Article FVSUM_1, MML version 4.99.1005
:: FVSUM_1:th 2
theorem
for b1 being non empty Abelian addLoopStr holds
the addF of b1 is commutative(the carrier of b1);
:: FVSUM_1:th 3
theorem
for b1 being non empty add-associative addLoopStr holds
the addF of b1 is associative(the carrier of b1);
:: FVSUM_1:th 4
theorem
for b1 being non empty commutative multMagma holds
the multF of b1 is commutative(the carrier of b1);
:: FVSUM_1:funcreg 1
registration
let a1 be non empty Abelian addLoopStr;
cluster the addF of a1 -> Function-like quasi_total commutative;
end;
:: FVSUM_1:funcreg 2
registration
let a1 be non empty add-associative addLoopStr;
cluster the addF of a1 -> Function-like quasi_total associative;
end;
:: FVSUM_1:funcreg 3
registration
let a1 be non empty commutative multMagma;
cluster the multF of a1 -> Function-like quasi_total commutative;
end;
:: FVSUM_1:th 6
theorem
for b1 being non empty commutative left_unital multLoopStr holds
1. b1 is_a_unity_wrt the multF of b1;
:: FVSUM_1:th 7
theorem
for b1 being non empty commutative left_unital multLoopStr holds
the_unity_wrt the multF of b1 = 1. b1;
:: FVSUM_1:th 8
theorem
for b1 being non empty left_zeroed right_zeroed addLoopStr holds
0. b1 is_a_unity_wrt the addF of b1;
:: FVSUM_1:th 9
theorem
for b1 being non empty left_zeroed right_zeroed addLoopStr holds
the_unity_wrt the addF of b1 = 0. b1;
:: FVSUM_1:th 10
theorem
for b1 being non empty left_zeroed right_zeroed addLoopStr holds
the addF of b1 is having_a_unity(the carrier of b1);
:: FVSUM_1:th 11
theorem
for b1 being non empty commutative left_unital multLoopStr holds
the multF of b1 is having_a_unity(the carrier of b1);
:: FVSUM_1:th 12
theorem
for b1 being non empty distributive doubleLoopStr holds
the multF of b1 is_distributive_wrt the addF of b1;
:: FVSUM_1:funcnot 1 => FVSUM_1:func 1
definition
let a1 be non empty multMagma;
let a2 be Element of the carrier of a1;
func A2 multfield -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 equals
(the multF of a1) [;](a2,id the carrier of a1);
end;
:: FVSUM_1:def 1
theorem
for b1 being non empty multMagma
for b2 being Element of the carrier of b1 holds
b2 multfield = (the multF of b1) [;](b2,id the carrier of b1);
:: FVSUM_1:funcnot 2 => FVSUM_1:func 2
definition
let a1 be non empty addLoopStr;
func diffield A1 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of a1 equals
(the addF of a1) *(id the carrier of a1,comp a1);
end;
:: FVSUM_1:def 2
theorem
for b1 being non empty addLoopStr holds
diffield b1 = (the addF of b1) *(id the carrier of b1,comp b1);
:: FVSUM_1:th 14
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of the carrier of b1 holds
(diffield b1) .(b2,b3) = b2 - b3;
:: FVSUM_1:th 15
theorem
for b1 being non empty distributive doubleLoopStr
for b2 being Element of the carrier of b1 holds
b2 multfield is_distributive_wrt the addF of b1;
:: FVSUM_1:th 16
theorem
for b1 being non empty right_complementable left_zeroed add-associative right_zeroed addLoopStr holds
comp b1 is_an_inverseOp_wrt the addF of b1;
:: FVSUM_1:th 17
theorem
for b1 being non empty right_complementable left_zeroed add-associative right_zeroed addLoopStr holds
the addF of b1 is having_an_inverseOp(the carrier of b1);
:: FVSUM_1:th 18
theorem
for b1 being non empty right_complementable left_zeroed add-associative right_zeroed addLoopStr holds
the_inverseOp_wrt the addF of b1 = comp b1;
:: FVSUM_1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr holds
comp b1 is_distributive_wrt the addF of b1;
:: FVSUM_1:funcnot 3 => FVSUM_1:func 3
definition
let a1 be non empty addLoopStr;
let a2, a3 be FinSequence of the carrier of a1;
