Article RLVECT_2, MML version 4.99.1005
:: RLVECT_2:funcnot 1 => RLVECT_2:func 1
definition
let a1 be 1-sorted;
let a2 be set;
assume a2 in a1;
func vector(A1,A2) -> Element of the carrier of a1 equals
a2;
end;
:: RLVECT_2:def 1
theorem
for b1 being 1-sorted
for b2 being set
st b2 in b1
holds vector(b1,b2) = b2;
:: RLVECT_2:th 3
theorem
for b1 being non empty 1-sorted
for b2 being Element of the carrier of b1 holds
vector(b1,b2) = b2;
:: RLVECT_2:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3, b4 being FinSequence of the carrier of b1
st len b2 = len b3 &
len b2 = len b4 &
(for b5 being Element of NAT
st b5 in dom b2
holds b4 . b5 = (b2 /. b5) + (b3 /. b5))
holds Sum b4 = (Sum b2) + Sum b3;
:: RLVECT_2:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of REAL
for b3, b4 being FinSequence of the carrier of b1
st len b3 = len b4 &
(for b5 being Element of NAT
st b5 in dom b3
holds b4 . b5 = b2 * (b3 /. b5))
holds Sum b4 = b2 * Sum b3;
:: RLVECT_2:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being FinSequence of the carrier of b1
st len b2 = len b3 &
(for b4 being Element of NAT
st b4 in dom b2
holds b3 . b4 = - (b2 /. b4))
holds Sum b3 = - Sum b2;
:: RLVECT_2:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3, b4 being FinSequence of the carrier of b1
st len b2 = len b3 &
len b2 = len b4 &
(for b5 being Element of NAT
st b5 in dom b2
holds b4 . b5 = (b2 /. b5) - (b3 /. b5))
holds Sum b4 = (Sum b2) - Sum b3;
:: RLVECT_2:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being FinSequence of the carrier of b1
for b4 being Function-like quasi_total bijective Relation of dom b2,dom b2
st len b2 = len b3 &
(for b5 being Element of NAT
st b5 in dom b3
holds b3 . b5 = b2 . (b4 . b5))
holds Sum b2 = Sum b3;
:: RLVECT_2:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being FinSequence of the carrier of b1
for b4 being Function-like quasi_total bijective Relation of dom b2,dom b2
st b3 = b2 * b4
holds Sum b2 = Sum b3;
:: RLVECT_2:exreg 1
registration
let a1 be 1-sorted;
cluster empty finite Element of bool the carrier of a1;
end;
:: RLVECT_2:funcnot 2 => RLVECT_2:func 2
definition
let a1 be non empty addLoopStr;
let a2 be finite Element of bool the carrier of a1;
assume a1 is Abelian & a1 is add-associative & a1 is right_zeroed;
func Sum A2 -> Element of the carrier of a1 means
ex b1 being FinSequence of the carrier of a1 st
rng b1 = a2 & b1 is one-to-one & it = Sum b1;
end;
:: RLVECT_2:def 4
theorem
for b1 being non empty addLoopStr
for b2 being finite Element of bool the carrier of b1
st b1 is Abelian & b1 is add-associative & b1 is right_zeroed
for b3 being Element of the carrier of b1 holds
b3 = Sum b2
iff
ex b4 being FinSequence of the carrier of b1 st
rng b4 = b2 & b4 is one-to-one & b3 = Sum b4;
:: RLVECT_2:exreg 2
registration
let a1 be non empty 1-sorted;
cluster non empty finite Element of bool the carrier of a1;
end;
:: RLVECT_2:th 14
theorem
for b1 being non empty Abelian add-associative right_zeroed addLoopStr holds
Sum {} b1 = 0. b1;
:: RLVECT_2:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
Sum {b2} = b2;
:: RLVECT_2:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds Sum {b2,b3} = b2 + b3;
:: RLVECT_2:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3, b4 being Element of the carrier of b1
st b2 <> b3 & b3 <> b4 & b2 <> b4
holds Sum {b2,b3,b4} = (b2 + b3) + b4;
:: RLVECT_2:th 18
theorem
for b1 being non empty Abelian add-associative right_zeroed addLoopStr
for b2, b3 being finite Element of bool the carrier of b1
st b3 misses b2
holds Sum (b3 \/ b2) = (Sum b3) + Sum b2;
:: RLVECT_2:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being finite Element of bool the carrier of b1 holds
Sum (b3 \/ b2) = ((Sum b3) + Sum b2) - Sum (b3 /\ b2);
:: RLVECT_2:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being finite Element of bool the carrier of b1 holds
Sum (b3 /\ b2) = ((Sum b3) + Sum b2) - Sum (b3 \/ b2);
:: RLVECT_2:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being finite Element of bool the carrier of b1 holds
