Article RUSUB_5, MML version 4.99.1005
:: RUSUB_5:prednot 1 => RUSUB_5:pred 1
definition
let a1 be non empty RLSStruct;
let a2, a3 be Affine Element of bool the carrier of a1;
pred A2 is_parallel_to A3 means
ex b1 being Element of the carrier of a1 st
a2 = a3 + {b1};
end;
:: RUSUB_5:dfs 1
definiens
let a1 be non empty RLSStruct;
let a2, a3 be Affine Element of bool the carrier of a1;
To prove
a2 is_parallel_to a3
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
a2 = a3 + {b1};
:: RUSUB_5:def 1
theorem
for b1 being non empty RLSStruct
for b2, b3 being Affine Element of bool the carrier of b1 holds
b2 is_parallel_to b3
iff
ex b4 being Element of the carrier of b1 st
b2 = b3 + {b4};
:: RUSUB_5:th 1
theorem
for b1 being non empty right_zeroed RLSStruct
for b2 being Affine Element of bool the carrier of b1 holds
b2 is_parallel_to b2;
:: RUSUB_5:th 2
theorem
for b1 being non empty right_complementable add-associative right_zeroed RLSStruct
for b2, b3 being Affine Element of bool the carrier of b1
st b2 is_parallel_to b3
holds b3 is_parallel_to b2;
:: RUSUB_5:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RLSStruct
for b2, b3, b4 being Affine Element of bool the carrier of b1
st b2 is_parallel_to b3 & b3 is_parallel_to b4
holds b2 is_parallel_to b4;
:: RUSUB_5:funcnot 1 => RUSUB_5:func 1
definition
let a1 be non empty addLoopStr;
let a2, a3 be Element of bool the carrier of a1;
func A2 - A3 -> Element of bool the carrier of a1 equals
{b1 - b2 where b1 is Element of the carrier of a1, b2 is Element of the carrier of a1: b1 in a2 & b2 in a3};
end;
:: RUSUB_5:def 2
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of bool the carrier of b1 holds
b2 - b3 = {b4 - b5 where b4 is Element of the carrier of b1, b5 is Element of the carrier of b1: b4 in b2 & b5 in b3};
:: RUSUB_5:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Affine Element of bool the carrier of b1 holds
b2 - b3 is Affine(b1);
:: RUSUB_5:th 5
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of bool the carrier of b1
st (b2 is empty or b3 is empty)
holds b2 + b3 is empty;
:: RUSUB_5:th 6
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty Element of bool the carrier of b1 holds
b2 + b3 is not empty;
:: RUSUB_5:th 7
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of bool the carrier of b1
st (b2 is empty or b3 is empty)
holds b2 - b3 is empty;
:: RUSUB_5:th 8
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty Element of bool the carrier of b1 holds
b2 - b3 is not empty;
:: RUSUB_5:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1 holds
b2 = b3 + {b4}
iff
b2 - {b4} = b3;
:: RUSUB_5:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
st b4 in b3
holds b2 + {b4} c= b2 + b3;
:: RUSUB_5:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
st b4 in b3
holds b2 - {b4} c= b2 - b3;
:: RUSUB_5:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Element of bool the carrier of b1 holds
0. b1 in b2 - b2;
:: RUSUB_5:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Affine Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b2 is Subspace-like(b1) & b3 in b2
holds b2 + {b3} c= b2;
:: RUSUB_5:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being non empty Affine Element of bool the carrier of b1
st b3 is Subspace-like(b1) & b4 is Subspace-like(b1) & b2 is_parallel_to b3 & b2 is_parallel_to b4
holds b3 = b4;
:: RUSUB_5:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Affine Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2
holds 0. b1 in b2 - {b3};
:: RUSUB_5:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Affine Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2
holds ex b4 being non empty Affine Element of bool the carrier of b1 st
b4 = b2 - {b3} & b2 is_parallel_to b4 & b4 is Subspace-like(b1);
:: RUSUB_5:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Affine Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2
holds b2 - {b4} = b2 - {b3};
:: RUSUB_5:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Affine Element of bool the carrier of b1 holds
b2 - b2 = union {b2 - {b3} where b3 is Element of the carrier of b1: b3 in b2};
