Article LMOD_7, MML version 4.99.1005
:: LMOD_7:sch 1
scheme LMOD_7:sch 1
{F1 -> set}:
for b1, b2 being Element of F1()
st (for b3 being set holds
b3 in b1
iff
P1[b3]) &
(for b3 being set holds
b3 in b2
iff
P1[b3])
holds b1 = b2
:: LMOD_7:sch 2
scheme LMOD_7:sch 2
{F1 -> non empty set,
F2 -> set}:
for b1, b2 being Function-like quasi_total Relation of F1(),F1()
st (for b3 being Element of F1() holds
b1 . b3 = F2(b3)) &
(for b3 being Element of F1() holds
b2 . b3 = F2(b3))
holds b1 = b2
:: LMOD_7:sch 3
scheme LMOD_7:sch 3
{F1 -> non empty set,
F2 -> set}:
for b1, b2 being Function-like quasi_total Relation of [:F1(),F1(),F1():],F1()
st (for b3, b4, b5 being Element of F1() holds
b1 .(b3,b4,b5) = F2(b3, b4, b5)) &
(for b3, b4, b5 being Element of F1() holds
b2 .(b3,b4,b5) = F2(b3, b4, b5))
holds b1 = b2
:: LMOD_7:sch 4
scheme LMOD_7:sch 4
{F1 -> non empty set,
F2 -> set}:
for b1, b2 being Function-like quasi_total Relation of [:F1(),F1(),F1(),F1():],F1()
st (for b3, b4, b5, b6 being Element of F1() holds
b1 .(b3,b4,b5,b6) = F2(b3, b4, b5, b6)) &
(for b3, b4, b5, b6 being Element of F1() holds
b2 .(b3,b4,b5,b6) = F2(b3, b4, b5, b6))
holds b1 = b2
:: LMOD_7:sch 5
scheme LMOD_7:sch 5
{F1 -> non empty set,
F2 -> non empty set,
F3 -> Element of F2()}:
ex b1 being Element of bool F2() st
b1 = {F3(b2) where b2 is Element of F1(): P1[b2]}
:: LMOD_7:sch 6
scheme LMOD_7:sch 6
{F1 -> non empty set,
F2 -> Element of F1()}:
P1[F2()]
provided
F2() in {b1 where b1 is Element of F1(): P1[b1]};
:: LMOD_7:sch 7
scheme LMOD_7:sch 7
{F1 -> set,
F2 -> non empty set,
F3 -> Element of F2()}:
(F3() in F1()
iff
P1[F3()])
provided
F1() = {b1 where b1 is Element of F2(): P1[b1]};
:: LMOD_7:sch 8
scheme LMOD_7:sch 8
{F1 -> set,
F2 -> non empty set,
F3 -> Element of F2()}:
P1[F3()]
provided
F3() in F1()
and
F1() = {b1 where b1 is Element of F2(): P1[b1]};
:: LMOD_7:sch 9
scheme LMOD_7:sch 9
{F1 -> set,
F2 -> set,
F3 -> non empty set}:
(F1() in F2()
iff
ex b1 being Element of F3() st
F1() = b1 & P1[b1])
provided
F2() = {b1 where b1 is Element of F3(): P1[b1]};
:: LMOD_7:sch 10
scheme LMOD_7:sch 10
{F1 -> non empty set,
F2 -> non empty set,
F3 -> set,
F4 -> Element of F1(),
F5 -> set}:
(F4() in F5(F3())
iff
for b1 being Element of F2()
st b1 in F3()
holds P1[F4(), b1])
provided
F5(F3()) = {b1 where b1 is Element of F1(): P2[b1, F3()]}
and
(P2[F4(), F3()]
iff
for b1 being Element of F2()
st b1 in F3()
holds P1[F4(), b1]);
:: LMOD_7:th 1
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
st b1 is not trivial & b3 is linearly-independent(b1, b2)
holds not 0. b2 in b3;
:: LMOD_7:th 2
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Element of bool the carrier of b2
for b5 being Linear_Combination of b4
st not b3 in b4
holds b5 . b3 = 0. b1;
:: LMOD_7:th 3
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
st b1 is trivial
holds (for b4 being Linear_Combination of b3 holds
Carrier b4 = {}) &
Lin b3 is trivial;
:: LMOD_7:th 4
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
st b2 is not trivial
for b3 being Element of bool the carrier of b2
st b3 is base(b1, b2)
holds b3 <> {};
:: LMOD_7:th 5
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
st b3 \/ b4 is linearly-independent(b1, b2) & b3 misses b4
holds (Lin b3) /\ Lin b4 = (0). b2;
:: LMOD_7:th 6
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Element of bool the carrier of b2
st b3 is base(b1, b2) & b3 = b4 \/ b5 & b4 misses b5
holds b2 is_the_direct_sum_of Lin b4,Lin b5;
:: LMOD_7:modenot 1 => LMOD_7:mode 1
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
mode SUBMODULE_DOMAIN of A2 -> non empty set means
for b1 being set
st b1 in it
holds b1 is strict Subspace of a2;
end;
