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;