Article TOPALG_1, MML version 4.99.1005
:: TOPALG_1:th 1
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
st b3 * (b3 /") = id b2 & b3 /" * b3 = id b1
holds b3 is being_isomorphism(b1, b2);
:: TOPALG_1:th 2
theorem
for b1 being Element of bool the carrier of I[01]
for b2 being Element of the carrier of I[01]
st b1 = ].b2,1.]
holds b1 ` = [.0,b2.];
:: TOPALG_1:th 3
theorem
for b1 being Element of bool the carrier of I[01]
for b2 being Element of the carrier of I[01]
st b1 = [.0,b2.[
holds b1 ` = [.b2,1.];
:: TOPALG_1:th 4
theorem
for b1 being Element of bool the carrier of I[01]
for b2 being Element of the carrier of I[01]
st b1 = ].b2,1.]
holds b1 is open(I[01]);
:: TOPALG_1:th 5
theorem
for b1 being Element of bool the carrier of I[01]
for b2 being Element of the carrier of I[01]
st b1 = [.0,b2.[
holds b1 is open(I[01]);
:: TOPALG_1:th 6
theorem
for b1 being real set
for b2 being Element of NAT
for b3 being Element of b2 -tuples_on REAL holds
b1 * - b3 = - (b1 * b3);
:: TOPALG_1:th 7
theorem
for b1 being real set
for b2 being Element of NAT
for b3, b4 being Element of b2 -tuples_on REAL holds
b1 * (b3 - b4) = (b1 * b3) - (b1 * b4);
:: TOPALG_1:th 8
theorem
for b1, b2 being real set
for b3 being Element of NAT
for b4 being Element of b3 -tuples_on REAL holds
(b1 - b2) * b4 = (b1 * b4) - (b2 * b4);
:: TOPALG_1:th 9
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of b1 -tuples_on REAL holds
(b2 + b3) - (b4 + b5) = (b2 - b4) + (b3 - b5);
:: TOPALG_1:th 10
theorem
for b1 being real set
for b2 being Element of NAT
for b3 being Element of REAL b2
st 0 <= b1 & b1 <= 1
holds |.b1 * b3.| <= |.b3.|;
:: TOPALG_1:th 11
theorem
for b1 being Element of NAT
for b2 being Element of REAL b1
for b3 being Element of the carrier of I[01] holds
|.b3 * b2.| <= |.b2.|;
:: TOPALG_1:th 12
theorem
for b1, b2 being real set
for b3 being Element of NAT
for b4, b5, b6, b7, b8, b9 being Element of the carrier of Euclid b3
for b10, b11, b12, b13 being Element of the carrier of TOP-REAL b3
st b4 = b10 & b5 = b11 & b6 = b12 & b7 = b13 & b8 = b10 + b12 & b9 = b11 + b13 & dist(b4,b5) < b1 & dist(b6,b7) < b2
holds dist(b8,b9) < b1 + b2;
:: TOPALG_1:th 13
theorem
for b1, b2 being real set
for b3 being Element of NAT
for b4, b5, b6, b7 being Element of the carrier of Euclid b3
for b8, b9 being Element of the carrier of TOP-REAL b3
st b4 = b8 & b5 = b9 & b6 = b1 * b8 & b7 = b1 * b9 & dist(b4,b5) < b2 & b1 <> 0
holds dist(b6,b7) < (abs b1) * b2;
:: TOPALG_1:th 14
theorem
for b1, b2, b3, b4 being real set
for b5 being Element of NAT
for b6, b7, b8, b9, b10, b11 being Element of the carrier of Euclid b5
for b12, b13, b14, b15 being Element of the carrier of TOP-REAL b5
st b6 = b12 & b7 = b13 & b8 = b14 & b9 = b15 & b10 = (b1 * b12) + (b2 * b14) & b11 = (b1 * b13) + (b2 * b15) & dist(b6,b7) < b3 & dist(b8,b9) < b4 & b1 <> 0 & b2 <> 0
holds dist(b10,b11) < ((abs b1) * b3) + ((abs b2) * b4);
:: TOPALG_1:th 16
theorem
for b1 being real set
for b2 being Element of NAT
for b3 being non empty TopSpace-like TopStruct
for b4, b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of TOP-REAL b2
st b4 is continuous(b3, TOP-REAL b2) &
(for b6 being Element of the carrier of b3 holds
b5 . b6 = b1 * (b4 . b6))
holds b5 is continuous(b3, TOP-REAL b2);
:: TOPALG_1:th 17
theorem
for b1, b2 being real set
for b3 being Element of NAT
for b4 being non empty TopSpace-like TopStruct
for b5, b6, b7 being Function-like quasi_total Relation of the carrier of b4,the carrier of TOP-REAL b3
st b5 is continuous(b4, TOP-REAL b3) &
b6 is continuous(b4, TOP-REAL b3) &
(for b8 being Element of the carrier of b4 holds
b7 . b8 = (b1 * (b5 . b8)) + (b2 * (b6 . b8)))
holds b7 is continuous(b4, TOP-REAL b3);
:: TOPALG_1:th 18
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of [:TOP-REAL b1,I[01]:],the carrier of TOP-REAL b1
