Article RANKNULL, MML version 4.99.1005
:: RANKNULL:th 1
theorem
for b1, b2 being Relation-like Function-like set
st b2 is one-to-one & b1 | proj2 b2 is one-to-one & proj2 b2 c= proj1 b1
holds b2 * b1 is one-to-one;
:: RANKNULL:th 2
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set
st b2 c= b3 & b1 | b3 is one-to-one
holds b1 | b2 is one-to-one;
:: RANKNULL:th 3
theorem
for b1 being 1-sorted
for b2, b3 being Element of bool the carrier of b1 holds
b2 meets b3
iff
ex b4 being Element of the carrier of b1 st
b4 in b2 & b4 in b3;
:: RANKNULL:exreg 1
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over a1;
cluster finite Basis of a2;
end;
:: RANKNULL:exreg 2
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
cluster Relation-like Function-like non empty quasi_total linear total Relation of the carrier of a2,the carrier of a3;
end;
:: RANKNULL:th 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
st [#] b2 is finite
holds b2 is finite-dimensional(b1);
:: RANKNULL:th 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1
st Card [#] b2 = 1
holds dim b2 = {};
:: RANKNULL:th 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
st Card [#] b2 = 2
holds dim b2 = 1;
:: RANKNULL:modenot 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
mode linear-transformation of a2,a3 is Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
end;
:: RANKNULL:th 7
theorem
for b1, b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
dom b3 = [#] b1 & proj2 b3 c= [#] b2;
:: RANKNULL:th 8
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5, b6 being Element of the carrier of b3 holds
(b4 . b5) - (b4 . b6) = b4 . (b5 - b6);
:: RANKNULL:th 9
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3 holds
b4 . 0. b2 = 0. b3;
:: RANKNULL:funcnot 1 => RANKNULL:func 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
func ker A4 -> strict Subspace of a2 means
[#] it = {b1 where b1 is Element of the carrier of a2: a4 . b1 = 0. a3};
end;
:: RANKNULL:def 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b5 being strict Subspace of b2 holds
b5 = ker b4
iff
[#] b5 = {b6 where b6 is Element of the carrier of b2: b4 . b6 = 0. b3};
:: RANKNULL:th 10
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b5 being Element of the carrier of b2 holds
b5 in ker b4
iff
b4 . b5 = 0. b3;
:: RANKNULL:funcnot 2 => RANKNULL:func 2
definition
let a1, a2 be non empty 1-sorted;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be Element of bool the carrier of a1;
redefine func a3 .: a4 -> Element of bool the carrier of a2;
end;
:: RANKNULL:funcnot 3 => RANKNULL:func 3
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
func im A4 -> strict Subspace of a3 means
[#] it = a4 .: [#] a2;
end;
:: RANKNULL:def 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b5 being strict Subspace of b3 holds
b5 = im b4
iff
[#] b5 = b4 .: [#] b2;
:: RANKNULL:th 11
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2 holds
0. b3 in ker b4;
:: RANKNULL:th 12
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3 holds
b4 .: b5 is Element of bool the carrier of im b4;
:: RANKNULL:th 13
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of the carrier of b2 holds
b5 in im b4
iff
ex b6 being Element of the carrier of b3 st
b5 = b4 . b6;
:: RANKNULL:th 14
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b5 being Element of the carrier of ker b4 holds
b4 . b5 = 0. b3;
:: RANKNULL:th 15
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
st b4 is one-to-one
holds ker b4 = (0). b3;
:: RANKNULL:th 16
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1 holds
dim (0). b2 = {};
:: RANKNULL:th 17
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5, b6 being Element of the carrier of b3
st b4 . b5 = b4 . b6
holds b5 - b6 in ker b4;
:: RANKNULL:th 18
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4, b5 being Element of the carrier of b2
st b4 - b5 in Lin b3
holds b4 in Lin (b3 \/ {b5});
:: RANKNULL:th 19
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Element of bool the carrier of b2