:: LMOD_7:dfs 1
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be non empty set;
To prove
a3 is SUBMODULE_DOMAIN of a2
it is sufficient to prove
thus for b1 being set
st b1 in a3
holds b1 is strict Subspace of a2;
:: LMOD_7:def 1
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being non empty set holds
b3 is SUBMODULE_DOMAIN of b2
iff
for b4 being set
st b4 in b3
holds b4 is strict Subspace of b2;
:: LMOD_7:funcnot 1 => LMOD_7:func 1
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
redefine func Submodules a2 -> SUBMODULE_DOMAIN of a2;
end;
:: LMOD_7:modenot 2 => LMOD_7:mode 2
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be SUBMODULE_DOMAIN of a2;
redefine mode Element of a3 -> strict Subspace of a2;
end;
:: LMOD_7:exreg 1
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be SUBMODULE_DOMAIN of a2;
cluster non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed Element of a3;
end;
:: LMOD_7:modenot 3 => LMOD_7:mode 3
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
assume a2 is not trivial;
mode LINE of A2 -> strict Subspace of a2 means
ex b1 being Element of the carrier of a2 st
b1 <> 0. a2 & it = <:b1:>;
end;
:: LMOD_7:dfs 2
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be strict Subspace of a2;
To prove
a3 is LINE of a2
it is sufficient to prove
thus a2 is not trivial;
thus ex b1 being Element of the carrier of a2 st
b1 <> 0. a2 & a3 = <:b1:>;
:: LMOD_7:def 2
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
st b2 is not trivial
for b3 being strict Subspace of b2 holds
b3 is LINE of b2
iff
ex b4 being Element of the carrier of b2 st
b4 <> 0. b2 & b3 = <:b4:>;
:: LMOD_7:modenot 4 => LMOD_7:mode 4
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
mode LINE_DOMAIN of A2 -> non empty set means
for b1 being set
st b1 in it
holds b1 is LINE of a2;
end;
:: LMOD_7:dfs 3
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be non empty set;
To prove
a3 is LINE_DOMAIN of a2
it is sufficient to prove
thus for b1 being set
st b1 in a3
holds b1 is LINE of a2;
:: LMOD_7:def 3
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being non empty set holds
b3 is LINE_DOMAIN of b2
iff
for b4 being set
st b4 in b3
holds b4 is LINE of b2;
:: LMOD_7:funcnot 2 => LMOD_7:func 2
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
func lines A2 -> LINE_DOMAIN of a2 means
for b1 being set holds
b1 in it
iff
b1 is LINE of a2;
end;
:: LMOD_7:def 4
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being LINE_DOMAIN of b2 holds
b3 = lines b2
iff
for b4 being set holds
b4 in b3
iff
b4 is LINE of b2;
:: LMOD_7:modenot 5 => LMOD_7:mode 5
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be LINE_DOMAIN of a2;
redefine mode Element of a3 -> LINE of a2;
end;
:: LMOD_7:modenot 6 => LMOD_7:mode 6
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
assume a2 is not trivial & a2 is free(a1);
mode HIPERPLANE of A2 -> strict Subspace of a2 means
ex b1 being Element of the carrier of a2 st
b1 <> 0. a2 & a2 is_the_direct_sum_of <:b1:>,it;
end;
:: LMOD_7:dfs 5
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be strict Subspace of a2;
To prove
a3 is HIPERPLANE of a2
it is sufficient to prove
thus a2 is not trivial & a2 is free(a1);
thus ex b1 being Element of the carrier of a2 st
b1 <> 0. a2 & a2 is_the_direct_sum_of <:b1:>,a3;
:: LMOD_7:def 5
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
st b2 is not trivial & b2 is free(b1)
for b3 being strict Subspace of b2 holds
b3 is HIPERPLANE of b2
iff
ex b4 being Element of the carrier of b2 st
b4 <> 0. b2 & b2 is_the_direct_sum_of <:b4:>,b3;
:: LMOD_7:modenot 7 => LMOD_7:mode 7
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
mode HIPERPLANE_DOMAIN of A2 -> non empty set means