st for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of I[01] holds
b2 .(b3,b4) = (1 - b4) * b3
holds b2 is continuous([:TOP-REAL b1,I[01]:], TOP-REAL b1);
:: TOPALG_1:th 19
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of [:TOP-REAL b1,I[01]:],the carrier of TOP-REAL b1
st for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of I[01] holds
b2 .(b3,b4) = b4 * b3
holds b2 is continuous([:TOP-REAL b1,I[01]:], TOP-REAL b1);
:: TOPALG_1:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
st b2,b3 are_connected & b3,b4 are_connected
for b5, b6 being Path of b2,b3
for b7 being Path of b3,b4
st b5,b6 are_homotopic
holds b5,(b6 + b7) + - b7 are_homotopic;
:: TOPALG_1:th 21
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Path of b2,b3
for b7 being Path of b3,b4
st b5,b6 are_homotopic
holds b5,(b6 + b7) + - b7 are_homotopic;
:: TOPALG_1:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
st b2,b3 are_connected & b4,b3 are_connected
for b5, b6 being Path of b2,b3
for b7 being Path of b4,b3
st b5,b6 are_homotopic
holds b5,(b6 + - b7) + b7 are_homotopic;
:: TOPALG_1:th 23
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Path of b2,b3
for b7 being Path of b4,b3
st b5,b6 are_homotopic
holds b5,(b6 + - b7) + b7 are_homotopic;
:: TOPALG_1:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
st b2,b3 are_connected & b4,b2 are_connected
for b5, b6 being Path of b2,b3
for b7 being Path of b4,b2
st b5,b6 are_homotopic
holds b5,((- b7) + b7) + b6 are_homotopic;
:: TOPALG_1:th 25
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Path of b2,b3
for b7 being Path of b4,b2
st b5,b6 are_homotopic
holds b5,((- b7) + b7) + b6 are_homotopic;
:: TOPALG_1:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
st b2,b3 are_connected & b2,b4 are_connected
for b5, b6 being Path of b2,b3
for b7 being Path of b2,b4
st b5,b6 are_homotopic
holds b5,(b7 + - b7) + b6 are_homotopic;
:: TOPALG_1:th 27
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Path of b2,b3
for b7 being Path of b2,b4
st b5,b6 are_homotopic
holds b5,(b7 + - b7) + b6 are_homotopic;
:: TOPALG_1:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
st b2,b3 are_connected & b4,b3 are_connected
for b5, b6 being Path of b2,b3
for b7 being Path of b3,b4
st b5 + b7,b6 + b7 are_homotopic
holds b5,b6 are_homotopic;
:: TOPALG_1:th 29
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Path of b2,b3
for b7 being Path of b3,b4
st b5 + b7,b6 + b7 are_homotopic
holds b5,b6 are_homotopic;
:: TOPALG_1:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
st b2,b3 are_connected & b2,b4 are_connected
for b5, b6 being Path of b2,b3
for b7 being Path of b4,b2
st b7 + b5,b7 + b6 are_homotopic
holds b5,b6 are_homotopic;
:: TOPALG_1:th 31
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Path of b2,b3
for b7 being Path of b4,b2
st b7 + b5,b7 + b6 are_homotopic
holds b5,b6 are_homotopic;
:: TOPALG_1:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st b2,b3 are_connected & b3,b4 are_connected & b4,b5 are_connected & b5,b6 are_connected
for b7 being Path of b2,b3
for b8 being Path of b3,b4
for b9 being Path of b4,b5
for b10 being Path of b5,b6 holds
((b7 + b8) + b9) + b10,(b7 + (b8 + b9)) + b10 are_homotopic;
:: TOPALG_1:th 33
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
for b7 being Path of b2,b3
for b8 being Path of b3,b4
for b9 being Path of b4,b5
for b10 being Path of b5,b6 holds
((b7 + b8) + b9) + b10,(b7 + (b8 + b9)) + b10 are_homotopic;
:: TOPALG_1:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st b2,b3 are_connected & b3,b4 are_connected & b4,b5 are_connected & b5,b6 are_connected
for b7 being Path of b2,b3
for b8 being Path of b3,b4
for b9 being Path of b4,b5
for b10 being Path of b5,b6 holds
((b7 + b8) + b9) + b10,b7 + ((b8 + b9) + b10) are_homotopic;