Sum (b3 \ b2) = (Sum (b3 \/ b2)) - Sum b2;
:: RLVECT_2:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being finite Element of bool the carrier of b1 holds
Sum (b3 \ b2) = (Sum b3) - Sum (b3 /\ b2);
:: RLVECT_2:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being finite Element of bool the carrier of b1 holds
Sum (b3 \+\ b2) = (Sum (b3 \/ b2)) - Sum (b3 /\ b2);
:: RLVECT_2:th 24
theorem
for b1 being non empty Abelian add-associative right_zeroed addLoopStr
for b2, b3 being finite Element of bool the carrier of b1 holds
Sum (b3 \+\ b2) = (Sum (b3 \ b2)) + Sum (b2 \ b3);
:: RLVECT_2:modenot 1 => RLVECT_2:mode 1
definition
let a1 be non empty ZeroStr;
mode Linear_Combination of A1 -> Element of Funcs(the carrier of a1,REAL) means
ex b1 being finite Element of bool the carrier of a1 st
for b2 being Element of the carrier of a1
st not b2 in b1
holds it . b2 = 0;
end;
:: RLVECT_2:dfs 3
definiens
let a1 be non empty ZeroStr;
let a2 be Element of Funcs(the carrier of a1,REAL);
To prove
a2 is Linear_Combination of a1
it is sufficient to prove
thus ex b1 being finite Element of bool the carrier of a1 st
for b2 being Element of the carrier of a1
st not b2 in b1
holds a2 . b2 = 0;
:: RLVECT_2:def 5
theorem
for b1 being non empty ZeroStr
for b2 being Element of Funcs(the carrier of b1,REAL) holds
b2 is Linear_Combination of b1
iff
ex b3 being finite Element of bool the carrier of b1 st
for b4 being Element of the carrier of b1
st not b4 in b3
holds b2 . b4 = 0;
:: RLVECT_2:funcnot 3 => RLVECT_2:func 3
definition
let a1 be non empty addLoopStr;
let a2 be Linear_Combination of a1;
func Carrier A2 -> finite Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: a2 . b1 <> 0};
end;
:: RLVECT_2:def 6
theorem
for b1 being non empty addLoopStr
for b2 being Linear_Combination of b1 holds
Carrier b2 = {b3 where b3 is Element of the carrier of b1: b2 . b3 <> 0};
:: RLVECT_2:th 28
theorem
for b1 being non empty addLoopStr
for b2 being Linear_Combination of b1
for b3 being Element of the carrier of b1 holds
b2 . b3 = 0
iff
not b3 in Carrier b2;
:: RLVECT_2:funcnot 4 => RLVECT_2:func 4
definition
let a1 be non empty addLoopStr;
func ZeroLC A1 -> Linear_Combination of a1 means
Carrier it = {};
end;
:: RLVECT_2:def 7
theorem
for b1 being non empty addLoopStr
for b2 being Linear_Combination of b1 holds
b2 = ZeroLC b1
iff
Carrier b2 = {};
:: RLVECT_2:th 30
theorem
for b1 being non empty addLoopStr
for b2 being Element of the carrier of b1 holds
(ZeroLC b1) . b2 = 0;
:: RLVECT_2:modenot 2 => RLVECT_2:mode 2
definition
let a1 be non empty addLoopStr;
let a2 be Element of bool the carrier of a1;
mode Linear_Combination of A2 -> Linear_Combination of a1 means
Carrier it c= a2;
end;
:: RLVECT_2:dfs 6
definiens
let a1 be non empty addLoopStr;
let a2 be Element of bool the carrier of a1;
let a3 be Linear_Combination of a1;
To prove
a3 is Linear_Combination of a2
it is sufficient to prove
thus Carrier a3 c= a2;
:: RLVECT_2:def 8
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1
for b3 being Linear_Combination of b1 holds
b3 is Linear_Combination of b2
iff
Carrier b3 c= b2;
:: RLVECT_2:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Linear_Combination of b2
st b2 c= b3
holds b4 is Linear_Combination of b3;
:: RLVECT_2:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1 holds
ZeroLC b1 is Linear_Combination of b2;
:: RLVECT_2:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of {} the carrier of b1 holds
b2 = ZeroLC b1;
:: RLVECT_2:funcnot 5 => RLVECT_2:func 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be FinSequence of the carrier of a1;
let a3 be Function-like quasi_total Relation of the carrier of a1,REAL;
func A3 (#) A2 -> FinSequence of the carrier of a1 means
len it = len a2 &
(for b1 being Element of NAT
st b1 in dom it
holds it . b1 = (a3 . (a2 /. b1)) * (a2 /. b1));
end;
:: RLVECT_2:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being FinSequence of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL
for b4 being FinSequence of the carrier of b1 holds
