Article MODCAT_1, MML version 4.99.1005
:: MODCAT_1:modenot 1 => MODCAT_1:mode 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
mode LeftMod_DOMAIN of A1 -> non empty set means
for b1 being Element of it holds
b1 is non empty right_complementable Abelian add-associative right_zeroed strict VectSp-like VectSpStr over a1;
end;
:: MODCAT_1:dfs 1
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty set;
To prove
a2 is LeftMod_DOMAIN of a1
it is sufficient to prove
thus for b1 being Element of a2 holds
b1 is non empty right_complementable Abelian add-associative right_zeroed strict VectSp-like VectSpStr over a1;
:: MODCAT_1:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty set holds
b2 is LeftMod_DOMAIN of b1
iff
for b3 being Element of b2 holds
b3 is non empty right_complementable Abelian add-associative right_zeroed strict VectSp-like VectSpStr over b1;
:: MODCAT_1:modenot 2 => MODCAT_1:mode 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LeftMod_DOMAIN of a1;
redefine mode Element of a2 -> non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
end;
:: MODCAT_1:exreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LeftMod_DOMAIN of a1;
cluster non empty right_complementable Abelian add-associative right_zeroed strict VectSp-like Element of a2;
end;
:: MODCAT_1:modenot 3 => MODCAT_1:mode 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
mode LModMorphism_DOMAIN of A1 -> non empty set means
for b1 being Element of it holds
b1 is strict LModMorphism-like LModMorphismStr over a1;
end;
:: MODCAT_1:dfs 2
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty set;
To prove
a2 is LModMorphism_DOMAIN of a1
it is sufficient to prove
thus for b1 being Element of a2 holds
b1 is strict LModMorphism-like LModMorphismStr over a1;
:: MODCAT_1:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty set holds
b2 is LModMorphism_DOMAIN of b1
iff
for b3 being Element of b2 holds
b3 is strict LModMorphism-like LModMorphismStr over b1;
:: MODCAT_1:modenot 4 => MODCAT_1:mode 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LModMorphism_DOMAIN of a1;
redefine mode Element of a2 -> LModMorphism-like LModMorphismStr over a1;
end;
:: MODCAT_1:exreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LModMorphism_DOMAIN of a1;
cluster strict LModMorphism-like Element of a2;
end;
:: MODCAT_1:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being strict LModMorphism-like LModMorphismStr over b1 holds
{b2} is LModMorphism_DOMAIN of b1;
:: MODCAT_1:modenot 5 => MODCAT_1:mode 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
mode LModMorphism_DOMAIN of A2,A3 -> LModMorphism_DOMAIN of a1 means
for b1 being Element of it holds
b1 is strict Morphism of a2,a3;
end;
:: MODCAT_1:dfs 3
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be LModMorphism_DOMAIN of a1;
To prove
a4 is LModMorphism_DOMAIN of a2,a3
it is sufficient to prove
thus for b1 being Element of a4 holds
b1 is strict Morphism of a2,a3;
:: MODCAT_1:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being LModMorphism_DOMAIN of b1 holds
b4 is LModMorphism_DOMAIN of b2,b3
iff
for b5 being Element of b4 holds
b5 is strict Morphism of b2,b3;
:: MODCAT_1:th 4
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b2 holds
b1 is LModMorphism_DOMAIN of b3,b4
iff
for b5 being Element of b1 holds
b5 is strict Morphism of b3,b4;
:: MODCAT_1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being strict Morphism of b2,b3 holds
{b4} is LModMorphism_DOMAIN of b2,b3;
:: MODCAT_1:funcnot 1 => MODCAT_1:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