st b2 is Subspace of b3
holds b4 is Element of bool the carrier of b3;
:: RANKNULL:th 20
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
st b3 is linearly-independent(b1, b2)
holds b3 is Basis of Lin b3;
:: RANKNULL:th 21
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4 being Element of the carrier of b2
st b4 in Lin b3 & not b4 in b3
holds b3 \/ {b4} is linearly-dependent(b1, b2);
:: RANKNULL:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Basis of b3
st b5 is Basis of ker b4 & b5 c= b6
holds b4 | (b6 \ b5) is one-to-one;
:: RANKNULL:th 23
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4 being Linear_Combination of b3
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1 holds
b4 +*(b5,b6) is Linear_Combination of b3 \/ {b5};
:: RANKNULL:funcnot 4 => RANKNULL:func 4
definition
let a1 be 1-sorted;
let a2 be Element of bool the carrier of a1;
func A1 \ A2 -> Element of bool the carrier of a1 equals
([#] a1) \ a2;
end;
:: RANKNULL:def 3
theorem
for b1 being 1-sorted
for b2 being Element of bool the carrier of b1 holds
b1 \ b2 = ([#] b1) \ b2;
:: RANKNULL:funcnot 5 => RANKNULL:func 5
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a3 be Linear_Combination of a2;
let a4 be Element of bool the carrier of a2;
redefine func a3 .: a4 -> Element of bool the carrier of a1;
end;
:: RANKNULL:exreg 3
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
cluster linearly-dependent Element of bool the carrier of a2;
end;
:: RANKNULL:funcnot 6 => RANKNULL:func 6
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a3 be Linear_Combination of a2;
let a4 be Element of bool the carrier of a2;
func A3 ! A4 -> Linear_Combination of a4 equals
(a3 | a4) +* ((a2 \ a4) --> 0. a1);
end;
:: RANKNULL:def 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of bool the carrier of b2 holds
b3 ! b4 = (b3 | b4) +* ((b2 \ b4) --> 0. b1);
:: RANKNULL:th 24
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2 holds
b3 = b3 ! Carrier b3;
:: RANKNULL:th 25
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of bool the carrier of b2
for b5 being Element of the carrier of b2
st b5 in b4
holds (b3 ! b4) . b5 = b3 . b5;
:: RANKNULL:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of bool the carrier of b2
for b5 being Element of the carrier of b2
st not b5 in b4
holds (b3 ! b4) . b5 = 0. b1;
:: RANKNULL:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
for b5 being Linear_Combination of b4
st b3 c= b4
holds b5 = (b5 ! b3) + (b5 ! (b4 \ b3));
:: RANKNULL:funcreg 1
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a3 be Linear_Combination of a2;
let a4 be Element of bool the carrier of a2;
cluster a3 .: a4 -> finite;
end;
:: RANKNULL:funcnot 7 => RANKNULL:func 7
definition
let a1, a2 be non empty 1-sorted;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be Element of bool the carrier of a2;
redefine func a3 " a4 -> Element of bool the carrier of a1;
end;
:: RANKNULL:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of bool the carrier of b2
st b4 misses Carrier b3
holds b3 .: b4 c= {0. b1};
:: RANKNULL:funcnot 8 => RANKNULL:func 8
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Linear_Combination of a2;
let a5 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
func A5 @ A4 -> Linear_Combination of a3 means
for b1 being Element of the carrier of a3 holds
it . b1 = Sum (a4 .: (a5 " {b1}));
end;
:: RANKNULL:def 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Linear_Combination of b2
for b5 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b6 being Linear_Combination of b3 holds
b6 = b5 @ b4
iff
for b7 being Element of the carrier of b3 holds
b6 . b7 = Sum (b4 .: (b5 " {b7}));
:: RANKNULL:th 29
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Linear_Combination of b3 holds
b4 @ b5 is Linear_Combination of b4 .: Carrier b5;
:: RANKNULL:th 30
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Linear_Combination of b3 holds
Carrier (b4 @ b5) c= b4 .: Carrier b5;
:: RANKNULL:th 31