b4 = b3 (#) b2
iff
len b4 = len b2 &
(for b5 being Element of NAT
st b5 in dom b4
holds b4 . b5 = (b3 . (b2 /. b5)) * (b2 /. b5));
:: RLVECT_2:th 40
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of the carrier of b2
for b4 being FinSequence of the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,REAL
st b1 in dom b4 & b3 = b4 . b1
holds (b5 (#) b4) . b1 = (b5 . b3) * b3;
:: RLVECT_2:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 (#) <*> the carrier of b1 = <*> the carrier of b1;
:: RLVECT_2:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b3 (#) <*b2*> = <*(b3 . b2) * b2*>;
:: RLVECT_2:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b4 (#) <*b2,b3*> = <*(b4 . b2) * b2,(b4 . b3) * b3*>;
:: RLVECT_2:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b5 (#) <*b2,b3,b4*> = <*(b5 . b2) * b2,(b5 . b3) * b3,(b5 . b4) * b4*>;
:: RLVECT_2:funcnot 6 => RLVECT_2:func 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Linear_Combination of a1;
func Sum A2 -> Element of the carrier of a1 means
ex b1 being FinSequence of the carrier of a1 st
b1 is one-to-one & rng b1 = Carrier a2 & it = Sum (a2 (#) b1);
end;
:: RLVECT_2:def 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
for b3 being Element of the carrier of b1 holds
b3 = Sum b2
iff
ex b4 being FinSequence of the carrier of b1 st
b4 is one-to-one & rng b4 = Carrier b2 & b3 = Sum (b2 (#) b4);
:: RLVECT_2:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1 holds
b2 <> {} & b2 is linearly-closed(b1)
iff
for b3 being Linear_Combination of b2 holds
Sum b3 in b2;
:: RLVECT_2:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
Sum ZeroLC b1 = 0. b1;
:: RLVECT_2:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of {} the carrier of b1 holds
Sum b2 = 0. b1;
:: RLVECT_2:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Linear_Combination of {b2} holds
Sum b3 = (b3 . b2) * b2;
:: RLVECT_2:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st b2 <> b3
for b4 being Linear_Combination of {b2,b3} holds
Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);
:: RLVECT_2:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
st Carrier b2 = {}
holds Sum b2 = 0. b1;
:: RLVECT_2:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Linear_Combination of b1
st Carrier b3 = {b2}
holds Sum b3 = (b3 . b2) * b2;
:: RLVECT_2:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Linear_Combination of b1
st Carrier b4 = {b2,b3} & b2 <> b3
holds Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);
:: RLVECT_2:prednot 1 => RLVECT_2:pred 1
definition
let a1 be non empty addLoopStr;
let a2, a3 be Linear_Combination of a1;
redefine pred A2 = A3 means
for b1 being Element of the carrier of a1 holds
a2 . b1 = a3 . b1;
symmetry;
:: for a1 being non empty addLoopStr
:: for a2, a3 being Linear_Combination of a1
:: st a2 = a3
:: holds a3 = a2;
reflexivity;
:: for a1 being non empty addLoopStr
:: for a2 being Linear_Combination of a1 holds
:: a2 = a2;
end;
:: RLVECT_2:dfs 9
definiens
let a1 be non empty addLoopStr;
let a2, a3 be Linear_Combination of a1;
To prove
a2 = a3
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
a2 . b1 = a3 . b1;
:: RLVECT_2:def 11
theorem
for b1 being non empty addLoopStr
for b2, b3 being Linear_Combination of b1 holds
b2 = b3
iff
for b4 being Element of the carrier of b1 holds
b2 . b4 = b3 . b4;
:: RLVECT_2:funcnot 7 => RLVECT_2:func 7
definition
let a1 be non empty addLoopStr;
let a2, a3 be Linear_Combination of a1;
redefine func A2 + A3 -> Linear_Combination of a1 means
for b1 being Element of the carrier of a1 holds
it . b1 = (a2 . b1) + (a3 . b1);
commutativity;
:: for a1 being non empty addLoopStr
:: for a2, a3 being Linear_Combination of a1 holds
:: a2 + a3 = a3 + a2;
end;
:: RLVECT_2:def 12
theorem
for b1 being non empty addLoopStr
for b2, b3, b4 being Linear_Combination of b1 holds
b4 = b2 + b3
iff
for b5 being Element of the carrier of b1 holds
b4 . b5 = (b2 . b5) + (b3 . b5);
:: RLVECT_2:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Linear_Combination of b1 holds
