Article TOPALG_4, MML version 4.99.1005
:: TOPALG_4:th 1
theorem
for b1, b2 being non empty multMagma
for b3 being Element of the carrier of product <*b1,b2*> holds
ex b4 being Element of the carrier of b1 st
ex b5 being Element of the carrier of b2 st
b3 = <*b4,b5*>;
:: TOPALG_4:funcnot 1 => TOPALG_4:func 1
definition
let a1, a2, a3, a4 be non empty multMagma;
let a5 be Function-like quasi_total Relation of the carrier of a1,the carrier of a3;
let a6 be Function-like quasi_total Relation of the carrier of a2,the carrier of a4;
func Gr2Iso(A5,A6) -> Function-like quasi_total Relation of the carrier of product <*a1,a2*>,the carrier of product <*a3,a4*> means
for b1 being Element of the carrier of product <*a1,a2*> holds
ex b2 being Element of the carrier of a1 st
ex b3 being Element of the carrier of a2 st
b1 = <*b2,b3*> &
it . b1 = <*a5 . b2,a6 . b3*>;
end;
:: TOPALG_4:def 1
theorem
for b1, b2, b3, b4 being non empty multMagma
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
for b7 being Function-like quasi_total Relation of the carrier of product <*b1,b2*>,the carrier of product <*b3,b4*> holds
b7 = Gr2Iso(b5,b6)
iff
for b8 being Element of the carrier of product <*b1,b2*> holds
ex b9 being Element of the carrier of b1 st
ex b10 being Element of the carrier of b2 st
b8 = <*b9,b10*> &
b7 . b8 = <*b5 . b9,b6 . b10*>;
:: TOPALG_4:th 2
theorem
for b1, b2, b3, b4 being non empty multMagma
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
for b7 being Element of the carrier of b1
for b8 being Element of the carrier of b2 holds
(Gr2Iso(b5,b6)) . <*b7,b8*> = <*b5 . b7,b6 . b8*>;
:: TOPALG_4:funcnot 2 => TOPALG_4:func 2
definition
let a1, a2, a3, a4 be non empty Group-like associative multMagma;
let a5 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a3;
let a6 be Function-like quasi_total multiplicative Relation of the carrier of a2,the carrier of a4;
redefine func Gr2Iso(a5,a6) -> Function-like quasi_total multiplicative Relation of the carrier of product <*a1,a2*>,the carrier of product <*a3,a4*>;
end;
:: TOPALG_4:th 3
theorem
for b1, b2, b3, b4 being non empty multMagma
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
st b5 is one-to-one & b6 is one-to-one
holds Gr2Iso(b5,b6) is one-to-one;
:: TOPALG_4:th 4
theorem
for b1, b2, b3, b4 being non empty multMagma
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
st b5 is onto(the carrier of b1, the carrier of b3) & b6 is onto(the carrier of b2, the carrier of b4)
holds Gr2Iso(b5,b6) is onto(the carrier of product <*b1,b2*>, the carrier of product <*b3,b4*>);
:: TOPALG_4:th 5
theorem
for b1, b2, b3, b4 being non empty Group-like associative multMagma
for b5 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b4
st b5 is being_isomorphism(b1, b3) & b6 is being_isomorphism(b2, b4)
holds Gr2Iso(b5,b6) is being_isomorphism(product <*b1,b2*>, product <*b3,b4*>);
:: TOPALG_4:th 6
theorem
for b1, b2, b3, b4 being non empty Group-like associative multMagma
st b1,b3 are_isomorphic & b2,b4 are_isomorphic
holds product <*b1,b2*>,product <*b3,b4*> are_isomorphic;
:: TOPALG_4:th 7
theorem
for b1, b2 being Relation-like Function-like set
st proj1 b1 = proj1 b2
holds pr1 <:b1,b2:> = b1;
:: TOPALG_4:th 8
theorem
for b1, b2 being Relation-like Function-like set
st proj1 b1 = proj1 b2
holds pr2 <:b1,b2:> = b2;
:: TOPALG_4:funcnot 3 => TOPALG_4:func 3
definition
let a1, a2, a3 be non empty TopSpace-like TopStruct;