:: TOPALG_1:th 35
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
for b7 being Path of b2,b3
for b8 being Path of b3,b4
for b9 being Path of b4,b5
for b10 being Path of b5,b6 holds
((b7 + b8) + b9) + b10,b7 + ((b8 + b9) + b10) are_homotopic;
:: TOPALG_1:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st b2,b3 are_connected & b3,b4 are_connected & b4,b5 are_connected & b5,b6 are_connected
for b7 being Path of b2,b3
for b8 being Path of b3,b4
for b9 being Path of b4,b5
for b10 being Path of b5,b6 holds
(b7 + (b8 + b9)) + b10,(b7 + b8) + (b9 + b10) are_homotopic;
:: TOPALG_1:th 37
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
for b7 being Path of b2,b3
for b8 being Path of b3,b4
for b9 being Path of b4,b5
for b10 being Path of b5,b6 holds
(b7 + (b8 + b9)) + b10,(b7 + b8) + (b9 + b10) are_homotopic;
:: TOPALG_1:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 are_connected & b3,b4 are_connected & b3,b5 are_connected
for b6 being Path of b2,b3
for b7 being Path of b5,b3
for b8 being Path of b3,b4 holds
((b6 + - b7) + b7) + b8,b6 + b8 are_homotopic;
:: TOPALG_1:th 39
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Path of b2,b3
for b7 being Path of b4,b3
for b8 being Path of b3,b5 holds
((b6 + - b7) + b7) + b8,b6 + b8 are_homotopic;
:: TOPALG_1:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 are_connected & b2,b4 are_connected & b4,b5 are_connected
for b6 being Path of b2,b3
for b7 being Path of b4,b5
for b8 being Path of b2,b4 holds
(((b6 + - b6) + b8) + b7) + - b7,b8 are_homotopic;
:: TOPALG_1:th 41
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Path of b2,b3
for b7 being Path of b4,b5
for b8 being Path of b2,b4 holds
(((b6 + - b6) + b8) + b7) + - b7,b8 are_homotopic;
:: TOPALG_1:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 are_connected & b2,b4 are_connected & b5,b4 are_connected
for b6 being Path of b2,b3
for b7 being Path of b4,b5
for b8 being Path of b2,b4 holds
(b6 + (((- b6) + b8) + b7)) + - b7,b8 are_homotopic;
:: TOPALG_1:th 43
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Path of b2,b3
for b7 being Path of b4,b5
for b8 being Path of b2,b4 holds
(b6 + (((- b6) + b8) + b7)) + - b7,b8 are_homotopic;
:: TOPALG_1:th 44
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st b2,b3 are_connected & b3,b4 are_connected & b4,b5 are_connected & b5,b6 are_connected & b2,b7 are_connected
for b8 being Path of b2,b3
for b9 being Path of b3,b4
for b10 being Path of b4,b5
for b11 being Path of b5,b6
for b12 being Path of b7,b4 holds
(b8 + (b9 + b10)) + b11,((b8 + b9) + - b12) + ((b12 + b10) + b11) are_homotopic;
:: TOPALG_1:th 45
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
for b8 being Path of b2,b3
for b9 being Path of b3,b4
for b10 being Path of b4,b5
for b11 being Path of b5,b6
for b12 being Path of b7,b4 holds
(b8 + (b9 + b10)) + b11,((b8 + b9) + - b12) + ((b12 + b10) + b11) are_homotopic;
:: TOPALG_1:modenot 1
definition
let a1 be TopStruct;
let a2 be Element of the carrier of a1;
mode Loop of a2 is Path of a2,a2;
end;
:: TOPALG_1:funcnot 1 => TOPALG_1:func 1
definition
let a1 be non empty TopStruct;
let a2 be Element of the carrier of a1;
func Loops A2 -> set means
for b1 being set holds
b1 in it
iff
b1 is Path of a2,a2;
end;
:: TOPALG_1:def 1
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1
for b3 being set holds
b3 = Loops b2
iff
for b4 being set holds
b4 in b3
iff
b4 is Path of b2,b2;
:: TOPALG_1:funcreg 1
registration
let a1 be non empty TopStruct;
let a2 be Element of the carrier of a1;
cluster Loops a2 -> non empty;
end;
:: TOPALG_1:funcnot 2 => TOPALG_1:func 2
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
func EqRel(A1,A2) -> Relation of Loops a2,Loops a2 means
for b1, b2 being Path of a2,a2 holds
[b1,b2] in it
iff
b1,b2 are_homotopic;
end;