func Morphs(A2,A3) -> LModMorphism_DOMAIN of a2,a3 means
for b1 being set holds
b1 in it
iff
b1 is strict Morphism of a2,a3;
end;
:: MODCAT_1:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being LModMorphism_DOMAIN of b2,b3 holds
b4 = Morphs(b2,b3)
iff
for b5 being set holds
b5 in b4
iff
b5 is strict Morphism of b2,b3;
:: MODCAT_1:modenot 6 => MODCAT_1:mode 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be LModMorphism_DOMAIN of a2,a3;
redefine mode Element of a4 -> Morphism of a2,a3;
end;
:: MODCAT_1:prednot 1 => MODCAT_1:pred 1
definition
let a1, a2 be set;
let a3 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
pred GO A1,A2,A3 means
ex b1, b2 being set st
a1 = [b1,b2] &
(ex b3 being non empty right_complementable Abelian add-associative right_zeroed strict VectSp-like VectSpStr over a3 st
a2 = b3 &
b1 = addLoopStr(#the carrier of b3,the addF of b3,the ZeroF of b3#) &
b2 = the lmult of b3);
end;
:: MODCAT_1:dfs 5
definiens
let a1, a2 be set;
let a3 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
To prove
GO a1,a2,a3
it is sufficient to prove
thus ex b1, b2 being set st
a1 = [b1,b2] &
(ex b3 being non empty right_complementable Abelian add-associative right_zeroed strict VectSp-like VectSpStr over a3 st
a2 = b3 &
b1 = addLoopStr(#the carrier of b3,the addF of b3,the ZeroF of b3#) &
b2 = the lmult of b3);
:: MODCAT_1:def 5
theorem
for b1, b2 being set
for b3 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr holds
GO b1,b2,b3
iff
ex b4, b5 being set st
b1 = [b4,b5] &
(ex b6 being non empty right_complementable Abelian add-associative right_zeroed strict VectSp-like VectSpStr over b3 st
b2 = b6 &
b4 = addLoopStr(#the carrier of b6,the addF of b6,the ZeroF of b6#) &
b5 = the lmult of b6);
:: MODCAT_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3, b4 being set
st GO b2,b3,b1 & GO b2,b4,b1
holds b3 = b4;
:: MODCAT_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty universal set holds
ex b3 being set st
b3 in {[b4,b5] where b4 is Element of GroupObjects b2, b5 is Element of Funcs([:the carrier of b1,1:],1): TRUE} &
GO b3,TrivialLMod b1,b1;
:: MODCAT_1:funcnot 2 => MODCAT_1:func 2
definition
let a1 be non empty universal set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
func LModObjects(A1,A2) -> set means
for b1 being set holds
b1 in it
iff
ex b2 being set st
b2 in {[b3,b4] where b3 is Element of GroupObjects a1, b4 is Element of Funcs([:the carrier of a2,1:],1): TRUE} &
GO b2,b1,a2;
end;
:: MODCAT_1:def 6
theorem
for b1 being non empty universal set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3 being set holds
b3 = LModObjects(b1,b2)
iff
for b4 being set holds
b4 in b3
iff
ex b5 being set st
b5 in {[b6,b7] where b6 is Element of GroupObjects b1, b7 is Element of Funcs([:the carrier of b2,1:],1): TRUE} &
GO b5,b4,b2;
:: MODCAT_1:th 8
theorem
for b1 being non empty universal set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr holds
TrivialLMod b2 in LModObjects(b1,b2);
:: MODCAT_1:funcreg 1
registration
let a1 be non empty universal set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
cluster LModObjects(a1,a2) -> non empty;
end;
:: MODCAT_1:th 9
theorem
for b1 being non empty universal set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3 being Element of LModObjects(b1,b2) holds
b3 is non empty right_complementable Abelian add-associative right_zeroed strict VectSp-like VectSpStr over b2;
:: MODCAT_1:funcnot 3 => MODCAT_1:func 3
definition
let a1 be non empty universal set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
redefine func LModObjects(a1,a2) -> LeftMod_DOMAIN of a2;