for b1 being set
st b1 in it
holds b1 is HIPERPLANE of a2;
end;
:: LMOD_7:dfs 6
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be non empty set;
To prove
a3 is HIPERPLANE_DOMAIN of a2
it is sufficient to prove
thus for b1 being set
st b1 in a3
holds b1 is HIPERPLANE of a2;
:: LMOD_7:def 6
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being non empty set holds
b3 is HIPERPLANE_DOMAIN of b2
iff
for b4 being set
st b4 in b3
holds b4 is HIPERPLANE of b2;
:: LMOD_7:funcnot 3 => LMOD_7:func 3
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
func hiperplanes A2 -> HIPERPLANE_DOMAIN of a2 means
for b1 being set holds
b1 in it
iff
b1 is HIPERPLANE of a2;
end;
:: LMOD_7:def 7
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being HIPERPLANE_DOMAIN of b2 holds
b3 = hiperplanes b2
iff
for b4 being set holds
b4 in b3
iff
b4 is HIPERPLANE of b2;
:: LMOD_7:modenot 8 => LMOD_7:mode 8
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be HIPERPLANE_DOMAIN of a2;
redefine mode Element of a3 -> HIPERPLANE of a2;
end;
:: LMOD_7:funcnot 4 => LMOD_7:func 4
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be FinSequence of Subspaces a2;
func Sum A3 -> Element of Submodules a2 equals
(SubJoin a2) "**" a3;
end;
:: LMOD_7:def 8
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being FinSequence of Subspaces b2 holds
Sum b3 = (SubJoin b2) "**" b3;
:: LMOD_7:funcnot 5 => LMOD_7:func 5
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be FinSequence of Subspaces a2;
func /\ A3 -> Element of Submodules a2 equals
(SubMeet a2) "**" a3;
end;
:: LMOD_7:def 9
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being FinSequence of Subspaces b2 holds
/\ b3 = (SubMeet b2) "**" b3;
:: LMOD_7:th 14
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
SubJoin b2 is commutative(Subspaces b2) & SubJoin b2 is associative(Subspaces b2) & SubJoin b2 is having_a_unity(Subspaces b2) & (0). b2 = the_unity_wrt SubJoin b2;
:: LMOD_7:th 15
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
SubMeet b2 is commutative(Subspaces b2) & SubMeet b2 is associative(Subspaces b2) & SubMeet b2 is having_a_unity(Subspaces b2) & (Omega). b2 = the_unity_wrt SubMeet b2;
:: LMOD_7:funcnot 6 => LMOD_7:func 6
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3, a4 be Element of bool the carrier of a2;
func A3 + A4 -> Element of bool the carrier of a2 means
for b1 being set holds
b1 in it
iff
ex b2, b3 being Element of the carrier of a2 st
b2 in a3 & b3 in a4 & b1 = b2 + b3;
end;
:: LMOD_7:def 10
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Element of bool the carrier of b2 holds
b5 = b3 + b4
iff
for b6 being set holds
b6 in b5
iff
ex b7, b8 being Element of the carrier of b2 st
b7 in b3 & b8 in b4 & b6 = b7 + b8;
:: LMOD_7:modenot 9 => LMOD_7:mode 9
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Element of bool the carrier of a2;
assume a3 <> {};
mode Vector of A3 -> Element of the carrier of a2 means
it is Element of a3;
end;
:: LMOD_7:dfs 11
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Element of bool the carrier of a2;
let a4 be Element of the carrier of a2;
To prove
a4 is Vector of a3
it is sufficient to prove
thus a3 <> {};
thus a4 is Element of a3;
:: LMOD_7:def 11
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
st b3 <> {}
for b4 being Element of the carrier of b2 holds
b4 is Vector of b3
iff
b4 is Element of b3;
:: LMOD_7:th 16
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
st b3 <> {} & b3 c= b4
for b5 being set
st b5 is Vector of b3
holds b5 is Vector of b4;
:: LMOD_7:th 17
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
b3 in b4 + b5
iff
b4 - b3 in b5;
:: LMOD_7:th 18
