Article GRSOLV_1, MML version 4.99.1005

:: GRSOLV_1:attrnot 1 => GRSOLV_1:attr 1
definition
  let a1 be non empty Group-like associative multMagma;
  attr a1 is solvable means
    ex b1 being FinSequence of Subgroups a1 st
       0 < len b1 &
        b1 . 1 = (Omega). a1 &
        b1 . len b1 = (1). a1 &
        (for b2 being Element of NAT
           st b2 in dom b1 & b2 + 1 in dom b1
        for b3, b4 being strict Subgroup of a1
              st b3 = b1 . b2 & b4 = b1 . (b2 + 1)
           holds b4 is strict normal Subgroup of b3 &
            (for b5 being normal Subgroup of b3
                  st b5 = b4
               holds b3 ./. b5 is commutative));
end;

:: GRSOLV_1:dfs 1
definiens
  let a1 be non empty Group-like associative multMagma;
To prove
     a1 is solvable
it is sufficient to prove
  thus ex b1 being FinSequence of Subgroups a1 st
       0 < len b1 &
        b1 . 1 = (Omega). a1 &
        b1 . len b1 = (1). a1 &
        (for b2 being Element of NAT
           st b2 in dom b1 & b2 + 1 in dom b1
        for b3, b4 being strict Subgroup of a1
              st b3 = b1 . b2 & b4 = b1 . (b2 + 1)
           holds b4 is strict normal Subgroup of b3 &
            (for b5 being normal Subgroup of b3
                  st b5 = b4
               holds b3 ./. b5 is commutative));

:: GRSOLV_1:def 1
theorem
for b1 being non empty Group-like associative multMagma holds
      b1 is solvable
   iff
      ex b2 being FinSequence of Subgroups b1 st
         0 < len b2 &
          b2 . 1 = (Omega). b1 &
          b2 . len b2 = (1). b1 &
          (for b3 being Element of NAT
             st b3 in dom b2 & b3 + 1 in dom b2
          for b4, b5 being strict Subgroup of b1
                st b4 = b2 . b3 & b5 = b2 . (b3 + 1)
             holds b5 is strict normal Subgroup of b4 &
              (for b6 being normal Subgroup of b4
                    st b6 = b5
                 holds b4 ./. b6 is commutative));

:: GRSOLV_1:exreg 1
registration
  cluster non empty strict Group-like associative solvable multMagma;
end;

:: GRSOLV_1:th 1
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3, b4 being strict Subgroup of b1
      st b3 is normal Subgroup of b4
   holds b3 /\ b2 is normal Subgroup of b4 /\ b2;

:: GRSOLV_1:th 2
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being strict Subgroup of b1
for b3 being strict normal Subgroup of b2
for b4, b5 being Element of the carrier of b2 holds
(b4 * b3) * (b5 * b3) = (b4 * b5) * b3;

:: GRSOLV_1:th 3
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being strict Subgroup of b1
for b4 being strict normal Subgroup of b3
for b5 being strict Subgroup of b1
   st b5 = b2 /\ b3
for b6 being normal Subgroup of b5
      st b6 = b2 /\ b4
   holds ex b7 being Subgroup of b3 ./. b4 st
      b5 ./. b6,b7 are_isomorphic;

:: GRSOLV_1:th 4
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being strict Subgroup of b1
for b4 being strict normal Subgroup of b3
for b5 being strict Subgroup of b1
   st b5 = b3 /\ b2
for b6 being normal Subgroup of b5
      st b6 = b4 /\ b2
   holds ex b7 being Subgroup of b3 ./. b4 st
      b5 ./. b6,b7 are_isomorphic;

:: GRSOLV_1:th 5
theorem
for b1 being non empty strict Group-like associative solvable multMagma
for b2 being strict Subgroup of b1 holds
   b2 is solvable;

:: GRSOLV_1:th 6
theorem
for b1 being non empty strict Group-like associative multMagma
      st ex b2 being FinSequence of Subgroups b1 st
           0 < len b2 &
            b2 . 1 = (Omega). b1 &
            b2 . len b2 = (1). b1 &
            (for b3 being Element of NAT
               st b3 in dom b2 & b3 + 1 in dom b2
            for b4, b5 being strict Subgroup of b1
                  st b4 = b2 . b3 & b5 = b2 . (b3 + 1)
               holds b5 is strict normal Subgroup of b4 &
                (for b6 being normal Subgroup of b4
                      st b6 = b5
                   holds b4 ./. b6 is non empty Group-like associative cyclic multMagma))
   holds b1 is solvable;

:: GRSOLV_1:th 7
theorem
for b1 being non empty strict Group-like associative commutative multMagma holds
   b1 is solvable;

:: GRSOLV_1:funcnot 1 => GRSOLV_1:func 1
definition
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  let a4 be Subgroup of a1;
  func A3 | A4 -> Function-like quasi_total multiplicative Relation of the carrier of a4,the carrier of a2 equals
    a3 | the carrier of a4;
end;

:: GRSOLV_1:def 2
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
   b3 | b4 = b3 | the carrier of b4;

:: GRSOLV_1:funcnot 2 => GRSOLV_1:func 2
definition
  let a1, a2 be non empty strict Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  let a4 be Subgroup of a1;
  func A3 .: A4 -> strict Subgroup of a2 equals
    Image (a3 | a4);
end;

:: GRSOLV_1:def 3
theorem
for b1, b2 being non empty strict 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
   b3 .: b4 = Image (b3 | b4);

:: GRSOLV_1:th 9
theorem
for b1, b2 being non empty strict 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 strict Subgroup of b1 holds
   the carrier of b3 .: b4 = b3 .: the carrier of b4;

:: GRSOLV_1:th 10
theorem
for b1, b2 being non empty strict 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 strict Subgroup of b1 holds
   Image (b3 | b4) is strict Subgroup of Image b3;

:: GRSOLV_1:th 11
theorem
for b1, b2 being non empty strict 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 strict Subgroup of b1 holds
   b3 .: b4 is strict Subgroup of Image b3;

:: GRSOLV_1:th 12
theorem
for b1, b2 being non empty strict Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   b3 .: (1). b1 = (1). b2 & b3 .: (Omega). b1 = (Omega). Image b3;

:: GRSOLV_1:th 13
theorem
for b1, b2 being non empty strict 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 strict Subgroup of b1
      st b4 is Subgroup of b5
   holds b3 .: b4 is Subgroup of b3 .: b5;

:: GRSOLV_1:th 14
theorem
for b1, b2 being non empty strict 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 strict Subgroup of b1
for b5 being Element of the carrier of b1 holds
   (b3 . b5) * (b3 .: b4) = b3 .: (b5 * b4) &
    (b3 .: b4) * (b3 . b5) = b3 .: (b4 * b5);

:: GRSOLV_1:th 15
theorem
for b1, b2 being non empty strict 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 Element of bool the carrier of b1 holds
(b3 .: b4) * (b3 .: b5) = b3 .: (b4 * b5);

:: GRSOLV_1:th 16
theorem
for b1, b2 being non empty strict 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 strict Subgroup of b1
      st b4 is strict normal Subgroup of b5
   holds b3 .: b4 is strict normal Subgroup of b3 .: b5;

:: GRSOLV_1:th 17
theorem
for b1, b2 being non empty strict Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
      st b1 is non empty Group-like associative solvable multMagma
   holds Image b3 is solvable;