Carrier (b2 + b3) c= (Carrier b2) \/ Carrier b3;
:: RLVECT_2:th 59
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Linear_Combination of b1
st b3 is Linear_Combination of b2 & b4 is Linear_Combination of b2
holds b3 + b4 is Linear_Combination of b2;
:: RLVECT_2:th 60
theorem
for b1 being non empty addLoopStr
for b2, b3 being Linear_Combination of b1 holds
b2 + b3 = b3 + b2;
:: RLVECT_2:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Linear_Combination of b1 holds
b2 + (b3 + b4) = (b2 + b3) + b4;
:: RLVECT_2:th 62
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
b2 + ZeroLC b1 = b2 & (ZeroLC b1) + b2 = b2;
:: RLVECT_2:funcnot 8 => RLVECT_2:func 8
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of REAL;
let a3 be Linear_Combination of a1;
func A2 * A3 -> Linear_Combination of a1 means
for b1 being Element of the carrier of a1 holds
it . b1 = a2 * (a3 . b1);
end;
:: RLVECT_2:def 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of REAL
for b3, b4 being Linear_Combination of b1 holds
b4 = b2 * b3
iff
for b5 being Element of the carrier of b1 holds
b4 . b5 = b2 * (b3 . b5);
:: RLVECT_2:th 65
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of REAL
for b3 being Linear_Combination of b1
st b2 <> 0
holds Carrier (b2 * b3) = Carrier b3;
:: RLVECT_2:th 66
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
0 * b2 = ZeroLC b1;
:: RLVECT_2:th 67
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of REAL
for b3 being Element of bool the carrier of b1
for b4 being Linear_Combination of b1
st b4 is Linear_Combination of b3
holds b2 * b4 is Linear_Combination of b3;
:: RLVECT_2:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of REAL
for b4 being Linear_Combination of b1 holds
(b2 + b3) * b4 = (b2 * b4) + (b3 * b4);
:: RLVECT_2:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of REAL
for b3, b4 being Linear_Combination of b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4);
:: RLVECT_2:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of REAL
for b4 being Linear_Combination of b1 holds
b2 * (b3 * b4) = (b2 * b3) * b4;
:: RLVECT_2:th 71
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
1 * b2 = b2;
:: RLVECT_2:funcnot 9 => RLVECT_2:func 9
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Linear_Combination of a1;
func - A2 -> Linear_Combination of a1 equals
(- 1) * a2;
end;
:: RLVECT_2:def 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
- b2 = (- 1) * b2;
:: RLVECT_2:th 73
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Linear_Combination of b1 holds
(- b3) . b2 = - (b3 . b2);
:: RLVECT_2:th 74
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Linear_Combination of b1
st b2 + b3 = ZeroLC b1
holds b3 = - b2;
:: RLVECT_2:th 75
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
Carrier - b2 = Carrier b2;
:: RLVECT_2:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Linear_Combination of b1
st b3 is Linear_Combination of b2
holds - b3 is Linear_Combination of b2;
:: RLVECT_2:th 77
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
- - b2 = b2;
:: RLVECT_2:funcnot 10 => RLVECT_2:func 10
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3 be Linear_Combination of a1;
func A2 - A3 -> Linear_Combination of a1 equals
a2 + - a3;
end;
:: RLVECT_2:def 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Linear_Combination of b1 holds
b2 - b3 = b2 + - b3;
:: RLVECT_2:th 79
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being Linear_Combination of b1 holds
(b3 - b4) . b2 = (b3 . b2) - (b4 . b2);
:: RLVECT_2:th 80
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Linear_Combination of b1 holds
Carrier (b2 - b3) c= (Carrier b2) \/ Carrier b3;
:: RLVECT_2:th 81
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Linear_Combination of b1
st b3 is Linear_Combination of b2 & b4 is Linear_Combination of b2
holds b3 - b4 is Linear_Combination of b2;
:: RLVECT_2:th 82
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
b2 - b2 = ZeroLC b1;
:: RLVECT_2:funcnot 11 => RLVECT_2:func 11
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func LinComb A1 -> set means