:: TOPALG_1:def 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Relation of Loops b2,Loops b2 holds
b3 = EqRel(b1,b2)
iff
for b4, b5 being Path of b2,b2 holds
[b4,b5] in b3
iff
b4,b5 are_homotopic;
:: TOPALG_1:funcreg 2
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
cluster EqRel(a1,a2) -> non empty symmetric transitive total;
end;
:: TOPALG_1:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3, b4 being Path of b2,b2 holds
b4 in Class(EqRel(b1,b2),b3)
iff
b3,b4 are_homotopic;
:: TOPALG_1:th 47
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3, b4 being Path of b2,b2 holds
Class(EqRel(b1,b2),b3) = Class(EqRel(b1,b2),b4)
iff
b3,b4 are_homotopic;
:: TOPALG_1:funcnot 3 => TOPALG_1:func 3
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
func FundamentalGroup(A1,A2) -> strict multMagma means
the carrier of it = Class EqRel(a1,a2) &
(for b1, b2 being Element of the carrier of it holds
ex b3, b4 being Path of a2,a2 st
b1 = Class(EqRel(a1,a2),b3) &
b2 = Class(EqRel(a1,a2),b4) &
(the multF of it) .(b1,b2) = Class(EqRel(a1,a2),b3 + b4));
end;
:: TOPALG_1:def 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being strict multMagma holds
b3 = FundamentalGroup(b1,b2)
iff
the carrier of b3 = Class EqRel(b1,b2) &
(for b4, b5 being Element of the carrier of b3 holds
ex b6, b7 being Path of b2,b2 st
b4 = Class(EqRel(b1,b2),b6) &
b5 = Class(EqRel(b1,b2),b7) &
(the multF of b3) .(b4,b5) = Class(EqRel(b1,b2),b6 + b7));
:: TOPALG_1:funcnot 4 => TOPALG_1:func 3
notation
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
synonym pi_1(a1,a2) for FundamentalGroup(a1,a2);
end;
:: TOPALG_1:funcreg 3
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
cluster FundamentalGroup(a1,a2) -> non empty strict;
end;
:: TOPALG_1:th 48
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being set holds
b3 in the carrier of FundamentalGroup(b1,b2)
iff
ex b4 being Path of b2,b2 st
b3 = Class(EqRel(b1,b2),b4);
:: TOPALG_1:funcreg 4
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
cluster FundamentalGroup(a1,a2) -> strict Group-like associative;
end;
:: TOPALG_1:funcnot 5 => TOPALG_1:func 4
definition
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Element of the carrier of a1;
let a4 be Path of a2,a3;
assume a2,a3 are_connected;
func pi_1-iso A4 -> Function-like quasi_total Relation of the carrier of FundamentalGroup(a1,a3),the carrier of FundamentalGroup(a1,a2) means
for b1 being Path of a3,a3 holds
it . Class(EqRel(a1,a3),b1) = Class(EqRel(a1,a2),(a4 + b1) + - a4);
end;
:: TOPALG_1:def 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
st b2,b3 are_connected
for b5 being Function-like quasi_total Relation of the carrier of FundamentalGroup(b1,b3),the carrier of FundamentalGroup(b1,b2) holds
b5 = pi_1-iso b4
iff
for b6 being Path of b3,b3 holds
b5 . Class(EqRel(b1,b3),b6) = Class(EqRel(b1,b2),(b4 + b6) + - b4);
:: TOPALG_1:th 49
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Path of b2,b3
st b2,b3 are_connected & b4,b5 are_homotopic
holds pi_1-iso b4 = pi_1-iso b5;
:: TOPALG_1:th 50
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Path of b2,b3
st b4,b5 are_homotopic
holds pi_1-iso b4 = pi_1-iso b5;
:: TOPALG_1:th 51
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
st b2,b3 are_connected
holds pi_1-iso b4 is Function-like quasi_total multiplicative Relation of the carrier of FundamentalGroup(b1,b3),the carrier of FundamentalGroup(b1,b2);
:: TOPALG_1:funcnot 6 => TOPALG_1:func 5
definition
let a1 be non empty TopSpace-like arcwise_connected TopStruct;
let a2, a3 be Element of the carrier of a1;
let a4 be Path of a2,a3;
redefine func pi_1-iso a4 -> Function-like quasi_total multiplicative Relation of the carrier of FundamentalGroup(a1,a3),the carrier of FundamentalGroup(a1,a2);
end;
:: TOPALG_1:th 52