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
b3 + b5 = b4 + b5
iff
b3 - b4 in b5;
:: LMOD_7:funcnot 7 => LMOD_7:func 7
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
func A2 .. A3 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Element of the carrier of a2 st
b1 = b2 + a3;
end;
:: LMOD_7:def 12
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being set holds
b4 = b2 .. b3
iff
for b5 being set holds
b5 in b4
iff
ex b6 being Element of the carrier of b2 st
b5 = b6 + b3;
:: LMOD_7:funcreg 1
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
cluster a2 .. a3 -> non empty;
end;
:: LMOD_7:funcnot 8 => LMOD_7:func 8
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
let a4 be Element of the carrier of a2;
func A4 .. A3 -> Element of a2 .. a3 equals
a4 + a3;
end;
:: LMOD_7:def 13
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of the carrier of b2 holds
b4 .. b3 = b4 + b3;
:: LMOD_7:th 19
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of b2 .. b3 holds
ex b5 being Element of the carrier of b2 st
b4 = b5 .. b3;
:: LMOD_7:th 20
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
b3 .. b5 = b4 .. b5
iff
b3 - b4 in b5;
:: LMOD_7:funcnot 9 => LMOD_7:func 9
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
let a4 be Element of a2 .. a3;
func - A4 -> Element of a2 .. a3 means
for b1 being Element of the carrier of a2
st a4 = b1 .. a3
holds it = (- b1) .. a3;
end;
:: LMOD_7:def 14
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4, b5 being Element of b2 .. b3 holds
b5 = - b4
iff
for b6 being Element of the carrier of b2
st b4 = b6 .. b3
holds b5 = (- b6) .. b3;
:: LMOD_7:funcnot 10 => LMOD_7:func 10
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
let a4, a5 be Element of a2 .. a3;
func A4 + A5 -> Element of a2 .. a3 means
for b1, b2 being Element of the carrier of a2
st a4 = b1 .. a3 & a5 = b2 .. a3
holds it = (b1 + b2) .. a3;
end;
:: LMOD_7:def 15
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4, b5, b6 being Element of b2 .. b3 holds
b6 = b4 + b5
iff
for b7, b8 being Element of the carrier of b2
st b4 = b7 .. b3 & b5 = b8 .. b3
holds b6 = (b7 + b8) .. b3;
:: LMOD_7:funcnot 11 => LMOD_7:func 11
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
func COMPL A3 -> Function-like quasi_total Relation of a2 .. a3,a2 .. a3 means
for b1 being Element of a2 .. a3 holds
it . b1 = - b1;
end;
:: LMOD_7:def 16
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Function-like quasi_total Relation of b2 .. b3,b2 .. b3 holds
b4 = COMPL b3
iff
for b5 being Element of b2 .. b3 holds
b4 . b5 = - b5;
:: LMOD_7:funcnot 12 => LMOD_7:func 12
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
func ADD A3 -> Function-like quasi_total Relation of [:a2 .. a3,a2 .. a3:],a2 .. a3 means
for b1, b2 being Element of a2 .. a3 holds
it .(b1,b2) = b1 + b2;
end;
:: LMOD_7:def 17
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Function-like quasi_total Relation of [:b2 .. b3,b2 .. b3:],b2 .. b3 holds
b4 = ADD b3
iff
for b5, b6 being Element of b2 .. b3 holds
b4 .(b5,b6) = b5 + b6;
:: LMOD_7:funcnot 13 => LMOD_7:func 13
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
func A2 . A3 -> strict addLoopStr equals
addLoopStr(#a2 .. a3,ADD a3,(0. a2) .. a3#);
end;
:: LMOD_7:def 18
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
b2 . b3 = addLoopStr(#b2 .. b3,ADD b3,(0. b2) .. b3#);
:: LMOD_7:funcreg 2
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
cluster a2 . a3 -> non empty strict;
end;
:: LMOD_7:th 21
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2 holds
b3 .. b4 is Element of the carrier of b2 . b4;
:: LMOD_7:funcnot 14 => LMOD_7:func 14
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
