:: Matrices. Abelian Group of Matrices
:: by Katarzyna Jankowska
::
:: Received June 8, 1991
:: Copyright (c) 1991 Association of Mizar Users

begin

definition

let f be FinSequence;
attr f is tabular means :Def1: :: MATRIX_1:def 1
ex n being Nat st
for x being set st x in rng f holds
ex s being FinSequence st
( s = x & len s = n );

end;

:: deftheorem Def1 defines tabular MATRIX_1:def 1 :
for f being FinSequence holds
( f is tabular iff ex n being Nat st
for x being set st x in rng f holds
ex s being FinSequence st
( s = x & len s = n ) );

registration

cluster V1() Function-like V32() FinSequence-like FinSubsequence-like tabular set ;
existence
ex b₁ being FinSequence st b₁ is tabular

proof end;

end;

theorem Th1: :: MATRIX_1:1

for D being non empty set
for d being Element of D holds <*<*d*>*> is tabular

proof end;

theorem Th2: :: MATRIX_1:2

for x being set
for m, n being Nat holds m |-> (n |-> x) is tabular

proof end;

theorem Th3: :: MATRIX_1:3

for s being FinSequence holds <*s*> is tabular

proof end;

theorem Th4: :: MATRIX_1:4

for n being Nat
for s1, s2 being FinSequence st len s1 = n & len s2 = n holds
<*s1,s2*> is tabular

proof end;

theorem Th5: :: MATRIX_1:5

{} is tabular

proof end;

theorem :: MATRIX_1:6

<*{},{}*> is tabular by Th4, CARD_1:47;

theorem :: MATRIX_1:7

for D being non empty set
for a1, a2 being Element of D holds <*<*a1*>,<*a2*>*> is tabular

proof end;

theorem :: MATRIX_1:8

for D being non empty set
for a1, a2, b1, b2 being Element of D holds <*<*a1,a2*>,<*b1,b2*>*> is tabular

proof end;

registration

let D be set ;
cluster V1() V5(D * ) Function-like V32() FinSequence-like FinSubsequence-like tabular FinSequence of D * ;
existence
ex b₁ being FinSequence of D * st b₁ is tabular

proof end;

end;

definition

let D be set ;
mode Matrix of D is tabular FinSequence of D * ;

end;

registration

let D be non empty set ;
cluster V1() non empty-yielding V5(D * ) Function-like V32() FinSequence-like FinSubsequence-like tabular FinSequence of D * ;
existence
not for b₁ being Matrix of D holds b₁ is empty-yielding

proof end;

end;

theorem Th9: :: MATRIX_1:9

for D being non empty set
for s being FinSequence holds
( s is Matrix of D iff ex n being Nat st
for x being set st x in rng s holds
ex p being FinSequence of D st
( x = p & len p = n ) )

proof end;

definition

let D be non empty set ;
let m, n be Nat;
canceled;
mode Matrix of m,n,D -> Matrix of D means :Def3: :: MATRIX_1:def 3
( len it = m & ( for p being FinSequence of D st p in rng it holds
len p = n ) );
existence
ex b₁ being Matrix of D st
( len b₁ = m & ( for p being FinSequence of D st p in rng b₁ holds
len p = n ) )

proof end;

end;

:: deftheorem MATRIX_1:def 2 :
canceled;

:: deftheorem Def3 defines Matrix MATRIX_1:def 3 :
for D being non empty set
for m, n being Nat
for b₄ being Matrix of D holds
( b₄ is Matrix of m,n,D iff ( len b₄ = m & ( for p being FinSequence of D st p in rng b₄ holds
len p = n ) ) );

definition

let D be non empty set ;
let n be Nat;
mode Matrix of n,D is Matrix of n,n,D;

end;

definition

let K be non empty 1-sorted ;
mode Matrix of K is Matrix of the carrier of K;
let n be Nat;
mode Matrix of n,K is Matrix of n,n, the carrier of K;
let m be Nat;
mode Matrix of n,m,K is Matrix of n,m, the carrier of K;

