Article RLVECT_1, MML version 4.99.1005
:: RLVECT_1:structnot 1 => RLVECT_1:struct 1
definition
struct(addLoopStr) RLSStruct(#
carrier -> set,
ZeroF -> Element of the carrier of it,
addF -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
Mult -> Function-like quasi_total Relation of [:REAL,the carrier of it:],the carrier of it
#);
end;
:: RLVECT_1:attrnot 1 => RLVECT_1:attr 1
definition
let a1 be RLSStruct;
attr a1 is strict;
end;
:: RLVECT_1:exreg 1
registration
cluster strict RLSStruct;
end;
:: RLVECT_1:aggrnot 1 => RLVECT_1:aggr 1
definition
let a1 be set;
let a2 be Element of a1;
let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a4 be Function-like quasi_total Relation of [:REAL,a1:],a1;
aggr RLSStruct(#a1,a2,a3,a4#) -> strict RLSStruct;
end;
:: RLVECT_1:selnot 1 => RLVECT_1:sel 1
definition
let a1 be RLSStruct;
sel the Mult of a1 -> Function-like quasi_total Relation of [:REAL,the carrier of a1:],the carrier of a1;
end;
:: RLVECT_1:exreg 2
registration
cluster non empty RLSStruct;
end;
:: RLVECT_1:modenot 1
definition
let a1 be RLSStruct;
mode VECTOR of a1 is Element of the carrier of a1;
end;
:: RLVECT_1:th 3
theorem
for b1 being non empty 1-sorted
for b2 being Element of the carrier of b1 holds
b2 in b1;
:: RLVECT_1:funcnot 1 => RLVECT_1:func 1
definition
let a1 be non empty RLSStruct;
let a2 be Element of the carrier of a1;
let a3 be Element of REAL;
func A3 * A2 -> Element of the carrier of a1 equals
(the Mult of a1) .(a3,a2);
end;
:: RLVECT_1:def 4
theorem
for b1 being non empty RLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of REAL holds
b3 * b2 = (the Mult of b1) .(b3,b2);
:: RLVECT_1:th 5
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of the carrier of b1 holds
b2 + b3 = (the addF of b1) .(b2,b3);
:: RLVECT_1:funcreg 1
registration
let a1 be non empty set;
let a2 be Element of a1;
let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a4 be Function-like quasi_total Relation of [:REAL,a1:],a1;
cluster RLSStruct(#a1,a2,a3,a4#) -> non empty strict;
end;
:: RLVECT_1:attrnot 2 => RLVECT_1:attr 2
definition
let a1 be addLoopStr;
attr a1 is Abelian means
for b1, b2 being Element of the carrier of a1 holds
b1 + b2 = b2 + b1;
end;
:: RLVECT_1:dfs 2
definiens
let a1 be addLoopStr;
To prove
a1 is Abelian
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
b1 + b2 = b2 + b1;
:: RLVECT_1:def 5
theorem
for b1 being addLoopStr holds
b1 is Abelian
iff
for b2, b3 being Element of the carrier of b1 holds
b2 + b3 = b3 + b2;
:: RLVECT_1:attrnot 3 => RLVECT_1:attr 3
definition
let a1 be addLoopStr;
attr a1 is add-associative means
for b1, b2, b3 being Element of the carrier of a1 holds
(b1 + b2) + b3 = b1 + (b2 + b3);
end;
:: RLVECT_1:dfs 3
definiens
let a1 be addLoopStr;
To prove
a1 is add-associative
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1 holds
(b1 + b2) + b3 = b1 + (b2 + b3);
:: RLVECT_1:def 6
theorem
for b1 being addLoopStr holds
b1 is add-associative
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4);
:: RLVECT_1:attrnot 4 => RLVECT_1:attr 4
definition
let a1 be addLoopStr;
attr a1 is right_zeroed means
for b1 being Element of the carrier of a1 holds
b1 + 0. a1 = b1;
end;
:: RLVECT_1:dfs 4
definiens
let a1 be addLoopStr;
To prove
a1 is right_zeroed
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
b1 + 0. a1 = b1;
:: RLVECT_1:def 7
theorem
for b1 being addLoopStr holds
b1 is right_zeroed
iff
for b2 being Element of the carrier of b1 holds
b2 + 0. b1 = b2;
:: RLVECT_1:attrnot 5 => RLVECT_1:attr 5
definition
let a1 be non empty RLSStruct;
attr a1 is RealLinearSpace-like means
(for b1 being Element of REAL
for b2, b3 being Element of the carrier of a1 holds
b1 * (b2 + b3) = (b1 * b2) + (b1 * b3)) &
(for b1, b2 being Element of REAL
for b3 being Element of the carrier of a1 holds
(b1 + b2) * b3 = (b1 * b3) + (b2 * b3)) &
