Article LATSUBGR, MML version 4.99.1005

:: LATSUBGR:th 1
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Subgroup of b1 holds
the carrier of b2 /\ b3 = (the carrier of b2) /\ the carrier of b3;

:: LATSUBGR:th 2
theorem
for b1 being non empty Group-like associative multMagma
for b2 being set holds
      b2 in Subgroups b1
   iff
      ex b3 being strict Subgroup of b1 st
         b2 = b3;

:: LATSUBGR:th 3
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
for b3 being strict Subgroup of b1
      st b2 = the carrier of b3
   holds gr b2 = b3;

:: LATSUBGR:th 4
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Subgroup of b1
for b4 being Element of bool the carrier of b1
      st b4 = (the carrier of b2) \/ the carrier of b3
   holds b2 "\/" b3 = gr b4;

:: LATSUBGR:th 5
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Subgroup of b1
for b4 being Element of the carrier of b1
      st (b4 in b2 or b4 in b3)
   holds b4 in b2 "\/" b3;

:: LATSUBGR:th 6
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
for b4 being Subgroup of b1 holds
   ex b5 being strict Subgroup of b2 st
      the carrier of b5 = b3 .: the carrier of b4;

:: LATSUBGR:th 7
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
for b4 being Subgroup of b2 holds
   ex b5 being strict Subgroup of b1 st
      the carrier of b5 = b3 " the carrier of b4;

:: LATSUBGR:th 10
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
for b4, b5 being Subgroup of b1
for b6, b7 being Subgroup of b2
      st the carrier of b6 = b3 .: the carrier of b4 & the carrier of b7 = b3 .: the carrier of b5 & b4 is Subgroup of b5
   holds b6 is Subgroup of b7;

:: LATSUBGR:th 11
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
for b4, b5 being Subgroup of b2
for b6, b7 being Subgroup of b1
      st the carrier of b6 = b3 " the carrier of b4 & the carrier of b7 = b3 " the carrier of b5 & b4 is Subgroup of b5
   holds b6 is Subgroup of b7;

:: LATSUBGR:th 12
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1 holds
   b3 .: b4 c= b3 .: the carrier of gr b4;

:: LATSUBGR:th 13
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3, b4 being Subgroup of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Element of bool the carrier of b1
      st b6 = (the carrier of b3) \/ the carrier of b4
   holds b5 .: the carrier of b3 "\/" b4 = b5 .: the carrier of gr b6;

:: LATSUBGR:th 14
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
      st b2 = {1_ b1}
   holds gr b2 = (1). b1;

:: LATSUBGR:funcnot 1 => LATSUBGR:func 1
definition
  let a1 be non empty Group-like associative multMagma;
  func carr A1 -> Function-like quasi_total Relation of Subgroups a1,bool the carrier of a1 means
    for b1 being strict Subgroup of a1 holds
       it . b1 = the carrier of b1;
end;

:: LATSUBGR:def 1
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Function-like quasi_total Relation of Subgroups b1,bool the carrier of b1 holds
      b2 = carr b1
   iff
      for b3 being strict Subgroup of b1 holds
         b2 . b3 = the carrier of b3;

:: LATSUBGR:th 18
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1
for b3 being Element of the carrier of b1 holds
      b3 in (carr b1) . b2
   iff
      b3 in b2;

:: LATSUBGR:th 19
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
   1_ b1 in (carr b1) . b2;

:: LATSUBGR:th 20
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
   (carr b1) . b2 <> {};

:: LATSUBGR:th 21
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1
for b3, b4 being Element of the carrier of b1
      st b3 in (carr b1) . b2 & b4 in (carr b1) . b2
   holds b3 * b4 in (carr b1) . b2;

:: LATSUBGR:th 22
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1
for b3 being Element of the carrier of b1
      st b3 in (carr b1) . b2
   holds b3 " in (carr b1) . b2;

:: LATSUBGR:th 23
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict Subgroup of b1 holds
the carrier of b2 /\ b3 = ((carr b1) . b2) /\ ((carr b1) . b3);