func A2 + A3 -> FinSequence of the carrier of a1 equals
(the addF of a1) .:(a2,a3);
end;
:: FVSUM_1:def 3
theorem
for b1 being non empty addLoopStr
for b2, b3 being FinSequence of the carrier of b1 holds
b2 + b3 = (the addF of b1) .:(b2,b3);
:: FVSUM_1:th 21
theorem
for b1 being non empty addLoopStr
for b2, b3 being FinSequence of the carrier of b1
for b4, b5 being Element of the carrier of b1
for b6 being Element of NAT
st b6 in dom (b2 + b3) & b4 = b2 . b6 & b5 = b3 . b6
holds (b2 + b3) . b6 = b4 + b5;
:: FVSUM_1:funcnot 4 => FVSUM_1:func 4
definition
let a1 be Element of NAT;
let a2 be non empty addLoopStr;
let a3, a4 be Element of a1 -tuples_on the carrier of a2;
redefine func a3 + a4 -> Element of a1 -tuples_on the carrier of a2;
end;
:: FVSUM_1:th 22
theorem
for b1, b2 being Element of NAT
for b3 being non empty addLoopStr
for b4, b5 being Element of the carrier of b3
for b6, b7 being Element of b1 -tuples_on the carrier of b3
st b2 in Seg b1 & b4 = b6 . b2 & b5 = b7 . b2
holds (b6 + b7) . b2 = b4 + b5;
:: FVSUM_1:th 24
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of the carrier of b1 holds
<*b2*> + <*b3*> = <*b2 + b3*>;
:: FVSUM_1:th 25
theorem
for b1 being Element of NAT
for b2 being non empty addLoopStr
for b3, b4 being Element of the carrier of b2 holds
(b1 |-> b3) + (b1 |-> b4) = b1 |-> (b3 + b4);
:: FVSUM_1:th 28
theorem
for b1 being Element of NAT
for b2 being non empty left_zeroed Abelian right_zeroed addLoopStr
for b3 being Element of b1 -tuples_on the carrier of b2 holds
b3 + (b1 |-> 0. b2) = b3 & b3 = (b1 |-> 0. b2) + b3;
:: FVSUM_1:funcnot 5 => FVSUM_1:func 5
definition
let a1 be non empty addLoopStr;
let a2 be FinSequence of the carrier of a1;
func - A2 -> FinSequence of the carrier of a1 equals
(comp a1) * a2;
end;
:: FVSUM_1:def 4
theorem
for b1 being non empty addLoopStr
for b2 being FinSequence of the carrier of b1 holds
- b2 = (comp b1) * b2;
:: FVSUM_1:th 30
theorem
for b1 being Element of NAT
for b2 being non empty addLoopStr
for b3 being Element of the carrier of b2
for b4 being FinSequence of the carrier of b2
st b1 in dom - b4 & b3 = b4 . b1
holds (- b4) . b1 = - b3;
:: FVSUM_1:funcnot 6 => FVSUM_1:func 6
definition
let a1 be Element of NAT;
let a2 be non empty addLoopStr;
let a3 be Element of a1 -tuples_on the carrier of a2;
redefine func - a3 -> Element of a1 -tuples_on the carrier of a2;
end;
:: FVSUM_1:th 31
theorem
for b1, b2 being Element of NAT
for b3 being non empty addLoopStr
for b4 being Element of the carrier of b3
for b5 being Element of b2 -tuples_on the carrier of b3
st b1 in Seg b2 & b4 = b5 . b1
holds (- b5) . b1 = - b4;
:: FVSUM_1:th 33
theorem
for b1 being non empty addLoopStr
for b2 being Element of the carrier of b1 holds
- <*b2*> = <*- b2*>;
:: FVSUM_1:th 34
theorem
for b1 being Element of NAT
for b2 being non empty addLoopStr
for b3 being Element of the carrier of b2 holds
- (b1 |-> b3) = b1 |-> - b3;
:: FVSUM_1:th 35
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3 being Element of b1 -tuples_on the carrier of b2 holds
b3 + - b3 = b1 |-> 0. b2 & (- b3) + b3 = b1 |-> 0. b2;
:: FVSUM_1:th 36
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable left_zeroed add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2
st b3 + b4 = b1 |-> 0. b2
holds b3 = - b4 & b4 = - b3;
:: FVSUM_1:th 37
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable left_zeroed add-associative right_zeroed addLoopStr
for b3 being Element of b1 -tuples_on the carrier of b2 holds
- - b3 = b3;
:: FVSUM_1:th 38
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable left_zeroed add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2