(for b1, b2 being Element of REAL
for b3 being Element of the carrier of a1 holds
(b1 * b2) * b3 = b1 * (b2 * b3)) &
(for b1 being Element of the carrier of a1 holds
1 * b1 = b1);
end;
:: RLVECT_1:dfs 5
definiens
let a1 be non empty RLSStruct;
To prove
a1 is RealLinearSpace-like
it is sufficient to prove
thus (for b1 being Element of REAL
for b2, b3 being Element of the carrier of a1 holds
b1 * (b2 + b3) = (b1 * b2) + (b1 * b3)) &
(for b1, b2 being Element of REAL
for b3 being Element of the carrier of a1 holds
(b1 + b2) * b3 = (b1 * b3) + (b2 * b3)) &
(for b1, b2 being Element of REAL
for b3 being Element of the carrier of a1 holds
(b1 * b2) * b3 = b1 * (b2 * b3)) &
(for b1 being Element of the carrier of a1 holds
1 * b1 = b1);
:: RLVECT_1:def 9
theorem
for b1 being non empty RLSStruct holds
b1 is RealLinearSpace-like
iff
(for b2 being Element of REAL
for b3, b4 being Element of the carrier of b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4)) &
(for b2, b3 being Element of REAL
for b4 being Element of the carrier of b1 holds
(b2 + b3) * b4 = (b2 * b4) + (b3 * b4)) &
(for b2, b3 being Element of REAL
for b4 being Element of the carrier of b1 holds
(b2 * b3) * b4 = b2 * (b3 * b4)) &
(for b2 being Element of the carrier of b1 holds
1 * b2 = b2);
:: RLVECT_1:funcnot 2 => RLVECT_1:func 2
definition
func Trivial-RLSStruct -> strict RLSStruct equals
RLSStruct(#1,op0,op2,pr2(REAL,1)#);
end;
:: RLVECT_1:def 10
theorem
Trivial-RLSStruct = RLSStruct(#1,op0,op2,pr2(REAL,1)#);
:: RLVECT_1:funcreg 2
registration
cluster Trivial-RLSStruct -> non empty trivial strict;
end;
:: RLVECT_1:exreg 3
registration
cluster non empty strict right_complementable Abelian add-associative right_zeroed addLoopStr;
end;
:: RLVECT_1:exreg 4
registration
cluster non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
end;
:: RLVECT_1:modenot 2
definition
mode RealLinearSpace is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
end;
:: RLVECT_1:funcnot 3 => RLVECT_1:func 3
definition
let a1 be non empty Abelian addLoopStr;
let a2, a3 be Element of the carrier of a1;
redefine func a2 + a3 -> Element of the carrier of a1;
commutativity;
:: for a1 being non empty Abelian addLoopStr
:: for a2, a3 being Element of the carrier of a1 holds
:: a2 + a3 = a3 + a2;
end;
:: RLVECT_1:th 9
theorem
for b1 being right_complementable add-associative right_zeroed addLoopStr holds
b1 is right_add-cancelable;
:: RLVECT_1:th 10
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
b2 + 0. b1 = b2 & (0. b1) + b2 = b2;
:: RLVECT_1:funcnot 4 => ALGSTR_0:func 4
definition
let a1 be addLoopStr;
let a2 be Element of the carrier of a1;
assume a1 is add-associative & a1 is right_zeroed & a1 is right_complementable;
func - A2 -> Element of the carrier of a1 means
a2 + it = 0. a1;
end;
:: RLVECT_1:def 11
theorem
for b1 being non empty addLoopStr
for b2 being Element of the carrier of b1
st b1 is add-associative & b1 is right_zeroed & b1 is right_complementable
for b3 being Element of the carrier of b1 holds
b3 = - b2
iff
b2 + b3 = 0. b1;
:: RLVECT_1:funcnot 5 => RLVECT_1:func 4
definition
let a1 be addLoopStr;
let a2, a3 be Element of the carrier of a1;
redefine func A2 - A3 -> Element of the carrier of a1 equals
a2 + - a3;
end;
:: RLVECT_1:def 12
theorem
for b1 being addLoopStr
for b2, b3 being Element of the carrier of b1 holds
b2 - b3 = b2 + - b3;
:: RLVECT_1:th 16
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
b2 + - b2 = 0. b1 & (- b2) + b2 = 0. b1;
:: RLVECT_1:th 19
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1
st b2 + b3 = 0. b1
holds b2 = - b3;
:: RLVECT_1:th 20
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
b2 + b4 = b3;
:: RLVECT_1:th 21
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3, b4, b5 being Element of the carrier of b1
st (b2 + b4 = b2 + b5 or b4 + b2 = b5 + b2)
holds b4 = b5;
:: RLVECT_1:th 22