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
st b2,b3 are_connected
holds pi_1-iso b4 is one-to-one;
:: TOPALG_1:th 53
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
st b2,b3 are_connected
holds pi_1-iso b4 is onto(the carrier of FundamentalGroup(b1,b3), the carrier of FundamentalGroup(b1,b2));
:: TOPALG_1:funcreg 5
registration
let a1 be non empty TopSpace-like arcwise_connected TopStruct;
let a2, a3 be Element of the carrier of a1;
let a4 be Path of a2,a3;
cluster pi_1-iso a4 -> Function-like one-to-one quasi_total onto;
end;
:: TOPALG_1:th 54
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
st b2,b3 are_connected
holds (pi_1-iso b4) /" = pi_1-iso - b4;
:: TOPALG_1:th 55
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3 holds
(pi_1-iso b4) /" = pi_1-iso - b4;
:: TOPALG_1:th 56
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
st b2,b3 are_connected
for b5 being Function-like quasi_total multiplicative Relation of the carrier of FundamentalGroup(b1,b3),the carrier of FundamentalGroup(b1,b2)
st b5 = pi_1-iso b4
holds b5 is being_isomorphism(FundamentalGroup(b1,b3), FundamentalGroup(b1,b2));
:: TOPALG_1:th 57
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3 holds
pi_1-iso b4 is being_isomorphism(FundamentalGroup(b1,b3), FundamentalGroup(b1,b2));
:: TOPALG_1:th 58
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
st b2,b3 are_connected
holds FundamentalGroup(b1,b2),FundamentalGroup(b1,b3) are_isomorphic;
:: TOPALG_1:th 59
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1 holds
FundamentalGroup(b1,b2),FundamentalGroup(b1,b3) are_isomorphic;
:: TOPALG_1:funcnot 7 => TOPALG_1:func 6
definition
let a1 be Element of NAT;
let a2, a3 be Element of the carrier of TOP-REAL a1;
let a4, a5 be Path of a2,a3;
func RealHomotopy(A4,A5) -> Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of TOP-REAL a1 means
for b1, b2 being Element of the carrier of I[01] holds
it .(b1,b2) = ((1 - b2) * (a4 . b1)) + (b2 * (a5 . b1));
end;
:: TOPALG_1:def 5
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4, b5 being Path of b2,b3
for b6 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of TOP-REAL b1 holds
b6 = RealHomotopy(b4,b5)
iff
for b7, b8 being Element of the carrier of I[01] holds
b6 .(b7,b8) = ((1 - b8) * (b4 . b7)) + (b8 * (b5 . b7));
:: TOPALG_1:th 60
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4, b5 being Path of b2,b3 holds
b4,b5 are_homotopic;
:: TOPALG_1:funcnot 8 => TOPALG_1:func 7
definition
let a1 be Element of NAT;
let a2, a3 be Element of the carrier of TOP-REAL a1;
let a4, a5 be Path of a2,a3;
redefine func RealHomotopy(a4,a5) -> Homotopy of a4,a5;
end;
:: TOPALG_1:condreg 1
registration
let a1 be Element of NAT;
let a2, a3 be Element of the carrier of TOP-REAL a1;
let a4, a5 be Path of a2,a3;
cluster -> continuous (Homotopy of a4,a5);
end;
:: TOPALG_1:th 61
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being Path of b2,b2 holds
the carrier of FundamentalGroup(TOP-REAL b1,b2) = {Class(EqRel(TOP-REAL b1,b2),b3)};
:: TOPALG_1:funcreg 6
registration
let a1 be Element of NAT;
let a2 be Element of the carrier of TOP-REAL a1;
cluster FundamentalGroup(TOP-REAL a1,a2) -> trivial strict;
end;
:: TOPALG_1:th 62
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of the carrier of FundamentalGroup(b1,b2)
for b5, b6 being Path of b2,b2
st b3 = Class(EqRel(b1,b2),b5) & b4 = Class(EqRel(b1,b2),b6)
holds b3 * b4 = Class(EqRel(b1,b2),b5 + b6);
:: TOPALG_1:th 63
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being constant Path of b2,b2 holds
1_ FundamentalGroup(b1,b2) = Class(EqRel(b1,b2),b3);
:: TOPALG_1:th 64
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of the carrier of FundamentalGroup(b1,b2)
for b5 being Path of b2,b2
st b3 = Class(EqRel(b1,b2),b5) & b4 = Class(EqRel(b1,b2),- b5)
holds b3 " = b4;