Article JORDAN1K, MML version 4.99.1005
:: JORDAN1K:th 1
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
st b3 is onto(b1, b2)
for b4 being Element of b2 holds
ex b5 being set st
b5 in b1 & b4 = b3 . b5;
:: JORDAN1K:th 2
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
st b3 is onto(b1, b2)
for b4 being Element of b2 holds
ex b5 being Element of b1 st
b4 = b3 . b5;
:: JORDAN1K:th 3
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of bool b1
st b3 is onto(b1, b2)
holds (b3 .: b4) ` c= b3 .: (b4 `);
:: JORDAN1K:th 4
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of bool b1
st b3 is one-to-one
holds b3 .: (b4 `) c= (b3 .: b4) `;
:: JORDAN1K:th 5
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of bool b1
st b3 is bijective(b1, b2)
holds (b3 .: b4) ` = b3 .: (b4 `);
:: JORDAN1K:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is_a_component_of {} b1
iff
b2 is empty;
:: JORDAN1K:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 c= b3 & b2 is_a_component_of b4 & b3 is_a_component_of b4
holds b2 = b3;
:: JORDAN1K:th 8
theorem
for b1 being Element of NAT
st 1 <= b1
for b2 being Element of bool the carrier of Euclid b1
st b2 is bounded(Euclid b1)
holds b2 ` is bounded(not Euclid b1);
:: JORDAN1K:th 9
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being non empty Element of bool the carrier of TopSpaceMetr b1
for b3 being Element of the carrier of TopSpaceMetr b1 holds
0 <= (dist_min b2) . b3;
:: JORDAN1K:th 10
theorem
for b1 being Element of REAL
for b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being non empty Element of bool the carrier of TopSpaceMetr b2
for b4 being Element of the carrier of b2
st for b5 being Element of the carrier of b2
st b5 in b3
holds b1 <= dist(b4,b5)
holds b1 <= (dist_min b3) . b4;
:: JORDAN1K:th 11
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2, b3 being non empty Element of bool the carrier of TopSpaceMetr b1 holds
0 <= min_dist_min(b2,b3);
:: JORDAN1K:th 12
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2, b3 being compact Element of bool the carrier of TopSpaceMetr b1
st b2 meets b3
holds min_dist_min(b2,b3) = 0;
:: JORDAN1K:th 13
theorem
for b1 being Element of REAL
for b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3, b4 being non empty Element of bool the carrier of TopSpaceMetr b2
st for b5, b6 being Element of the carrier of b2
st b5 in b3 & b6 in b4
holds b1 <= dist(b5,b6)
holds b1 <= min_dist_min(b3,b4);
:: JORDAN1K:th 14
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
st b2 is_a_component_of b3 ` & not b2 is_inside_component_of b3
holds b2 is_outside_component_of b3;
:: JORDAN1K:th 15
theorem
for b1 being Element of NAT
st 1 <= b1
holds BDD {} TOP-REAL b1 = {} TOP-REAL b1;
:: JORDAN1K:th 16
theorem
for b1 being Element of NAT holds
BDD [#] TOP-REAL b1 = {} TOP-REAL b1;
:: JORDAN1K:th 17
theorem
for b1 being Element of NAT
st 1 <= b1
holds UBD {} TOP-REAL b1 = [#] TOP-REAL b1;
:: JORDAN1K:th 18
theorem
for b1 being Element of NAT holds
UBD [#] TOP-REAL b1 = {} TOP-REAL b1;
:: JORDAN1K:th 19
theorem
for b1 being Element of NAT
for b2 being connected Element of bool the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of TOP-REAL b1
st b2 misses b3 & not b2 c= UBD b3
holds b2 c= BDD b3;
:: JORDAN1K:th 20
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of REAL holds
dist(|[0,0]|,b2 * b1) = (abs b2) * dist(|[0,0]|,b1);
:: JORDAN1K:th 21
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2 holds
dist(b1 + b2,b3 + b2) = dist(b1,b3);
:: JORDAN1K:th 22
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
st b1 <> b2
holds 0 < dist(b1,b2);
:: JORDAN1K:th 23