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1
st (b2 + b3 = b2 or b3 + b2 = b2)
holds b3 = 0. b1;
:: RLVECT_1:th 23
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of the carrier of b2
st (b1 = 0 or b3 = 0. b2)
holds b1 * b3 = 0. b2;
:: RLVECT_1:th 24
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of the carrier of b2
st b1 * b3 = 0. b2 & b1 <> 0
holds b3 = 0. b2;
:: RLVECT_1:th 25
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr holds
- 0. b1 = 0. b1;
:: RLVECT_1:th 26
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
b2 - 0. b1 = b2;
:: RLVECT_1:th 27
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
(0. b1) - b2 = - b2;
:: RLVECT_1:th 28
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
b2 - b2 = 0. b1;
:: RLVECT_1:th 29
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 holds
- b2 = (- 1) * b2;
:: RLVECT_1:th 30
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
- - b2 = b2;
:: RLVECT_1:th 31
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1
st - b2 = - b3
holds b2 = b3;
:: RLVECT_1:th 33
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
st b2 = - b2
holds b2 = 0. b1;
:: RLVECT_1:th 34
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
st b2 + b2 = 0. b1
holds b2 = 0. b1;
:: RLVECT_1:th 35
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1
st b2 - b3 = 0. b1
holds b2 = b3;
:: RLVECT_1:th 36
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
b3 - b4 = b2;
:: RLVECT_1:th 37
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3, b4 being Element of the carrier of b1
st b2 - b3 = b2 - b4
holds b3 = b4;
:: RLVECT_1:th 38
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of the carrier of b2 holds
b1 * - b3 = (- b1) * b3;
:: RLVECT_1:th 39
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of the carrier of b2 holds
b1 * - b3 = - (b1 * b3);
:: RLVECT_1:th 40
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of the carrier of b2 holds
(- b1) * - b3 = b1 * b3;
:: RLVECT_1:th 41
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 - (b3 + b4) = (b2 - b4) - b3;
:: RLVECT_1:th 42
theorem
for b1 being non empty add-associative addLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 + b3) - b4 = b2 + (b3 - b4);
:: RLVECT_1:th 43
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 holds
b2 - (b3 - b4) = (b2 - b3) + b4;
:: RLVECT_1:th 44
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
- (b2 + b3) = (- b3) - b2;
:: RLVECT_1:th 45
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
- (b2 + b3) = (- b3) + - b2;
:: RLVECT_1:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
(- b2) - b3 = (- b3) - b2;
:: RLVECT_1:th 47
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
- (b2 - b3) = b3 + - b2;
:: RLVECT_1:th 48
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of b2 holds
b1 * (b3 - b4) = (b1 * b3) - (b1 * b4);
:: RLVECT_1:th 49
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b4 being Element of the carrier of b3 holds
(b1 - b2) * b4 = (b1 * b4) - (b2 * b4);
:: RLVECT_1:th 50
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of b2
st b1 <> 0 & b1 * b3 = b1 * b4
holds b3 = b4;
:: RLVECT_1:th 51
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b4 being Element of the carrier of b3
st b4 <> 0. b3 & b1 * b4 = b2 * b4
holds b1 = b2;
:: RLVECT_1:funcnot 6 => RLVECT_1:func 5
definition
let a1 be non empty 1-sorted;
let a2, a3 be Element of the carrier of a1;
redefine func <*a2, a3*> -> FinSequence of the carrier of a1;
end;
:: RLVECT_1:funcnot 7 => RLVECT_1:func 6
definition
let a1 be non empty 1-sorted;
let a2, a3, a4 be Element of the carrier of a1;
redefine func <*a2, a3, a4*> -> FinSequence of the carrier of a1;
end;
:: RLVECT_1:funcnot 8 => RLVECT_1:func 7
definition
let a1 be non empty addLoopStr;
let a2 be FinSequence of the carrier of a1;
func Sum A2 -> Element of the carrier of a1 means