st - b3 = - b4
holds b3 = b4;
:: FVSUM_1:th 39
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4, b5 being Element of b1 -tuples_on the carrier of b2
st (b4 + b3 = b5 + b3 or b4 + b3 = b3 + b5)
holds b4 = b5;
:: FVSUM_1:th 40
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2 holds
- (b3 + b4) = (- b3) + - b4;
:: FVSUM_1:funcnot 7 => FVSUM_1:func 7
definition
let a1 be non empty addLoopStr;
let a2, a3 be FinSequence of the carrier of a1;
func A2 - A3 -> FinSequence of the carrier of a1 equals
(diffield a1) .:(a2,a3);
end;
:: FVSUM_1:def 5
theorem
for b1 being non empty addLoopStr
for b2, b3 being FinSequence of the carrier of b1 holds
b2 - b3 = (diffield b1) .:(b2,b3);
:: FVSUM_1:th 42
theorem
for b1 being Element of NAT
for b2 being non empty addLoopStr
for b3, b4 being Element of the carrier of b2
for b5, b6 being FinSequence of the carrier of b2
st b1 in dom (b5 - b6) & b3 = b5 . b1 & b4 = b6 . b1
holds (b5 - b6) . b1 = b3 - b4;
:: FVSUM_1:funcnot 8 => FVSUM_1:func 8
definition
let a1 be Element of NAT;
let a2 be non empty addLoopStr;
let a3, a4 be Element of a1 -tuples_on the carrier of a2;
redefine func a3 - a4 -> Element of a1 -tuples_on the carrier of a2;
end;
:: FVSUM_1:th 43
theorem
for b1, b2 being Element of NAT
for b3 being non empty addLoopStr
for b4, b5 being Element of the carrier of b3
for b6, b7 being Element of b2 -tuples_on the carrier of b3
st b1 in Seg b2 & b4 = b6 . b1 & b5 = b7 . b1
holds (b6 - b7) . b1 = b4 - b5;
:: FVSUM_1:th 45
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of the carrier of b1 holds
<*b2*> - <*b3*> = <*b2 - b3*>;
:: FVSUM_1:th 46
theorem
for b1 being Element of NAT
for b2 being non empty addLoopStr
for b3, b4 being Element of the carrier of b2 holds
(b1 |-> b3) - (b1 |-> b4) = b1 |-> (b3 - b4);
:: FVSUM_1:th 48
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable left_zeroed add-associative right_zeroed addLoopStr
for b3 being Element of b1 -tuples_on the carrier of b2 holds
b3 - (b1 |-> 0. b2) = b3;
:: FVSUM_1:th 49
theorem
for b1 being Element of NAT
for b2 being non empty left_zeroed Abelian right_zeroed addLoopStr
for b3 being Element of b1 -tuples_on the carrier of b2 holds
(b1 |-> 0. b2) - b3 = - b3;
:: FVSUM_1:th 50
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable left_zeroed add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2 holds
b3 - - b4 = b3 + b4;
:: FVSUM_1:th 51
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2 holds
- (b3 - b4) = b4 - b3;
:: FVSUM_1:th 52
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2 holds
- (b3 - b4) = (- b3) + b4;
:: FVSUM_1:th 53
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3 being Element of b1 -tuples_on the carrier of b2 holds
b3 - b3 = b1 |-> 0. b2;
:: FVSUM_1:th 54
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2
st b3 - b4 = b1 |-> 0. b2
holds b3 = b4;
:: FVSUM_1:th 55
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4, b5 being Element of b1 -tuples_on the carrier of b2 holds
(b3 - b4) - b5 = b3 - (b4 + b5);
:: FVSUM_1:th 56
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4, b5 being Element of b1 -tuples_on the carrier of b2 holds
b3 + (b4 - b5) = (b3 + b4) - b5;
:: FVSUM_1:th 57
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4, b5 being Element of b1 -tuples_on the carrier of b2 holds
b3 - (b4 - b5) = (b3 - b4) + b5;
:: FVSUM_1:th 58
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2 holds
b3 = (b3 + b4) - b4;
:: FVSUM_1:th 59
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2 holds
b3 = (b3 - b4) + b4;
:: FVSUM_1:th 60
theorem