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2 holds
dist(b1 - b2,b3 - b2) = dist(b1,b3);
:: JORDAN1K:th 24
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
dist(b1,b2) = dist(- b1,- b2);
:: JORDAN1K:th 25
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2 holds
dist(b1 - b2,b1 - b3) = dist(b2,b3);
:: JORDAN1K:th 26
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of REAL holds
dist(b3 * b1,b3 * b2) = (abs b3) * dist(b1,b2);
:: JORDAN1K:th 27
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of REAL
st b3 <= 1
holds dist(b1,(b3 * b1) + ((1 - b3) * b2)) = (1 - b3) * dist(b1,b2);
:: JORDAN1K:th 28
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of REAL
st 0 <= b3
holds dist(b1,(b3 * b2) + ((1 - b3) * b1)) = b3 * dist(b2,b1);
:: JORDAN1K:th 29
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b1 in LSeg(b2,b3)
holds (dist(b2,b1)) + dist(b1,b3) = dist(b2,b3);
:: JORDAN1K:th 30
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b1 in LSeg(b2,b3) & b1 <> b2
holds dist(b1,b3) < dist(b2,b3);
:: JORDAN1K:th 31
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of Euclid 2
st b2 = |[0,0]|
holds Ball(b2,b1) = {b3 where b3 is Element of the carrier of TOP-REAL 2: |.b3.| < b1};
:: JORDAN1K:th 32
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3, b4 being Element of REAL holds
(AffineMap(b2,b3,b2,b4)) . b1 = (b2 * b1) + |[b3,b4]|;
:: JORDAN1K:th 33
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of REAL holds
(AffineMap(b3,b1 `1,b3,b1 `2)) . b2 = (b3 * b2) + b1;
:: JORDAN1K:th 34
theorem
for b1, b2, b3, b4 being Element of REAL
st 0 < b1 & 0 < b2
holds (AffineMap(b1,b3,b2,b4)) * AffineMap(1 / b1,- (b3 / b1),1 / b2,- (b4 / b2)) = id REAL 2;
:: JORDAN1K:th 35
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of REAL
for b3, b4 being Element of the carrier of Euclid 2
st b3 = |[0,0]| & b4 = b1 & 0 < b2
holds (AffineMap(b2,b1 `1,b2,b1 `2)) .: Ball(b3,1) = Ball(b4,b2);
:: JORDAN1K:th 36
theorem
for b1, b2, b3, b4 being Element of REAL
st 0 < b1 & 0 < b3
holds AffineMap(b1,b2,b3,b4) is onto(the carrier of TOP-REAL 2, the carrier of TOP-REAL 2);
:: JORDAN1K:th 37
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of Euclid 2 holds
(Ball(b2,b1)) ` is connected Element of bool the carrier of TOP-REAL 2;
:: JORDAN1K:funcnot 1 => JORDAN1K:func 1
definition
let a1 be Element of NAT;
let a2, a3 be Element of bool the carrier of TOP-REAL a1;
func dist_min(A2,A3) -> Element of REAL means
ex b1, b2 being Element of bool the carrier of TopSpaceMetr Euclid a1 st
a2 = b1 & a3 = b2 & it = min_dist_min(b1,b2);
end;
:: JORDAN1K:def 1
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of REAL holds
b4 = dist_min(b2,b3)
iff
ex b5, b6 being Element of bool the carrier of TopSpaceMetr Euclid b1 st
b2 = b5 & b3 = b6 & b4 = min_dist_min(b5,b6);
:: JORDAN1K:funcnot 2 => JORDAN1K:func 2
definition
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
let a2, a3 be non empty compact Element of bool the carrier of TopSpaceMetr a1;
redefine func min_dist_min(a2,a3) -> Element of REAL;
commutativity;
:: for a1 being non empty Reflexive discerning symmetric triangle MetrStruct
:: for a2, a3 being non empty compact Element of bool the carrier of TopSpaceMetr a1 holds
:: min_dist_min(a2,a3) = min_dist_min(a3,a2);
end;
:: JORDAN1K:funcnot 3 => JORDAN1K:func 3
definition
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
let a2, a3 be non empty compact Element of bool the carrier of TopSpaceMetr a1;
redefine func max_dist_max(a2,a3) -> Element of REAL;
commutativity;
:: for a1 being non empty Reflexive discerning symmetric triangle MetrStruct
