Article CONNSP_3, MML version 4.99.1005
:: CONNSP_3:funcnot 1 => CONNSP_3:func 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
func Component_of A2 -> Element of bool the carrier of a1 means
ex b1 being Element of bool bool the carrier of a1 st
(for b2 being Element of bool the carrier of a1 holds
b2 in b1
iff
b2 is connected(a1) & a2 c= b2) &
union b1 = it;
end;
:: CONNSP_3:def 1
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b3 = Component_of b2
iff
ex b4 being Element of bool bool the carrier of b1 st
(for b5 being Element of bool the carrier of b1 holds
b5 in b4
iff
b5 is connected(b1) & b2 c= b5) &
union b4 = b3;
:: CONNSP_3:th 1
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st ex b3 being Element of bool the carrier of b1 st
b3 is connected(b1) & b2 c= b3
holds b2 c= Component_of b2;
:: CONNSP_3:th 2
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st for b3 being Element of bool the carrier of b1
st b3 is connected(b1)
holds not b2 c= b3
holds Component_of b2 = {};
:: CONNSP_3:th 3
theorem
for b1 being non empty TopSpace-like TopStruct holds
Component_of {} b1 = the carrier of b1;
:: CONNSP_3:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is connected(b1)
holds Component_of b2 <> {};
:: CONNSP_3:th 5
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is connected(b1) & b2 <> {}
holds Component_of b2 is connected(b1);
:: CONNSP_3:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is connected(b1) & b3 is connected(b1) & Component_of b2 c= b3
holds b3 = Component_of b2;
:: CONNSP_3:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is_a_component_of b1
holds Component_of b2 = b2;
:: CONNSP_3:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is_a_component_of b1
iff
ex b3 being Element of bool the carrier of b1 st
b3 is connected(b1) & b3 <> {} & b2 = Component_of b3;
:: CONNSP_3:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is connected(b1) & b2 <> {}
holds Component_of b2 is_a_component_of b1;
:: CONNSP_3:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is_a_component_of b1 & b3 is connected(b1) & b3 c= b2 & b3 <> {}
holds b2 = Component_of b3;
:: CONNSP_3:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is connected(b1) & b2 <> {}
holds Component_of Component_of b2 = Component_of b2;
:: CONNSP_3:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is connected(b1) & b3 is connected(b1) & b2 <> {} & b2 c= b3
holds Component_of b2 = Component_of b3;
:: CONNSP_3:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is connected(b1) & b3 is connected(b1) & b2 <> {} & b2 c= b3
holds b3 c= Component_of b2;
:: CONNSP_3:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is connected(b1) & b2 \/ b3 is connected(b1) & b2 <> {}
holds b2 \/ b3 c= Component_of b2;
:: CONNSP_3:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b2 is connected(b1) & b3 in b2
holds Component_of b3 = Component_of b2;
:: CONNSP_3:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is connected(b1) & b3 is connected(b1) & b2 meets b3
holds b2 \/ b3 c= Component_of b2 & b2 \/ b3 c= Component_of b3 & b2 c= Component_of b3 & b3 c= Component_of b2;
:: CONNSP_3:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is connected(b1) & b2 <> {}
holds Cl b2 c= Component_of b2;
:: CONNSP_3:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is_a_component_of b1 & b3 is connected(b1) & b3 <> {} & b2 misses b3
holds b2 misses Component_of b3;
:: CONNSP_3:modenot 1 => CONNSP_3:mode 1
definition
let a1 be TopStruct;
mode a_union_of_components of A1 -> Element of bool the carrier of a1 means
ex b1 being Element of bool bool the carrier of a1 st
(for b2 being Element of bool the carrier of a1
st b2 in b1
holds b2 is_a_component_of a1) &
it = union b1;
end;
:: CONNSP_3:dfs 2
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is a_union_of_components of a1
it is sufficient to prove
thus ex b1 being Element of bool bool the carrier of a1 st
(for b2 being Element of bool the carrier of a1
st b2 in b1
holds b2 is_a_component_of a1) &
a2 = union b1;
:: CONNSP_3:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is a_union_of_components of b1