let a4 be Element of the carrier of a2;
func A4 . A3 -> Element of the carrier of a2 . a3 equals
a4 .. a3;
end;
:: LMOD_7:def 19
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of the carrier of b2 holds
b4 . b3 = b4 .. b3;
:: LMOD_7:th 22
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of the carrier of b2 . b3 holds
ex b5 being Element of the carrier of b2 st
b4 = b5 . b3;
:: LMOD_7:th 23
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
b3 . b5 = b4 . b5
iff
b3 - b4 in b5;
:: LMOD_7:th 24
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
(b3 . b5) + (b4 . b5) = (b3 + b4) . b5 &
0. (b2 . b5) = (0. b2) . b5;
:: LMOD_7:funcreg 3
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
cluster a2 . a3 -> strict right_complementable Abelian add-associative right_zeroed;
end;
:: LMOD_7:funcnot 15 => LMOD_7:func 15
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
let a4 be Element of the carrier of a1;
let a5 be Element of the carrier of a2 . a3;
func A4 * A5 -> Element of the carrier of a2 . a3 means
for b1 being Element of the carrier of a2
st a5 = b1 . a3
holds it = (a4 * b1) . a3;
end;
:: LMOD_7:def 20
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2 . b3 holds
b6 = b4 * b5
iff
for b7 being Element of the carrier of b2
st b5 = b7 . b3
holds b6 = (b4 * b7) . b3;
:: LMOD_7:funcnot 16 => LMOD_7:func 16
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
func LMULT A3 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a2 . a3:],the carrier of a2 . a3 means
for b1 being Element of the carrier of a1
for b2 being Element of the carrier of a2 . a3 holds
it .(b1,b2) = b1 * b2;
end;
:: LMOD_7:def 21
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2 . b3:],the carrier of b2 . b3 holds
b4 = LMULT b3
iff
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2 . b3 holds
b4 .(b5,b6) = b5 * b6;
:: LMOD_7:funcnot 17 => LMOD_7:func 17
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
func A2 / A3 -> strict VectSpStr over a1 equals
VectSpStr(#the carrier of a2 . a3,the addF of a2 . a3,(0. a2) . a3,LMULT a3#);
end;
:: LMOD_7:def 22
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
b2 / b3 = VectSpStr(#the carrier of b2 . b3,the addF of b2 . b3,(0. b2) . b3,LMULT b3#);
:: LMOD_7:funcreg 4
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
cluster a2 / a3 -> non empty strict;
end;
:: LMOD_7:th 26
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2 holds
b3 . b4 is Element of the carrier of b2 / b4;
:: LMOD_7:th 27
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of the carrier of b2 / b3 holds
b4 is Element of the carrier of b2 . b3;
:: LMOD_7:funcnot 18 => LMOD_7:func 18
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
let a4 be Element of the carrier of a2;
func A4 / A3 -> Element of the carrier of a2 / a3 equals
a4 . a3;
end;
:: LMOD_7:def 23
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of the carrier of b2 holds
b4 / b3 = b4 . b3;
:: LMOD_7:th 28
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of the carrier of b2 / b3 holds
ex b5 being Element of the carrier of b2 st
b4 = b5 / b3;
:: LMOD_7:th 29
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
b3 / b5 = b4 / b5
iff
b3 - b4 in b5;
:: LMOD_7:th 30
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4, b5 being Element of the carrier of b3
for b6 being Subspace of b3 holds
(b4 / b6) + (b5 / b6) = (b4 + b5) / b6 &
b2 * (b4 / b6) = (b2 * b4) / b6;
:: LMOD_7:th 31
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
b2 / b3 is non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1;
:: LMOD_7:funcreg 5
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Subspace of a2;
cluster a2 / a3 -> strict VectSp-like;
end;