:: RUSUB_5:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Affine Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2
holds b2 - {b3} = union {b2 - {b4} where b4 is Element of the carrier of b1: b4 in b2};
:: RUSUB_5:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Affine Element of bool the carrier of b1 holds
ex b3 being non empty Affine Element of bool the carrier of b1 st
b3 = b2 - b2 & b3 is Subspace-like(b1) & b2 is_parallel_to b3;
:: RUSUB_5:funcnot 2 => RUSUB_5:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Subspace of a1;
func Ort_Comp A2 -> strict Subspace of a1 means
the carrier of it = {b1 where b1 is Element of the carrier of a1: for b2 being Element of the carrier of a1
st b2 in a2
holds b2,b1 are_orthogonal};
end;
:: RUSUB_5:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being strict Subspace of b1 holds
b3 = Ort_Comp b2
iff
the carrier of b3 = {b4 where b4 is Element of the carrier of b1: for b5 being Element of the carrier of b1
st b5 in b2
holds b5,b4 are_orthogonal};
:: RUSUB_5:funcnot 3 => RUSUB_5:func 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be non empty Element of bool the carrier of a1;
func Ort_Comp A2 -> strict Subspace of a1 means
the carrier of it = {b1 where b1 is Element of the carrier of a1: for b2 being Element of the carrier of a1
st b2 in a2
holds b2,b1 are_orthogonal};
end;
:: RUSUB_5:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being non empty Element of bool the carrier of b1
for b3 being strict Subspace of b1 holds
b3 = Ort_Comp b2
iff
the carrier of b3 = {b4 where b4 is Element of the carrier of b1: for b5 being Element of the carrier of b1
st b5 in b2
holds b5,b4 are_orthogonal};
:: RUSUB_5:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
0. b1 in Ort_Comp b2;
:: RUSUB_5:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
Ort_Comp (0). b1 = (Omega). b1;
:: RUSUB_5:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
Ort_Comp (Omega). b1 = (0). b1;
:: RUSUB_5:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
st b3 <> 0. b1 & b3 in b2
holds not b3 in Ort_Comp b2;
:: RUSUB_5:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being non empty Element of bool the carrier of b1 holds
b2 c= the carrier of Ort_Comp Ort_Comp b2;
:: RUSUB_5:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being non empty Element of bool the carrier of b1
st b2 c= b3
holds the carrier of Ort_Comp b3 c= the carrier of Ort_Comp b2;
:: RUSUB_5:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being non empty Element of bool the carrier of b1
st b3 = the carrier of b2
holds Ort_Comp b3 = Ort_Comp b2;
:: RUSUB_5:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being non empty Element of bool the carrier of b1 holds
Ort_Comp b2 = Ort_Comp Ort_Comp Ort_Comp b2;
:: RUSUB_5:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 + b3.|| ^2 = (||.b2.|| ^2 + (2 * (b2 .|. b3))) + (||.b3.|| ^2) &
||.b2 - b3.|| ^2 = (||.b2.|| ^2 - (2 * (b2 .|. b3))) + (||.b3.|| ^2);
:: RUSUB_5:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
st b2,b3 are_orthogonal
holds ||.b2 + b3.|| ^2 = ||.b2.|| ^2 + (||.b3.|| ^2);
:: RUSUB_5:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 + b3.|| ^2 + (||.b2 - b3.|| ^2) = (2 * (||.b2.|| ^2)) + (2 * (||.b3.|| ^2));
:: RUSUB_5:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
ex b3 being Subspace of b1 st
the carrier of b3 = {b4 where b4 is Element of the carrier of b1: b4 .|. b2 = 0};
:: RUSUB_5:funcnot 4 => RUSUB_5:func 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
func Family_open_set A1 -> Element of bool bool the carrier of a1 means
for b1 being Element of bool the carrier of a1 holds
b1 in it
iff
for b2 being Element of the carrier of a1
st b2 in b1
holds ex b3 being Element of REAL st
0 < b3 & Ball(b2,b3) c= b1;
end;
:: RUSUB_5:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool bool the carrier of b1 holds
b2 = Family_open_set b1
iff
for b3 being Element of bool the carrier of b1 holds
b3 in b2
iff
for b4 being Element of the carrier of b1
st b4 in b3
holds ex b5 being Element of REAL st
0 < b5 & Ball(b4,b5) c= b3;