let a4 be Function-like quasi_total Relation of the carrier of a3,the carrier of a1;
let a5 be Function-like quasi_total Relation of the carrier of a3,the carrier of a2;
redefine func <:a4, a5:> -> Function-like quasi_total Relation of the carrier of a3,the carrier of [:a1,a2:];
end;
:: TOPALG_4:funcnot 4 => TOPALG_4:func 4
definition
let a1, a2, a3 be non empty TopSpace-like TopStruct;
let a4 be Function-like quasi_total Relation of the carrier of a3,the carrier of [:a1,a2:];
redefine func pr1 a4 -> Function-like quasi_total Relation of the carrier of a3,the carrier of a1;
end;
:: TOPALG_4:funcnot 5 => TOPALG_4:func 5
definition
let a1, a2, a3 be non empty TopSpace-like TopStruct;
let a4 be Function-like quasi_total Relation of the carrier of a3,the carrier of [:a1,a2:];
redefine func pr2 a4 -> Function-like quasi_total Relation of the carrier of a3,the carrier of a2;
end;
:: TOPALG_4:th 9
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of [:b2,b3:] holds
pr1 b4 is continuous(b1, b2);
:: TOPALG_4:th 10
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of [:b2,b3:] holds
pr2 b4 is continuous(b1, b3);
:: TOPALG_4:th 11
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st [b3,b5],[b4,b6] are_connected
holds b3,b4 are_connected;
:: TOPALG_4:th 12
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st [b3,b5],[b4,b6] are_connected
holds b5,b6 are_connected;
:: TOPALG_4:th 13
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st [b3,b5],[b4,b6] are_connected
for b7 being Path of [b3,b5],[b4,b6] holds
pr1 b7 is Path of b3,b4;
:: TOPALG_4:th 14
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st [b3,b5],[b4,b6] are_connected
for b7 being Path of [b3,b5],[b4,b6] holds
pr2 b7 is Path of b5,b6;
:: TOPALG_4:th 15
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st b3,b4 are_connected & b5,b6 are_connected
holds [b3,b5],[b4,b6] are_connected;
:: TOPALG_4:th 16
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st b3,b4 are_connected & b5,b6 are_connected
for b7 being Path of b3,b4
for b8 being Path of b5,b6 holds
<:b7,b8:> is Path of [b3,b5],[b4,b6];
:: TOPALG_4:funcnot 6 => TOPALG_4:func 6
definition
let a1, a2 be non empty TopSpace-like arcwise_connected TopStruct;
let a3, a4 be Element of the carrier of a1;
let a5, a6 be Element of the carrier of a2;
let a7 be Path of a3,a4;
let a8 be Path of a5,a6;
redefine func <:a7, a8:> -> Path of [a3,a5],[a4,a6];
end;
:: TOPALG_4:funcnot 7 => TOPALG_4:func 7
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of a1;
let a4 be Element of the carrier of a2;
let a5 be Path of a3,a3;
let a6 be Path of a4,a4;
redefine func <:a5, a6:> -> Path of [a3,a4],[a3,a4];
end;
:: TOPALG_4:funcreg 1
registration
let a1, a2 be non empty TopSpace-like arcwise_connected TopStruct;
cluster [:a1,a2:] -> strict TopSpace-like arcwise_connected;
end;
:: TOPALG_4:funcnot 8 => TOPALG_4:func 8
definition
let a1, a2 be non empty TopSpace-like arcwise_connected TopStruct;
let a3, a4 be Element of the carrier of a1;
let a5, a6 be Element of the carrier of a2;
let a7 be Path of [a3,a5],[a4,a6];
redefine func pr1 a7 -> Path of a3,a4;
end;
:: TOPALG_4:funcnot 9 => TOPALG_4:func 9
definition
let a1, a2 be non empty TopSpace-like arcwise_connected TopStruct;
let a3, a4 be Element of the carrier of a1;
let a5, a6 be Element of the carrier of a2;
let a7 be Path of [a3,a5],[a4,a6];
redefine func pr2 a7 -> Path of a5,a6;
end;
:: TOPALG_4:funcnot 10 => TOPALG_4:func 10
