1 "\/" H2 = gr A

proof end;

theorem Th5: :: LATSUBGR:5

for G being Group
for H1, H2 being Subgroup of G
for g being Element of G st ( g in H1 or g in H2 ) holds
g in H1 "\/" H2

proof end;

theorem :: LATSUBGR:6

for G1, G2 being Group
for f being Homomorphism of G1,G2
for H1 being Subgroup of G1 ex H2 being strict Subgroup of G2 st the carrier of H2 = f .: the carrier of H1

proof end;

theorem :: LATSUBGR:7

for G1, G2 being Group
for f being Homomorphism of G1,G2
for H2 being Subgroup of G2 ex H1 being strict Subgroup of G1 st the carrier of H1 = f " the carrier of H2

proof end;

theorem :: LATSUBGR:8

canceled;

theorem :: LATSUBGR:9

canceled;

theorem :: LATSUBGR:10

for G1, G2 being Group
for f being Homomorphism of G1,G2
for H1, H2 being Subgroup of G1
for H3, H4 being Subgroup of G2 st the carrier of H3 = f .: the carrier of H1 & the carrier of H4 = f .: the carrier of H2 & H1 is Subgroup of H2 holds
H3 is Subgroup of H4

proof end;

theorem :: LATSUBGR:11

for G1, G2 being Group
for f being Homomorphism of G1,G2
for H1, H2 being Subgroup of G2
for H3, H4 being Subgroup of G1 st the carrier of H3 = f " the carrier of H1 & the carrier of H4 = f " the carrier of H2 & H1 is Subgroup of H2 holds
H3 is Subgroup of H4

proof end;

theorem :: LATSUBGR:12

for G1, G2 being Group
for f being Function of the carrier of G1,the carrier of G2
for A being Subset of G1 holds f .: A c= f .: the carrier of (gr A)

proof end;

theorem :: LATSUBGR:13

for G1, G2 being Group
for H1, H2 being Subgroup of G1
for f being Function of the carrier of G1,the carrier of G2
for A being Subset of G1 st A = the carrier of H1 \/ the carrier of H2 holds
f .: the carrier of (H1 "\/" H2) = f .: the carrier of (gr A) by Th4;

theorem Th14: :: LATSUBGR:14

for G being Group
for A being Subset of G st A = {(1. G)} holds
gr A = (1). G

proof end;

definition

let G be Group;
func carr G -> Function of Subgroups G, bool the carrier of G means :Def1: :: LATSUBGR:def 1
for H being strict Subgroup of G holds it . H = the carrier of H;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines carr LATSUBGR:def 1 :

theorem :: LATSUBGR:15

canceled;

theorem :: LATSUBGR:16

canceled;

theorem :: LATSUBGR:17

canceled;

theorem Th18: :: LATSUBGR:18

for G being Group
for H being strict Subgroup of G
for x being Element of G holds
( x in (carr G) . H iff x in H )

proof end;

theorem :: LATSUBGR:19

for G being Group
for H being strict Subgroup of G holds 1. G in (carr G) . H

proof end;

theorem :: LATSUBGR:20

for G being Group
for H being strict Subgroup of G holds (carr G) . H <> {}

proof end;

theorem :: LATSUBGR:21

for G being Group
for H being strict Subgroup of G
for g1, g2 being Element of G st g1 in (carr G) . H & g2 in (carr G) . H holds
g1 * g2 in (carr G) . H

proof end;

theorem :: LATSUBGR:22

for G being Group
for H being strict Subgroup of G
for g being Element of G st g in (carr G) . H holds
g " in (carr G) . H

proof end;

theorem Th23: :: LATSUBGR:23

for G being Group
for H1, H2 being strict Subgroup of G holds the carrier of (H1 /\ H2) = ((carr G) . H1) /\ ((carr G) . H2)

proof end;

theorem :: LATSUBGR:24

for G being Group
for H1, H2 being strict Subgroup of G holds (carr G) . (H1 /\ H2) = ((carr G) . H1) /\ ((carr G) . H2)

proof end;

definition

let G be Group;
let F be non empty Subset of (Subgroups G);
func meet F -> strict Subgroup of G means :Def2: :: LATSUBGR:def 2
the carrier of it = meet ((carr G) .: F);
existence

proof end;

uniqueness by GROUP_2:68;

end;

:: deftheorem Def2 defines meet LATSUBGR:def 2 :

theorem :: LATSUBGR:25

for G being Group
for F being non empty Subset of (Subgroups G) st (1). G in F holds
meet F = (1). G

proof end;

theorem :: LATSUBGR:26

for G being Group
for h being Element of Subgroups G
for F being non empty Subset of (Subgroups G) st F = {h} holds
meet F = h

proof end;

theorem Th27: :: LATSUBGR:27

for G being Group
for H1, H2 being Subgroup of G
for h1, h2 being Element of (lattice G) st h1 = H1 & h2 = H2 holds
h1 "\/" h2 = H1 "\/" H2

proof end;

theorem Th28: :: LATSUBGR:28

for G being Group
for H1, H2 being Subgroup of G
for h1, h2 being Element of (lattice G) st h1 = H1 & h2 = H2 holds
h1 "/\" h2 = H1 /\ H2

proof end;

theorem Th29: :: LATSUBGR:29

for G being Group
for p being Element of (lattice G)
for H being Subgroup of G st p = H holds
H is strict Subgroup of G

proof end;

theorem Th30: :: LATSUBGR:30

for G being Group
for H1, H2 being Subgroup of G
for p, q being Element of (lattice G) st p = H1 & q = H2 holds
( p [= q iff the carrier of H1 c= the carrier of H2 )

proof end;

theorem :: LATSUBGR:31

for G being Group
for H1, H2 being Subgroup of G
for p, q being Element of (lattice G) st p = H1 & q = H2 holds
( p [= q iff H1 is Subgroup of H2 )

proof end;

theorem :: LATSUBGR:32

for G being Group holds lattice G is complete

proof end;

definition

let G1, G2 be Group;
let f be Function of the carrier of G1,the carrier of G2;
func FuncLatt f -> Function of the carrier of (lattice G1),the carrier of (lattice G2) means :Def3: :: LATSUBGR:def 3
for H being strict Subgroup of G1
for A being Subset of G2 st A = f .: the carrier of H holds
it . H = gr A;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines FuncLatt LATSUBGR:def 3 :

theorem :: LATSUBGR:33

for G being Group holds FuncLatt (id the carrier of G) = id the carrier of (lattice G)

proof end;

theorem :: LATSUBGR:34

for G1, G2 being Group
for f being Homomorphism of G1,G2 st f is one-to-one holds
FuncLatt f is one-to-one

proof end;

theorem :: LATSUBGR:35

for G1, G2 being Group
for f being Homomorphism of G1,G2 holds (FuncLatt f) . ((1). G1) = (1). G2

proof end;

theorem Th36: :: LATSUBGR:36

for G1, G2 being Group
for f being Homomorphism of G1,G2 st f is one-to-one holds
FuncLatt f is Semilattice-Homomorphism of lattice G1, lattice G2

proof end;

theorem Th37: :: LATSUBGR:37

for G1, G2 being Group
for f being Homomorphism of G1,G2 holds FuncLatt f is sup-Semilattice-Homomorphism of lattice G1, lattice G2

proof end;

theorem :: LATSUBGR:38

for G1, G2 being Group
for f being Homomorphism of G1,G2 st f is one-to-one holds
FuncLatt f is Homomorphism of lattice G1, lattice G2

proof end;