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Linear_Combination of b2
st Carrier b3 misses Carrier b4
holds Carrier (b3 + b4) = (Carrier b3) \/ Carrier b4;
:: RANKNULL:th 32
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Linear_Combination of b2
st Carrier b3 misses Carrier b4
holds Carrier (b3 - b4) = (Carrier b3) \/ Carrier b4;
:: RANKNULL:th 33
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
st b3 c= b4 & b4 is Basis of b2
holds b2 is_the_direct_sum_of Lin b3,Lin (b4 \ b3);
:: RANKNULL:th 34
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Linear_Combination of b5
for b7 being Element of the carrier of b3
st b4 | b5 is one-to-one & b7 in b5
holds ex b8 being Element of bool the carrier of b3 st
b8 misses b5 &
b4 " {b4 . b7} = {b7} \/ b8;
:: RANKNULL:th 35
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of bool the carrier of b2
st b4 misses Carrier b3 & b4 <> {}
holds b3 .: b4 = {0. b1};
:: RANKNULL:th 36
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b5 being Linear_Combination of b2
for b6 being Element of the carrier of b3
st b6 in Carrier (b4 @ b5)
holds b4 " {b6} meets Carrier b5;
:: RANKNULL:th 37
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Linear_Combination of b3
for b6 being Element of the carrier of b3
st b4 | Carrier b5 is one-to-one & b6 in Carrier b5
holds (b4 @ b5) . (b4 . b6) = b5 . b6;
:: RANKNULL:th 38
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Linear_Combination of b3
for b6 being FinSequence of the carrier of b3
st proj2 b6 = Carrier b5 & b4 | Carrier b5 is one-to-one
holds b4 * (b5 (#) b6) = (b4 @ b5) (#) (b4 * b6);
:: RANKNULL:th 39
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Linear_Combination of b3
st b4 | Carrier b5 is one-to-one
holds b4 .: Carrier b5 = Carrier (b4 @ b5);
:: RANKNULL:th 40
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Basis of b3
for b7 being Linear_Combination of b6 \ b5
st b5 is Basis of ker b4 & b5 c= b6
holds b4 . Sum b7 = Sum (b4 @ b7);
:: RANKNULL:th 41
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
st b3 is linearly-dependent(b1, b2)
holds ex b4 being Linear_Combination of b3 st
Carrier b4 <> {} & Sum b4 = 0. b2;
:: RANKNULL:funcnot 9 => RANKNULL:func 9
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Element of bool the carrier of a2;
let a5 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
let a6 be Linear_Combination of a5 .: a4;
assume a5 | a4 is one-to-one;
func A5 # A6 -> Linear_Combination of a4 equals
(a6 * a5) +* ((a2 \ a4) --> 0. a1);
end;
:: RANKNULL:def 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Element of bool the carrier of b2
for b5 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b6 being Linear_Combination of b5 .: b4
st b5 | b4 is one-to-one
holds b5 # b6 = (b6 * b5) +* ((b2 \ b4) --> 0. b1);
:: RANKNULL:th 42
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Linear_Combination of b4 .: b5
for b7 being Element of the carrier of b3
st b7 in b5 & b4 | b5 is one-to-one
holds (b4 # b6) . b7 = b6 . (b4 . b7);
:: RANKNULL:th 43
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Linear_Combination of b4 .: b5
st b4 | b5 is one-to-one
holds b4 @ (b4 # b6) = b6;
:: RANKNULL:funcnot 10 => RANKNULL:func 10
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over a1;
let a4 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
func rank A4 -> natural set equals
dim im a4;
end;
:: RANKNULL:def 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3 holds
rank b4 = dim im b4;
:: RANKNULL:funcnot 11 => RANKNULL:func 11
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over a1;
let a4 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
func nullity A4 -> natural set equals
dim ker a4;
end;
:: RANKNULL:def 8
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3 holds
nullity b4 = dim ker b4;
:: RANKNULL:th 44
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3 holds
dim b2 = (rank b4) + nullity b4;
:: RANKNULL:th 45
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
st b4 is one-to-one
holds dim b2 = rank b4;