ex b1 being Function-like quasi_total Relation of NAT,the carrier of a1 st
it = b1 . len a2 &
b1 . 0 = 0. a1 &
(for b2 being Element of NAT
for b3 being Element of the carrier of a1
st b2 < len a2 & b3 = a2 . (b2 + 1)
holds b1 . (b2 + 1) = (b1 . b2) + b3);
end;
:: RLVECT_1:def 13
theorem
for b1 being non empty addLoopStr
for b2 being FinSequence of the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 = Sum b2
iff
ex b4 being Function-like quasi_total Relation of NAT,the carrier of b1 st
b3 = b4 . len b2 &
b4 . 0 = 0. b1 &
(for b5 being Element of NAT
for b6 being Element of the carrier of b1
st b5 < len b2 & b6 = b2 . (b5 + 1)
holds b4 . (b5 + 1) = (b4 . b5) + b6);
:: RLVECT_1:th 54
theorem
for b1 being non empty addLoopStr
for b2 being FinSequence of the carrier of b1
for b3, b4 being Element of NAT
st b3 in Seg b4 & len b2 = b4
holds b2 . b3 is Element of the carrier of b1;
:: RLVECT_1:th 55
theorem
for b1 being non empty addLoopStr
for b2, b3 being FinSequence of the carrier of b1
for b4 being Element of the carrier of b1
st len b2 = (len b3) + 1 & b3 = b2 | dom b3 & b4 = b2 . len b2
holds Sum b2 = (Sum b3) + b4;
:: RLVECT_1:th 56
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being FinSequence of the carrier of b2
st len b3 = len b4 &
(for b5 being Element of NAT
for b6 being Element of the carrier of b2
st b5 in dom b3 & b6 = b4 . b5
holds b3 . b5 = b1 * b6)
holds Sum b3 = b1 * Sum b4;
:: RLVECT_1:th 57
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
for b5 being Element of the carrier of b1
st b4 in dom b2 & b5 = b3 . b4
holds b2 . b4 = - b5)
holds Sum b2 = - Sum b3;
:: RLVECT_1:th 58
theorem
for b1 being non empty add-associative right_zeroed addLoopStr
for b2, b3 being FinSequence of the carrier of b1 holds
Sum (b2 ^ b3) = (Sum b2) + Sum b3;
:: RLVECT_1:th 59
theorem
for b1 being non empty Abelian add-associative right_zeroed addLoopStr
for b2, b3 being FinSequence of the carrier of b1
st proj2 b2 = proj2 b3 & b2 is one-to-one & b3 is one-to-one
holds Sum b2 = Sum b3;
:: RLVECT_1:th 60
theorem
for b1 being non empty addLoopStr holds
Sum <*> the carrier of b1 = 0. b1;
:: RLVECT_1:th 61
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
Sum <*b2*> = b2;
:: RLVECT_1:th 62
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
Sum <*b2,b3*> = b2 + b3;
:: RLVECT_1:th 63
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
Sum <*b2,b3,b4*> = (b2 + b3) + b4;
:: RLVECT_1:th 64
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of REAL holds
b2 * Sum <*> the carrier of b1 = 0. b1;
:: RLVECT_1:th 66
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 Element of the carrier of b1 holds
b2 * Sum <*b3,b4*> = (b2 * b3) + (b2 * b4);
:: RLVECT_1: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, b4, b5 being Element of the carrier of b1 holds
b2 * Sum <*b3,b4,b5*> = ((b2 * b3) + (b2 * b4)) + (b2 * b5);
:: RLVECT_1:th 68
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr holds
- Sum <*> the carrier of b1 = 0. b1;
:: RLVECT_1:th 69
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
- Sum <*b2*> = - b2;
:: RLVECT_1:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
- Sum <*b2,b3*> = (- b2) - b3;
:: RLVECT_1:th 71
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 holds
- Sum <*b2,b3,b4*> = ((- b2) - b3) - b4;
:: RLVECT_1:th 72
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
Sum <*b2,b3*> = Sum <*b3,b2*>;
:: RLVECT_1:th 73
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
Sum <*b2,b3*> = (Sum <*b2*>) + Sum <*b3*>;
:: RLVECT_1:th 74
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr holds
Sum <*0. b1,0. b1*> = 0. b1;
:: RLVECT_1:th 75
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
Sum <*0. b1,b2*> = b2 & Sum <*b2,0. b1*> = b2;
:: RLVECT_1:th 76
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
Sum <*b2,- b2*> = 0. b1 &
Sum <*- b2,b2*> = 0. b1;