end;
:: MODCAT_1:funcnot 4 => MODCAT_1:func 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LeftMod_DOMAIN of a1;
func Morphs A2 -> LModMorphism_DOMAIN of a1 means
for b1 being set holds
b1 in it
iff
ex b2, b3 being strict Element of a2 st
b1 is strict Morphism of b2,b3;
end;
:: MODCAT_1:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3 being LModMorphism_DOMAIN of b1 holds
b3 = Morphs b2
iff
for b4 being set holds
b4 in b3
iff
ex b5, b6 being strict Element of b2 st
b4 is strict Morphism of b5,b6;
:: MODCAT_1:funcnot 5 => MODCAT_1:func 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LeftMod_DOMAIN of a1;
let a3 be Element of Morphs a2;
func dom' A3 -> Element of a2 equals
dom a3;
end;
:: MODCAT_1:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3 being Element of Morphs b2 holds
dom' b3 = dom b3;
:: MODCAT_1:funcnot 6 => MODCAT_1:func 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LeftMod_DOMAIN of a1;
let a3 be Element of Morphs a2;
func cod' A3 -> Element of a2 equals
cod a3;
end;
:: MODCAT_1:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3 being Element of Morphs b2 holds
cod' b3 = cod b3;
:: MODCAT_1:funcnot 7 => MODCAT_1:func 7
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LeftMod_DOMAIN of a1;
let a3 be Element of a2;
func ID A3 -> strict Element of Morphs a2 equals
ID a3;
end;
:: MODCAT_1:def 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3 being Element of b2 holds
ID b3 = ID b3;
:: MODCAT_1:funcnot 8 => MODCAT_1:func 8
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LeftMod_DOMAIN of a1;
func dom A2 -> Function-like quasi_total Relation of Morphs a2,a2 means
for b1 being Element of Morphs a2 holds
it . b1 = dom' b1;
end;
:: MODCAT_1:def 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3 being Function-like quasi_total Relation of Morphs b2,b2 holds
b3 = dom b2
iff
for b4 being Element of Morphs b2 holds
b3 . b4 = dom' b4;
:: MODCAT_1:funcnot 9 => MODCAT_1:func 9
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LeftMod_DOMAIN of a1;
func cod A2 -> Function-like quasi_total Relation of Morphs a2,a2 means
for b1 being Element of Morphs a2 holds
it . b1 = cod' b1;
end;
:: MODCAT_1:def 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3 being Function-like quasi_total Relation of Morphs b2,b2 holds
b3 = cod b2
iff
for b4 being Element of Morphs b2 holds
b3 . b4 = cod' b4;
:: MODCAT_1:funcnot 10 => MODCAT_1:func 10
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LeftMod_DOMAIN of a1;
func ID A2 -> Function-like quasi_total Relation of a2,Morphs a2 means
for b1 being Element of a2 holds
it . b1 = ID b1;
end;
:: MODCAT_1:def 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3 being Function-like quasi_total Relation of b2,Morphs b2 holds
b3 = ID b2
iff
for b4 being Element of b2 holds
b3 . b4 = ID b4;
:: MODCAT_1:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3, b4 being Element of Morphs b2
st dom' b3 = cod' b4
holds ex b5, b6, b7 being strict Element of b2 st
b3 is Morphism of b6,b7 & b4 is Morphism of b5,b6;
:: MODCAT_1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3, b4 being Element of Morphs b2
st dom' b3 = cod' b4
holds b3 * b4 in Morphs b2;
:: MODCAT_1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3, b4 being Element of Morphs b2
st dom b3 = cod b4
holds b3 * b4 in Morphs b2;
:: MODCAT_1:funcnot 11 => MODCAT_1:func 11
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be LeftMod_DOMAIN of a1;
func comp A2 -> Function-like Relation of [:Morphs a2,Morphs a2:],Morphs a2 means
(for b1, b2 being Element of Morphs a2 holds