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of a1;
let a4 be Element of the carrier of a2;
let a5 be Path of [a3,a4],[a3,a4];
redefine func pr1 a5 -> Path of a3,a3;
end;
:: TOPALG_4:funcnot 11 => TOPALG_4:func 11
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of a1;
let a4 be Element of the carrier of a2;
let a5 be Path of [a3,a4],[a3,a4];
redefine func pr2 a5 -> Path of a4,a4;
end;
:: TOPALG_4:th 17
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7, b8 being Path of [b3,b5],[b4,b6]
for b9 being Homotopy of b7,b8
for b10, b11 being Path of b3,b4
st b10 = pr1 b7 & b11 = pr1 b8 & b7,b8 are_homotopic
holds pr1 b9 is Homotopy of b10,b11;
:: TOPALG_4:th 18
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7, b8 being Path of [b3,b5],[b4,b6]
for b9 being Homotopy of b7,b8
for b10, b11 being Path of b5,b6
st b10 = pr2 b7 & b11 = pr2 b8 & b7,b8 are_homotopic
holds pr2 b9 is Homotopy of b10,b11;
:: TOPALG_4:th 19
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7, b8 being Path of [b3,b5],[b4,b6]
for b9, b10 being Path of b3,b4
st b9 = pr1 b7 & b10 = pr1 b8 & b7,b8 are_homotopic
holds b9,b10 are_homotopic;
:: TOPALG_4:th 20
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7, b8 being Path of [b3,b5],[b4,b6]
for b9, b10 being Path of b5,b6
st b9 = pr2 b7 & b10 = pr2 b8 & b7,b8 are_homotopic
holds b9,b10 are_homotopic;
:: TOPALG_4:th 21
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7, b8 being Path of [b3,b5],[b4,b6]
for b9, b10 being Path of b3,b4
for b11, b12 being Path of b5,b6
for b13 being Homotopy of b9,b10
for b14 being Homotopy of b11,b12
st b9 = pr1 b7 & b10 = pr1 b8 & b11 = pr2 b7 & b12 = pr2 b8 & b9,b10 are_homotopic & b11,b12 are_homotopic
holds <:b13,b14:> is Homotopy of b7,b8;
:: TOPALG_4:th 22
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7, b8 being Path of [b3,b5],[b4,b6]
for b9, b10 being Path of b3,b4
for b11, b12 being Path of b5,b6
st b9 = pr1 b7 & b10 = pr1 b8 & b11 = pr2 b7 & b12 = pr2 b8 & b9,b10 are_homotopic & b11,b12 are_homotopic
holds b7,b8 are_homotopic;
:: TOPALG_4:th 23
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8 being Element of the carrier of b2
for b9 being Path of [b3,b6],[b4,b7]
for b10 being Path of [b4,b7],[b5,b8]
for b11 being Path of b3,b4
for b12 being Path of b4,b5
st [b3,b6],[b4,b7] are_connected & [b4,b7],[b5,b8] are_connected & b11 = pr1 b9 & b12 = pr1 b10
holds pr1 (b9 + b10) = b11 + b12;
:: TOPALG_4:th 24
theorem
for b1, b2 being non empty TopSpace-like arcwise_connected TopStruct
for b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8 being Element of the carrier of b2
for b9 being Path of [b3,b6],[b4,b7]
for b10 being Path of [b4,b7],[b5,b8] holds
pr1 (b9 + b10) = (pr1 b9) + pr1 b10;
:: TOPALG_4:th 25
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8 being Element of the carrier of b2
for b9 being Path of [b3,b6],[b4,b7]
for b10 being Path of [b4,b7],[b5,b8]
for b11 being Path of b6,b7
for b12 being Path of b7,b8
st [b3,b6],[b4,b7] are_connected & [b4,b7],[b5,b8] are_connected & b11 = pr2 b9 & b12 = pr2 b10
holds pr2 (b9 + b10) = b11 + b12;
:: TOPALG_4:th 26
theorem
for b1, b2 being non empty TopSpace-like arcwise_connected TopStruct
for b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8 being Element of the carrier of b2
for b9 being Path of [b3,b6],[b4,b7]
for b10 being Path of [b4,b7],[b5,b8] holds
pr2 (b9 + b10) = (pr2 b9) + pr2 b10;