:: LATSUBGR:th 24
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict Subgroup of b1 holds
(carr b1) . (b2 /\ b3) = ((carr b1) . b2) /\ ((carr b1) . b3);

:: LATSUBGR:funcnot 2 => LATSUBGR:func 2
definition
  let a1 be non empty Group-like associative multMagma;
  let a2 be non empty Element of bool Subgroups a1;
  func meet A2 -> strict Subgroup of a1 means
    the carrier of it = meet ((carr a1) .: a2);
end;

:: LATSUBGR:def 2
theorem
for b1 being non empty Group-like associative multMagma
for b2 being non empty Element of bool Subgroups b1
for b3 being strict Subgroup of b1 holds
      b3 = meet b2
   iff
      the carrier of b3 = meet ((carr b1) .: b2);

:: LATSUBGR:th 25
theorem
for b1 being non empty Group-like associative multMagma
for b2 being non empty Element of bool Subgroups b1
      st (1). b1 in b2
   holds meet b2 = (1). b1;

:: LATSUBGR:th 26
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of Subgroups b1
for b3 being non empty Element of bool Subgroups b1
      st b3 = {b2}
   holds meet b3 = b2;

:: LATSUBGR:th 27
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Subgroup of b1
for b4, b5 being Element of the carrier of lattice b1
      st b4 = b2 & b5 = b3
   holds b4 "\/" b5 = b2 "\/" b3;

:: LATSUBGR:th 28
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Subgroup of b1
for b4, b5 being Element of the carrier of lattice b1
      st b4 = b2 & b5 = b3
   holds b4 "/\" b5 = b2 /\ b3;

:: LATSUBGR:th 29
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of lattice b1
for b3 being Subgroup of b1
      st b2 = b3
   holds b3 is strict Subgroup of b1;

:: LATSUBGR:th 30
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Subgroup of b1
for b4, b5 being Element of the carrier of lattice b1
      st b4 = b2 & b5 = b3
   holds    b4 [= b5
   iff
      the carrier of b2 c= the carrier of b3;

:: LATSUBGR:th 31
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Subgroup of b1
for b4, b5 being Element of the carrier of lattice b1
      st b4 = b2 & b5 = b3
   holds    b4 [= b5
   iff
      b2 is Subgroup of b3;

:: LATSUBGR:th 32
theorem
for b1 being non empty Group-like associative multMagma holds
   lattice b1 is complete;

:: LATSUBGR:funcnot 3 => LATSUBGR:func 3
definition
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  func FuncLatt A3 -> Function-like quasi_total Relation of the carrier of lattice a1,the carrier of lattice a2 means
    for b1 being strict Subgroup of a1
    for b2 being Element of bool the carrier of a2
          st b2 = a3 .: the carrier of b1
       holds it . b1 = gr b2;
end;

:: LATSUBGR:def 3
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of lattice b1,the carrier of lattice b2 holds
      b4 = FuncLatt b3
   iff
      for b5 being strict Subgroup of b1
      for b6 being Element of bool the carrier of b2
            st b6 = b3 .: the carrier of b5
         holds b4 . b5 = gr b6;

:: LATSUBGR:th 33
theorem
for b1 being non empty Group-like associative multMagma holds
   FuncLatt id the carrier of b1 = id the carrier of lattice b1;

:: LATSUBGR:th 34
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
      st b3 is one-to-one
   holds FuncLatt b3 is one-to-one;

:: LATSUBGR:th 35
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   (FuncLatt b3) . (1). b1 = (1). b2;

:: LATSUBGR:th 36
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
      st b3 is one-to-one
   holds FuncLatt b3 is Semilattice-Homomorphism of lattice b1,lattice b2;

:: LATSUBGR:th 37
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   FuncLatt b3 is sup-Semilattice-Homomorphism of lattice b1,lattice b2;

:: LATSUBGR:th 38
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
      st b3 is one-to-one
   holds FuncLatt b3 is Homomorphism of lattice b1,lattice b2;