:: for a2, a3 being non empty compact Element of bool the carrier of TopSpaceMetr a1 holds
:: max_dist_max(a2,a3) = max_dist_max(a3,a2);
end;
:: JORDAN1K:funcnot 4 => JORDAN1K:func 4
definition
let a1 be Element of NAT;
let a2, a3 be non empty compact Element of bool the carrier of TOP-REAL a1;
redefine func dist_min(a2,a3) -> Element of REAL;
commutativity;
:: for a1 being Element of NAT
:: for a2, a3 being non empty compact Element of bool the carrier of TOP-REAL a1 holds
:: dist_min(a2,a3) = dist_min(a3,a2);
end;
:: JORDAN1K:th 38
theorem
for b1 being Element of NAT
for b2, b3 being non empty Element of bool the carrier of TOP-REAL b1 holds
0 <= dist_min(b2,b3);
:: JORDAN1K:th 39
theorem
for b1 being Element of NAT
for b2, b3 being compact Element of bool the carrier of TOP-REAL b1
st b2 meets b3
holds dist_min(b2,b3) = 0;
:: JORDAN1K:th 40
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3, b4 being non empty Element of bool the carrier of TOP-REAL b1
st for b5, b6 being Element of the carrier of TOP-REAL b1
st b5 in b3 & b6 in b4
holds b2 <= dist(b5,b6)
holds b2 <= dist_min(b3,b4);
:: JORDAN1K:th 41
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being non empty Element of bool the carrier of TOP-REAL b1
st b4 c= b2
holds dist_min(b3,b2) <= dist_min(b3,b4);
:: JORDAN1K:th 42
theorem
for b1 being Element of NAT
for b2, b3 being non empty compact Element of bool the carrier of TOP-REAL b1 holds
ex b4, b5 being Element of the carrier of TOP-REAL b1 st
b4 in b2 & b5 in b3 & dist_min(b2,b3) = dist(b4,b5);
:: JORDAN1K:th 43
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
dist_min({b2},{b3}) = dist(b2,b3);
:: JORDAN1K:funcnot 5 => JORDAN1K:func 5
definition
let a1 be Element of NAT;
let a2 be Element of the carrier of TOP-REAL a1;
let a3 be Element of bool the carrier of TOP-REAL a1;
func dist(A2,A3) -> Element of REAL equals
dist_min({a2},a3);
end;
:: JORDAN1K:def 2
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of TOP-REAL b1 holds
dist(b2,b3) = dist_min({b2},b3);
:: JORDAN1K:th 44
theorem
for b1 being Element of NAT
for b2 being non empty Element of bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1 holds
0 <= dist(b3,b2);
:: JORDAN1K:th 45
theorem
for b1 being Element of NAT
for b2 being compact Element of bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
st b3 in b2
holds dist(b3,b2) = 0;
:: JORDAN1K:th 46
theorem
for b1 being Element of NAT
for b2 being non empty compact Element of bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1 holds
ex b4 being Element of the carrier of TOP-REAL b1 st
b4 in b2 & dist(b3,b2) = dist(b3,b4);
:: JORDAN1K:th 47
theorem
for b1 being Element of NAT
for b2 being non empty Element of bool the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of TOP-REAL b1
st b2 c= b3
for b4 being Element of the carrier of TOP-REAL b1 holds
dist(b4,b3) <= dist(b4,b2);
:: JORDAN1K:th 48
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being non empty Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1
st for b5 being Element of the carrier of TOP-REAL b1
st b5 in b3
holds b2 <= dist(b4,b5)
holds b2 <= dist(b4,b3);
:: JORDAN1K:th 49
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
dist(b2,{b3}) = dist(b2,b3);
:: JORDAN1K:th 50
theorem
for b1 being Element of NAT
for b2 being non empty Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
st b4 in b2
holds dist(b3,b2) <= dist(b3,b4);
:: JORDAN1K:th 51
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2 being open Element of bool the carrier of TOP-REAL 2
st b1 c= b2
for b3 being Element of the carrier of TOP-REAL 2
st not b3 in b2
holds dist(b3,b2) < dist(b3,b1);