for b1 being non empty multMagma
for b2, b3 being Element of the carrier of b1 holds
((the multF of b1) [;](b2,id the carrier of b1)) . b3 = b2 * b3;
:: FVSUM_1:th 61
theorem
for b1 being non empty multMagma
for b2, b3 being Element of the carrier of b1 holds
b2 multfield . b3 = b2 * b3;
:: FVSUM_1:funcnot 9 => FVSUM_1:func 9
definition
let a1 be non empty multMagma;
let a2 be FinSequence of the carrier of a1;
let a3 be Element of the carrier of a1;
func A3 * A2 -> FinSequence of the carrier of a1 equals
a3 multfield * a2;
end;
:: FVSUM_1:def 6
theorem
for b1 being non empty multMagma
for b2 being FinSequence of the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 * b2 = b3 multfield * b2;
:: FVSUM_1:th 62
theorem
for b1 being Element of NAT
for b2 being non empty multMagma
for b3, b4 being Element of the carrier of b2
for b5 being FinSequence of the carrier of b2
st b1 in dom (b3 * b5) & b4 = b5 . b1
holds (b3 * b5) . b1 = b3 * b4;
:: FVSUM_1:funcnot 10 => FVSUM_1:func 10
definition
let a1 be Element of NAT;
let a2 be non empty multMagma;
let a3 be Element of a1 -tuples_on the carrier of a2;
let a4 be Element of the carrier of a2;
redefine func a4 * a3 -> Element of a1 -tuples_on the carrier of a2;
end;
:: FVSUM_1:th 63
theorem
for b1, b2 being Element of NAT
for b3 being non empty multMagma
for b4, b5 being Element of the carrier of b3
for b6 being Element of b2 -tuples_on the carrier of b3
st b1 in Seg b2 & b4 = b6 . b1
holds (b5 * b6) . b1 = b5 * b4;
:: FVSUM_1:th 65
theorem
for b1 being non empty multMagma
for b2, b3 being Element of the carrier of b1 holds
b2 * <*b3*> = <*b2 * b3*>;
:: FVSUM_1:th 66
theorem
for b1 being Element of NAT
for b2 being non empty multMagma
for b3, b4 being Element of the carrier of b2 holds
b3 * (b1 |-> b4) = b1 |-> (b3 * b4);
:: FVSUM_1:th 67
theorem
for b1 being Element of NAT
for b2 being non empty associative multMagma
for b3, b4 being Element of the carrier of b2
for b5 being Element of b1 -tuples_on the carrier of b2 holds
(b3 * b4) * b5 = b3 * (b4 * b5);
:: FVSUM_1:th 68
theorem
for b1 being Element of NAT
for b2 being non empty distributive doubleLoopStr
for b3, b4 being Element of the carrier of b2
for b5 being Element of b1 -tuples_on the carrier of b2 holds
(b3 + b4) * b5 = (b3 * b5) + (b4 * b5);
:: FVSUM_1:th 69
theorem
for b1 being Element of NAT
for b2 being non empty distributive doubleLoopStr
for b3 being Element of the carrier of b2
for b4, b5 being Element of b1 -tuples_on the carrier of b2 holds
b3 * (b4 + b5) = (b3 * b4) + (b3 * b5);
:: FVSUM_1:th 70
theorem
for b1 being Element of NAT
for b2 being non empty commutative distributive left_unital doubleLoopStr
for b3 being Element of b1 -tuples_on the carrier of b2 holds
(1. b2) * b3 = b3;
:: FVSUM_1:th 71
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr
for b3 being Element of b1 -tuples_on the carrier of b2 holds
(0. b2) * b3 = b1 |-> 0. b2;
:: FVSUM_1:th 72
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable commutative add-associative right_zeroed distributive left_unital doubleLoopStr
for b3 being Element of b1 -tuples_on the carrier of b2 holds
(- 1. b2) * b3 = - b3;
:: FVSUM_1:funcnot 11 => FVSUM_1:func 11
definition
let a1 be non empty multMagma;
let a2, a3 be FinSequence of the carrier of a1;
func mlt(A2,A3) -> FinSequence of the carrier of a1 equals
(the multF of a1) .:(a2,a3);
end;
:: FVSUM_1:def 7
theorem
for b1 being non empty multMagma
for b2, b3 being FinSequence of the carrier of b1 holds
mlt(b2,b3) = (the multF of b1) .:(b2,b3);
:: FVSUM_1:th 73
theorem
for b1 being Element of NAT
for b2 being non empty multMagma