:: RLVECT_1:th 77
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
Sum <*b2,- b3*> = b2 - b3;
:: RLVECT_1:th 78
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
Sum <*- b2,- b3*> = - (b3 + b2);
:: RLVECT_1: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 holds
Sum <*b2,b2*> = 2 * b2;
:: RLVECT_1:th 80
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 holds
Sum <*- b2,- b2*> = (- 2) * b2;
:: RLVECT_1:th 81
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
Sum <*b2,b3,b4*> = ((Sum <*b2*>) + Sum <*b3*>) + Sum <*b4*>;
:: RLVECT_1:th 82
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
Sum <*b2,b3,b4*> = (Sum <*b2,b3*>) + b4;
:: RLVECT_1:th 83
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 holds
Sum <*b3,b2,b4*> = (Sum <*b2,b4*>) + b3;
:: RLVECT_1:th 84
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 holds
Sum <*b3,b2,b4*> = (Sum <*b3,b4*>) + b2;
:: RLVECT_1:th 85
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 holds
Sum <*b3,b2,b4*> = Sum <*b3,b4,b2*>;
:: RLVECT_1:th 86
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 holds
Sum <*b3,b2,b4*> = Sum <*b2,b3,b4*>;
:: RLVECT_1:th 87
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 holds
Sum <*b3,b2,b4*> = Sum <*b2,b4,b3*>;
:: RLVECT_1:th 89
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 holds
Sum <*b3,b2,b4*> = Sum <*b4,b2,b3*>;
:: RLVECT_1:th 90
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr holds
Sum <*0. b1,0. b1,0. b1*> = 0. b1;
:: RLVECT_1:th 91
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
Sum <*0. b1,0. b1,b2*> = b2 &
Sum <*0. b1,b2,0. b1*> = b2 &
Sum <*b2,0. b1,0. b1*> = b2;
:: RLVECT_1:th 92
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
Sum <*0. b1,b2,b3*> = b2 + b3 &
Sum <*b2,b3,0. b1*> = b2 + b3 &
Sum <*b2,0. b1,b3*> = b2 + b3;
:: RLVECT_1:th 93
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 holds
Sum <*b2,b2,b2*> = 3 * b2;
:: RLVECT_1:th 94
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1
st len b2 = 0
holds Sum b2 = 0. b1;
:: RLVECT_1:th 95
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1
st len b2 = 1
holds Sum b2 = b2 . 1;
:: RLVECT_1:th 96
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1
for b3, b4 being Element of the carrier of b1
st len b2 = 2 & b3 = b2 . 1 & b4 = b2 . 2
holds Sum b2 = b3 + b4;
:: RLVECT_1:th 97
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1
for b3, b4, b5 being Element of the carrier of b1
st len b2 = 3 & b3 = b2 . 1 & b4 = b2 . 2 & b5 = b2 . 3
holds Sum b2 = (b3 + b4) + b5;
:: RLVECT_1:attrnot 6 => RLVECT_1:attr 6
definition
let a1 be non empty ZeroStr;
let a2 be Element of the carrier of a1;
attr a2 is non-zero means
a2 <> 0. a1;
end;
:: RLVECT_1:dfs 10
definiens
let a1 be non empty ZeroStr;
let a2 be Element of the carrier of a1;
To prove
a2 is non-zero
it is sufficient to prove
thus a2 <> 0. a1;
:: RLVECT_1:def 14
theorem
for b1 being non empty ZeroStr
for b2 being Element of the carrier of b1 holds
b2 is non-zero(b1)
iff
b2 <> 0. b1;
:: RLVECT_1:attrnot 7 => RLVECT_1:attr 7
definition
let a1 be non empty addLoopStr;
attr a1 is zeroed means
for b1 being Element of the carrier of a1 holds
b1 + 0. a1 = b1 & (0. a1) + b1 = b1;
end;
:: RLVECT_1:dfs 11
definiens
let a1 be non empty addLoopStr;
To prove
a1 is zeroed
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
b1 + 0. a1 = b1 & (0. a1) + b1 = b1;
:: RLVECT_1:def 15
theorem
for b1 being non empty addLoopStr holds
b1 is zeroed
iff
for b2 being Element of the carrier of b1 holds
b2 + 0. b1 = b2 & (0. b1) + b2 = b2;
:: RLVECT_1:condreg 1
registration
cluster non empty zeroed -> right_zeroed (addLoopStr);
end;
:: RLVECT_1:condreg 2
registration
cluster non empty Abelian right_zeroed -> zeroed (addLoopStr);
end;
:: RLVECT_1:condreg 3
registration
cluster non empty right_complementable Abelian -> left_complementable (addLoopStr);
end;