iff
ex b3 being Element of bool bool the carrier of b1 st
(for b4 being Element of bool the carrier of b1
st b4 in b3
holds b4 is_a_component_of b1) &
b2 = union b3;
:: CONNSP_3:th 19
theorem
for b1 being non empty TopSpace-like TopStruct holds
{} b1 is a_union_of_components of b1;
:: CONNSP_3:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 = the carrier of b1
holds b2 is a_union_of_components of b1;
:: CONNSP_3:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2 & b2 is a_union_of_components of b1
holds Component_of b3 c= b2;
:: CONNSP_3:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is a_union_of_components of b1 & b3 is a_union_of_components of b1
holds b2 \/ b3 is a_union_of_components of b1 & b2 /\ b3 is a_union_of_components of b1;
:: CONNSP_3:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is a_union_of_components of b1
holds union b2 is a_union_of_components of b1;
:: CONNSP_3:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is a_union_of_components of b1
holds meet b2 is a_union_of_components of b1;
:: CONNSP_3:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is a_union_of_components of b1 & b3 is a_union_of_components of b1
holds b2 \ b3 is a_union_of_components of b1;
:: CONNSP_3:funcnot 2 => CONNSP_3:func 2
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Element of the carrier of a1;
assume a3 in a2;
func Down(A3,A2) -> Element of the carrier of a1 | a2 equals
a3;
end;
:: CONNSP_3:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2
holds Down(b3,b2) = b3;
:: CONNSP_3:funcnot 3 => CONNSP_3:func 3
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Element of the carrier of a1 | a2;
assume a2 <> {};
func Up A3 -> Element of the carrier of a1 equals
a3;
end;
:: CONNSP_3:def 4
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 | b2
st b2 <> {}
holds Up b3 = b3;
:: CONNSP_3:funcnot 4 => CONNSP_3:func 4
definition
let a1 be TopStruct;
let a2, a3 be Element of bool the carrier of a1;
func Down(A2,A3) -> Element of bool the carrier of a1 | a3 equals
a2 /\ a3;
end;
:: CONNSP_3:def 5
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Down(b2,b3) = b2 /\ b3;
:: CONNSP_3:funcnot 5 => CONNSP_3:func 5
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Element of bool the carrier of a1 | a2;
func Up A3 -> Element of bool the carrier of a1 equals
a3;
end;
:: CONNSP_3:def 6
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 | b2 holds
Up b3 = b3;
:: CONNSP_3:funcnot 6 => CONNSP_3:func 6
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Element of the carrier of a1;
assume a3 in a2;
func Component_of(A3,A2) -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 | a2
st b1 = a3
holds it = Component_of b1;
end;
:: CONNSP_3:def 7
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2
for b4 being Element of bool the carrier of b1 holds
b4 = Component_of(b3,b2)
iff
for b5 being Element of the carrier of b1 | b2
st b5 = b3
holds b4 = Component_of b5;
:: CONNSP_3:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2
holds b3 in Component_of(b3,b2);
:: CONNSP_3:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2
holds Component_of(b3,b2) = Component_of Down(b3,b2);
:: CONNSP_3:th 29
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is open(b1)
holds Down(b2,b3) is open(b1 | b3);
:: CONNSP_3:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3
holds Cl Down(b2,b3) = (Cl b2) /\ b3;
:: CONNSP_3:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 | b2 holds
Cl b3 = (Cl Up b3) /\ b2;
:: CONNSP_3:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3
holds Cl Down(b2,b3) c= Cl b2;
:: CONNSP_3:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 | b2
st b3 c= b2
holds Down(Up b3,b2) = b3;
:: CONNSP_3:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2
holds Component_of(b3,b2) is connected(b1);
:: CONNSP_3:exreg 1
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty a_union_of_components of a1;
end;
:: CONNSP_3:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_union_of_components of b1
st b2 is connected(b1)
holds b2 is_a_component_of b1;