for b3, b4 being Element of the carrier of b2
for b5, b6 being FinSequence of the carrier of b2
st b1 in dom mlt(b5,b6) & b3 = b5 . b1 & b4 = b6 . b1
holds (mlt(b5,b6)) . b1 = b3 * b4;
:: FVSUM_1:funcnot 12 => FVSUM_1:func 12
definition
let a1 be Element of NAT;
let a2 be non empty multMagma;
let a3, a4 be Element of a1 -tuples_on the carrier of a2;
redefine func mlt(a3,a4) -> Element of a1 -tuples_on the carrier of a2;
end;
:: FVSUM_1:th 74
theorem
for b1, b2 being Element of NAT
for b3 being non empty multMagma
for b4, b5 being Element of the carrier of b3
for b6, b7 being Element of b2 -tuples_on the carrier of b3
st b1 in Seg b2 & b4 = b6 . b1 & b5 = b7 . b1
holds (mlt(b6,b7)) . b1 = b4 * b5;
:: FVSUM_1:th 76
theorem
for b1 being non empty multMagma
for b2, b3 being Element of the carrier of b1 holds
mlt(<*b2*>,<*b3*>) = <*b2 * b3*>;
:: FVSUM_1:th 77
theorem
for b1 being Element of NAT
for b2 being non empty commutative multMagma
for b3, b4 being Element of b1 -tuples_on the carrier of b2 holds
mlt(b3,b4) = mlt(b4,b3);
:: FVSUM_1:th 78
theorem
for b1 being non empty commutative multMagma
for b2, b3 being FinSequence of the carrier of b1 holds
mlt(b2,b3) = mlt(b3,b2);
:: FVSUM_1:th 79
theorem
for b1 being Element of NAT
for b2 being non empty associative multMagma
for b3, b4, b5 being Element of b1 -tuples_on the carrier of b2 holds
mlt(b3,mlt(b4,b5)) = mlt(mlt(b3,b4),b5);
:: FVSUM_1:th 80
theorem
for b1 being Element of NAT
for b2 being non empty associative commutative multMagma
for b3 being Element of the carrier of b2
for b4 being Element of b1 -tuples_on the carrier of b2 holds
mlt(b1 |-> b3,b4) = b3 * b4 & mlt(b4,b1 |-> b3) = b3 * b4;
:: FVSUM_1:th 81
theorem
for b1 being Element of NAT
for b2 being non empty associative commutative multMagma
for b3, b4 being Element of the carrier of b2 holds
mlt(b1 |-> b3,b1 |-> b4) = b1 |-> (b3 * b4);
:: FVSUM_1:th 82
theorem
for b1 being Element of NAT
for b2 being non empty associative multMagma
for b3 being Element of the carrier of b2
for b4, b5 being Element of b1 -tuples_on the carrier of b2 holds
b3 * mlt(b4,b5) = mlt(b3 * b4,b5);
:: FVSUM_1:th 83
theorem
for b1 being Element of NAT
for b2 being non empty associative commutative multMagma
for b3 being Element of the carrier of b2
for b4, b5 being Element of b1 -tuples_on the carrier of b2 holds
b3 * mlt(b4,b5) = mlt(b3 * b4,b5) &
b3 * mlt(b4,b5) = mlt(b4,b3 * b5);
:: FVSUM_1:th 84
theorem
for b1 being Element of NAT
for b2 being non empty associative commutative multMagma
for b3 being Element of the carrier of b2
for b4 being Element of b1 -tuples_on the carrier of b2 holds
b3 * b4 = mlt(b1 |-> b3,b4);
:: FVSUM_1:condreg 1
registration
cluster non empty Abelian right_zeroed -> left_zeroed (addLoopStr);
end;
:: FVSUM_1:funcnot 13 => 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 equals
(the addF of a1) "**" a2;
end;
:: FVSUM_1:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1 holds
Sum b2 = (the addF of b1) "**" b2;
:: FVSUM_1:th 87
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1
for b3 being FinSequence of the carrier of b1 holds
Sum (b3 ^ <*b2*>) = (Sum b3) + b2;
:: FVSUM_1:th 89
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1
for b3 being FinSequence of the carrier of b1 holds
Sum (<*b2*> ^ b3) = b2 + Sum b3;
:: FVSUM_1:th 92
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed distributive doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being FinSequence of the carrier of b1 holds
Sum (b2 * b3) = b2 * Sum b3;
:: FVSUM_1:th 93
theorem
for b1 being non empty addLoopStr
for b2 being Element of 0 -tuples_on the carrier of b1 holds