:: RUSUB_5:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3, b4 being Element of REAL
st b3 <= b4
holds Ball(b2,b3) c= Ball(b2,b4);
:: RUSUB_5:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
ex b3 being Element of REAL st
0 < b3 & Ball(b2,b3) c= the carrier of b1;
:: RUSUB_5:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL
st b3 in Ball(b2,b4)
holds ex b5 being Element of REAL st
0 < b5 & Ball(b3,b5) c= Ball(b2,b4);
:: RUSUB_5:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Element of REAL
st b3 in (Ball(b2,b5)) /\ Ball(b4,b6)
holds ex b7 being Element of REAL st
Ball(b3,b7) c= Ball(b2,b5) & Ball(b3,b7) c= Ball(b4,b6);
:: RUSUB_5:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL holds
Ball(b2,b3) in Family_open_set b1;
:: RUSUB_5:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
the carrier of b1 in Family_open_set b1;
:: RUSUB_5:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of bool the carrier of b1
st b2 in Family_open_set b1 & b3 in Family_open_set b1
holds b2 /\ b3 in Family_open_set b1;
:: RUSUB_5:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool bool the carrier of b1
st b2 c= Family_open_set b1
holds union b2 in Family_open_set b1;
:: RUSUB_5:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
TopStruct(#the carrier of b1,Family_open_set b1#) is TopSpace-like TopStruct;
:: RUSUB_5:funcnot 5 => RUSUB_5:func 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
func TopUnitSpace A1 -> TopStruct equals
TopStruct(#the carrier of a1,Family_open_set a1#);
end;
:: RUSUB_5:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
TopUnitSpace b1 = TopStruct(#the carrier of b1,Family_open_set b1#);
:: RUSUB_5:funcreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
cluster TopUnitSpace a1 -> TopSpace-like;
end;
:: RUSUB_5:funcreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
cluster TopUnitSpace a1 -> non empty;
end;
:: RUSUB_5:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of TopUnitSpace b1
st b2 = [#] b1
holds b2 is open(TopUnitSpace b1) & b2 is closed(TopUnitSpace b1);
:: RUSUB_5:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of TopUnitSpace b1
st b2 = {} b1
holds b2 is open(TopUnitSpace b1) & b2 is closed(TopUnitSpace b1);
:: RUSUB_5:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL
st the carrier of b1 = {0. b1} & b3 <> 0
holds Sphere(b2,b3) is empty;
:: RUSUB_5:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL
st the carrier of b1 <> {0. b1} & 0 < b3
holds Sphere(b2,b3) is not empty;
:: RUSUB_5:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL
st b3 = 0
holds Ball(b2,b3) is empty;
:: RUSUB_5:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL
st the carrier of b1 = {0. b1} & 0 < b3
holds Ball(b2,b3) = {0. b1};
:: RUSUB_5:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL
st the carrier of b1 <> {0. b1} & 0 < b3
holds ex b4 being Element of the carrier of b1 st
b4 <> b2 & b4 in Ball(b2,b3);
:: RUSUB_5:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
the carrier of b1 = the carrier of TopUnitSpace b1 & the topology of TopUnitSpace b1 = Family_open_set b1;
:: RUSUB_5:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of TopUnitSpace b1
for b3 being Element of REAL
for b4 being Element of the carrier of b1
st b2 = Ball(b4,b3)
holds b2 is open(TopUnitSpace b1);
:: RUSUB_5:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of TopUnitSpace b1 holds
b2 is open(TopUnitSpace b1)
iff
for b3 being Element of the carrier of b1
st b3 in b2
holds ex b4 being Element of REAL st
0 < b4 & Ball(b3,b4) c= b2;
:: RUSUB_5:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of REAL holds
ex b6 being Element of the carrier of b1 st
ex b7 being Element of REAL st
(Ball(b2,b4)) \/ Ball(b3,b5) c= Ball(b6,b7);
:: RUSUB_5:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of TopUnitSpace b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = cl_Ball(b3,b4)
holds b2 is closed(TopUnitSpace b1);
:: RUSUB_5:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of TopUnitSpace b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = Sphere(b3,b4)
holds b2 is closed(TopUnitSpace b1);