for b1 being set holds
b1 in it
iff
b1 is Linear_Combination of a1;
end;
:: RLVECT_2:def 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set holds
b2 = LinComb b1
iff
for b3 being set holds
b3 in b2
iff
b3 is Linear_Combination of b1;
:: RLVECT_2:funcreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster LinComb a1 -> non empty;
end;
:: RLVECT_2:funcnot 12 => RLVECT_2:func 12
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of LinComb a1;
func @ A2 -> Linear_Combination of a1 equals
a2;
end;
:: RLVECT_2:def 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of LinComb b1 holds
@ b2 = b2;
:: RLVECT_2:funcnot 13 => RLVECT_2:func 13
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Linear_Combination of a1;
func @ A2 -> Element of LinComb a1 equals
a2;
end;
:: RLVECT_2:def 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
@ b2 = b2;
:: RLVECT_2:funcnot 14 => RLVECT_2:func 14
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func LCAdd A1 -> Function-like quasi_total Relation of [:LinComb a1,LinComb a1:],LinComb a1 means
for b1, b2 being Element of LinComb a1 holds
it .(b1,b2) = (@ b1) + @ b2;
end;
:: RLVECT_2:def 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of [:LinComb b1,LinComb b1:],LinComb b1 holds
b2 = LCAdd b1
iff
for b3, b4 being Element of LinComb b1 holds
b2 .(b3,b4) = (@ b3) + @ b4;
:: RLVECT_2:funcnot 15 => RLVECT_2:func 15
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func LCMult A1 -> Function-like quasi_total Relation of [:REAL,LinComb a1:],LinComb a1 means
for b1 being Element of REAL
for b2 being Element of LinComb a1 holds
it . [b1,b2] = b1 * @ b2;
end;
:: RLVECT_2:def 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of [:REAL,LinComb b1:],LinComb b1 holds
b2 = LCMult b1
iff
for b3 being Element of REAL
for b4 being Element of LinComb b1 holds
b2 . [b3,b4] = b3 * @ b4;
:: RLVECT_2:funcnot 16 => RLVECT_2:func 16
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func LC_RLSpace A1 -> RLSStruct equals
RLSStruct(#LinComb a1,@ ZeroLC a1,LCAdd a1,LCMult a1#);
end;
:: RLVECT_2:def 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
LC_RLSpace b1 = RLSStruct(#LinComb b1,@ ZeroLC b1,LCAdd b1,LCMult b1#);
:: RLVECT_2:funcreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster LC_RLSpace a1 -> non empty strict;
end;
:: RLVECT_2:funcreg 3
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster LC_RLSpace a1 -> right_complementable Abelian add-associative right_zeroed RealLinearSpace-like;
end;
:: RLVECT_2:th 92
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
the carrier of LC_RLSpace b1 = LinComb b1;
:: RLVECT_2:th 93
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
0. LC_RLSpace b1 = ZeroLC b1;
:: RLVECT_2:th 94
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
the addF of LC_RLSpace b1 = LCAdd b1;
:: RLVECT_2:th 95
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
the Mult of LC_RLSpace b1 = LCMult b1;
:: RLVECT_2:th 96
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Linear_Combination of b1 holds
(vector(LC_RLSpace b1,b2)) + vector(LC_RLSpace b1,b3) = b2 + b3;
:: RLVECT_2:th 97
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of REAL
for b3 being Linear_Combination of b1 holds
b2 * vector(LC_RLSpace b1,b3) = b2 * b3;
:: RLVECT_2:th 98
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
- vector(LC_RLSpace b1,b2) = - b2;
:: RLVECT_2:th 99
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Linear_Combination of b1 holds
(vector(LC_RLSpace b1,b2)) - vector(LC_RLSpace b1,b3) = b2 - b3;
:: RLVECT_2:funcnot 17 => RLVECT_2:func 17
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of bool the carrier of a1;
func LC_RLSpace A2 -> strict Subspace of LC_RLSpace a1 means
the carrier of it = {b1 where b1 is Linear_Combination of a2: TRUE};
end;
:: RLVECT_2:def 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being strict Subspace of LC_RLSpace b1 holds
b3 = LC_RLSpace b2
iff
the carrier of b3 = {b4 where b4 is Linear_Combination of b2: TRUE};