Sum b2 = 0. b1;
:: FVSUM_1:th 94
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1 holds
Sum - b2 = - Sum b2;
:: FVSUM_1:th 95
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2 holds
Sum (b3 + b4) = (Sum b3) + Sum b4;
:: FVSUM_1:th 96
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2 holds
Sum (b3 - b4) = (Sum b3) - Sum b4;
:: FVSUM_1:th 102
theorem
for b1 being non empty associative commutative well-unital doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being FinSequence of the carrier of b1 holds
Product (<*b2*> ^ b3) = b2 * Product b3;
:: FVSUM_1:th 104
theorem
for b1 being non empty associative commutative well-unital doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
Product <*b2,b3,b4*> = (b2 * b3) * b4;
:: FVSUM_1:th 105
theorem
for b1 being non empty associative commutative well-unital doubleLoopStr
for b2 being Element of 0 -tuples_on the carrier of b1 holds
Product b2 = 1. b1;
:: FVSUM_1:th 106
theorem
for b1 being Element of NAT
for b2 being non empty associative commutative well-unital doubleLoopStr holds
Product (b1 |-> 1_ b2) = 1_ b2;
:: FVSUM_1:th 107
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being FinSequence of the carrier of b1 holds
ex b3 being Element of NAT st
b3 in dom b2 & b2 . b3 = 0. b1
iff
Product b2 = 0. b1;
:: FVSUM_1:th 108
theorem
for b1, b2 being Element of NAT
for b3 being non empty associative commutative well-unital doubleLoopStr
for b4 being Element of the carrier of b3 holds
Product ((b1 + b2) |-> b4) = (Product (b1 |-> b4)) * Product (b2 |-> b4);
:: FVSUM_1:th 109
theorem
for b1, b2 being Element of NAT
for b3 being non empty associative commutative well-unital doubleLoopStr
for b4 being Element of the carrier of b3 holds
Product ((b1 * b2) |-> b4) = Product (b2 |-> Product (b1 |-> b4));
:: FVSUM_1:th 110
theorem
for b1 being Element of NAT
for b2 being non empty associative commutative well-unital doubleLoopStr
for b3, b4 being Element of the carrier of b2 holds
Product (b1 |-> (b3 * b4)) = (Product (b1 |-> b3)) * Product (b1 |-> b4);
:: FVSUM_1:th 111
theorem
for b1 being Element of NAT
for b2 being non empty associative commutative well-unital doubleLoopStr
for b3, b4 being Element of b1 -tuples_on the carrier of b2 holds
Product mlt(b3,b4) = (Product b3) * Product b4;
:: FVSUM_1:th 112
theorem
for b1 being Element of NAT
for b2 being non empty associative commutative well-unital doubleLoopStr
for b3 being Element of the carrier of b2
for b4 being Element of b1 -tuples_on the carrier of b2 holds
Product (b3 * b4) = (Product (b1 |-> b3)) * Product b4;
:: FVSUM_1:funcnot 14 => FVSUM_1:func 13
definition
let a1 be non empty doubleLoopStr;
let a2, a3 be FinSequence of the carrier of a1;
func A2 "*" A3 -> Element of the carrier of a1 equals
Sum mlt(a2,a3);
end;
:: FVSUM_1:def 10
theorem
for b1 being non empty doubleLoopStr
for b2, b3 being FinSequence of the carrier of b1 holds
b2 "*" b3 = Sum mlt(b2,b3);
:: FVSUM_1:th 113
theorem
for b1 being non empty right_complementable associative commutative Abelian add-associative right_zeroed left_unital doubleLoopStr
for b2, b3 being Element of the carrier of b1 holds
<*b2*> "*" <*b3*> = b2 * b3;
:: FVSUM_1:th 114
theorem
for b1 being non empty right_complementable associative commutative Abelian add-associative right_zeroed left_unital doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
<*b2,b3*> "*" <*b4,b5*> = (b2 * b4) + (b3 * b5);
:: FVSUM_1:th 115
theorem
for b1 being non empty associative commutative well-unital doubleLoopStr
for b2, b3 being FinSequence of the carrier of b1 holds
b2 "*" b3 = b3 "*" b2;