:: TOPALG_4:funcnot 12 => TOPALG_4:func 12
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of a1;
let a4 be Element of the carrier of a2;
func FGPrIso(A3,A4) -> Function-like quasi_total Relation of the carrier of FundamentalGroup([:a1,a2:],[a3,a4]),the carrier of product <*FundamentalGroup(a1,a3),FundamentalGroup(a2,a4)*> means
for b1 being Element of the carrier of FundamentalGroup([:a1,a2:],[a3,a4]) holds
ex b2 being Path of [a3,a4],[a3,a4] st
b1 = Class(EqRel([:a1,a2:],[a3,a4]),b2) &
it . b1 = <*Class(EqRel(a1,a3),pr1 b2),Class(EqRel(a2,a4),pr2 b2)*>;
end;
:: TOPALG_4:def 2
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of FundamentalGroup([:b1,b2:],[b3,b4]),the carrier of product <*FundamentalGroup(b1,b3),FundamentalGroup(b2,b4)*> holds
b5 = FGPrIso(b3,b4)
iff
for b6 being Element of the carrier of FundamentalGroup([:b1,b2:],[b3,b4]) holds
ex b7 being Path of [b3,b4],[b3,b4] st
b6 = Class(EqRel([:b1,b2:],[b3,b4]),b7) &
b5 . b6 = <*Class(EqRel(b1,b3),pr1 b7),Class(EqRel(b2,b4),pr2 b7)*>;
:: TOPALG_4:th 27
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Element of the carrier of FundamentalGroup([:b1,b2:],[b3,b4])
for b6 being Path of [b3,b4],[b3,b4]
st b5 = Class(EqRel([:b1,b2:],[b3,b4]),b6)
holds (FGPrIso(b3,b4)) . b5 = <*Class(EqRel(b1,b3),pr1 b6),Class(EqRel(b2,b4),pr2 b6)*>;
:: TOPALG_4:th 28
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Path of [b3,b4],[b3,b4] holds
(FGPrIso(b3,b4)) . Class(EqRel([:b1,b2:],[b3,b4]),b5) = <*Class(EqRel(b1,b3),pr1 b5),Class(EqRel(b2,b4),pr2 b5)*>;
:: TOPALG_4:funcreg 2
registration
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of a1;
let a4 be Element of the carrier of a2;
cluster FGPrIso(a3,a4) -> Function-like one-to-one quasi_total onto;
end;
:: TOPALG_4:funcnot 13 => TOPALG_4:func 13
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of a1;
let a4 be Element of the carrier of a2;
redefine func FGPrIso(a3,a4) -> Function-like quasi_total multiplicative Relation of the carrier of FundamentalGroup([:a1,a2:],[a3,a4]),the carrier of product <*FundamentalGroup(a1,a3),FundamentalGroup(a2,a4)*>;
end;
:: TOPALG_4:th 29
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
FGPrIso(b3,b4) is being_isomorphism(FundamentalGroup([:b1,b2:],[b3,b4]), product <*FundamentalGroup(b1,b3),FundamentalGroup(b2,b4)*>);
:: TOPALG_4:th 30
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
FundamentalGroup([:b1,b2:],[b3,b4]),product <*FundamentalGroup(b1,b3),FundamentalGroup(b2,b4)*> are_isomorphic;
:: TOPALG_4:th 31
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7 being Function-like quasi_total multiplicative Relation of the carrier of FundamentalGroup(b1,b3),the carrier of FundamentalGroup(b1,b4)
for b8 being Function-like quasi_total multiplicative Relation of the carrier of FundamentalGroup(b2,b5),the carrier of FundamentalGroup(b2,b6)
st b7 is being_isomorphism(FundamentalGroup(b1,b3), FundamentalGroup(b1,b4)) & b8 is being_isomorphism(FundamentalGroup(b2,b5), FundamentalGroup(b2,b6))
holds (Gr2Iso(b7,b8)) * FGPrIso(b3,b5) is being_isomorphism(FundamentalGroup([:b1,b2:],[b3,b5]), product <*FundamentalGroup(b1,b4),FundamentalGroup(b2,b6)*>);
:: TOPALG_4:th 32
theorem
for b1, b2 being non empty TopSpace-like arcwise_connected TopStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2 holds
FundamentalGroup([:b1,b2:],[b3,b5]),product <*FundamentalGroup(b1,b4),FundamentalGroup(b2,b6)*> are_isomorphic;