[b1,b2] in proj1 it
iff
dom' b1 = cod' b2) &
(for b1, b2 being Element of Morphs a2
st [b1,b2] in proj1 it
holds it .(b1,b2) = b1 * b2);
end;
:: MODCAT_1:def 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3 being Function-like Relation of [:Morphs b2,Morphs b2:],Morphs b2 holds
b3 = comp b2
iff
(for b4, b5 being Element of Morphs b2 holds
[b4,b5] in proj1 b3
iff
dom' b4 = cod' b5) &
(for b4, b5 being Element of Morphs b2
st [b4,b5] in proj1 b3
holds b3 .(b4,b5) = b4 * b5);
:: MODCAT_1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being LeftMod_DOMAIN of b1
for b3, b4 being Element of Morphs b2 holds
[b3,b4] in proj1 comp b2
iff
dom b3 = cod b4;
:: MODCAT_1:funcnot 12 => MODCAT_1:func 12
definition
let a1 be non empty universal set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
func LModCat(A1,A2) -> strict CatStr equals
CatStr(#LModObjects(a1,a2),Morphs LModObjects(a1,a2),dom LModObjects(a1,a2),cod LModObjects(a1,a2),comp LModObjects(a1,a2),ID LModObjects(a1,a2)#);
end;
:: MODCAT_1:def 15
theorem
for b1 being non empty universal set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr holds
LModCat(b1,b2) = CatStr(#LModObjects(b1,b2),Morphs LModObjects(b1,b2),dom LModObjects(b1,b2),cod LModObjects(b1,b2),comp LModObjects(b1,b2),ID LModObjects(b1,b2)#);
:: MODCAT_1:funcreg 2
registration
let a1 be non empty universal set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
cluster LModCat(a1,a2) -> strict non void;
end;
:: MODCAT_1:th 14
theorem
for b1 being non empty universal set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3, b4 being Element of the Edges of LModCat(b1,b2) holds
[b4,b3] in proj1 the Comp of LModCat(b1,b2)
iff
dom b4 = cod b3;
:: MODCAT_1:th 15
theorem
for b1 being non empty universal set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3 being Element of the Edges of LModCat(b1,b2)
for b4 being Element of Morphs LModObjects(b1,b2)
for b5 being Element of the Vertices of LModCat(b1,b2)
for b6 being Element of LModObjects(b1,b2) holds
b3 is strict Element of Morphs LModObjects(b1,b2) & b4 is Element of the Edges of LModCat(b1,b2) & b5 is strict Element of LModObjects(b1,b2) & b6 is Element of the Vertices of LModCat(b1,b2);
:: MODCAT_1:th 16
theorem
for b1 being non empty universal set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3 being Element of the Vertices of LModCat(b1,b2)
for b4 being Element of LModObjects(b1,b2)
st b3 = b4
holds id b3 = ID b4;
:: MODCAT_1:th 17
theorem
for b1 being non empty universal set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3 being Element of the Edges of LModCat(b1,b2)
for b4 being Element of Morphs LModObjects(b1,b2)
st b3 = b4
holds dom b3 = dom b4 & cod b3 = cod b4;
:: MODCAT_1:th 18
theorem
for b1 being non empty universal set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3, b4 being Element of the Edges of LModCat(b1,b2)
for b5, b6 being Element of Morphs LModObjects(b1,b2)
st b3 = b5 & b4 = b6
holds (dom b4 = cod b3 implies dom b6 = cod b5) &
(dom b6 = cod b5 implies dom b4 = cod b3) &
(dom b4 = cod b3 implies [b6,b5] in proj1 comp LModObjects(b1,b2)) &
([b6,b5] in proj1 comp LModObjects(b1,b2) implies dom b4 = cod b3) &
(dom b4 = cod b3 implies b4 * b3 = b6 * b5) &
(dom b3 = dom b4 implies dom b5 = dom b6) &
(dom b5 = dom b6 implies dom b3 = dom b4) &
(cod b3 = cod b4 implies cod b5 = cod b6) &
(cod b5 = cod b6 implies cod b3 = cod b4);
:: MODCAT_1:funcreg 3
registration
let a1 be non empty universal set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
cluster LModCat(a1,a2) -> strict Category-like;
end;