end;

theorem Th10: :: MATRIX_1:10

for m, n being Nat
for D being non empty set
for a being Element of D holds m |-> (n |-> a) is Matrix of m,n,D

proof end;

theorem Th11: :: MATRIX_1:11

for D being non empty set
for p being FinSequence of D holds <*p*> is Matrix of 1, len p,D

proof end;

theorem Th12: :: MATRIX_1:12

for n being Nat
for D being non empty set
for p1, p2 being FinSequence of D st len p1 = n & len p2 = n holds
<*p1,p2*> is Matrix of 2,n,D

proof end;

theorem Th13: :: MATRIX_1:13

for m being Nat
for D being non empty set holds {} is Matrix of 0 ,m,D

proof end;

theorem :: MATRIX_1:14

for D being non empty set holds <*{}*> is Matrix of 1, 0 ,D

proof end;

theorem :: MATRIX_1:15

for D being non empty set
for a being Element of D holds <*<*a*>*> is Matrix of 1,D

proof end;

theorem :: MATRIX_1:16

for D being non empty set holds <*{},{}*> is Matrix of 2, 0 ,D

proof end;

theorem :: MATRIX_1:17

for D being non empty set
for a1, a2 being Element of D holds <*<*a1,a2*>*> is Matrix of 1,2,D

proof end;

theorem :: MATRIX_1:18

for D being non empty set
for a1, a2 being Element of D holds <*<*a1*>,<*a2*>*> is Matrix of 2,1,D

proof end;

theorem :: MATRIX_1:19

for D being non empty set
for a1, a2, b1, b2 being Element of D holds <*<*a1,a2*>,<*b1,b2*>*> is Matrix of 2,D

proof end;

definition

let M be tabular FinSequence;
func width M -> Element of NAT means :Def4: :: MATRIX_1:def 4
ex s being FinSequence st
( s in rng M & len s = it ) if len M > 0
otherwise it = 0 ;
existence
( ( len M > 0 implies ex b₁ being Element of NAT ex s being FinSequence st
( s in rng M & len s = b₁ ) ) & ( not len M > 0 implies ex b₁ being Element of NAT st b₁ = 0 ) )

proof end;

uniqueness
for b₁, b₂ being Element of NAT holds
( ( len M > 0 & ex s being FinSequence st
( s in rng M & len s = b₁ ) & ex s being FinSequence st
( s in rng M & len s = b₂ ) implies b₁ = b₂ ) & ( not len M > 0 & b₁ = 0 & b₂ = 0 implies b₁ = b₂ ) )

proof end;

consistency
for b₁ being Element of NAT holds verum ;

end;

:: deftheorem Def4 defines width MATRIX_1:def 4 :
for M being tabular FinSequence
for b₂ being Element of NAT holds
( ( len M > 0 implies ( b₂ = width M iff ex s being FinSequence st
( s in rng M & len s = b₂ ) ) ) & ( not len M > 0 implies ( b₂ = width M iff b₂ = 0 ) ) );

theorem Th20: :: MATRIX_1:20

for D being non empty set
for M being Matrix of D st len M > 0 holds
for n being Nat holds
( M is Matrix of len M,n,D iff n = width M )

proof end;

definition

let M be tabular FinSequence;
func Indices M -> set equals :: MATRIX_1:def 5
[:(dom M),(Seg (width M)):];
coherence
[:(dom M),(Seg (width M)):] is set ;

end;

:: deftheorem defines Indices MATRIX_1:def 5 :
for M being tabular FinSequence holds Indices M = [:(dom M),(Seg (width M)):];

definition

let D be set ;
let M be Matrix of D;
let i, j be Nat;
assume A1: [i,j] in Indices M ;
func M * (i,j) -> Element of D means :Def6: :: MATRIX_1:def 6
ex p being FinSequence of D st
( p = M . i & it = p . j );
existence
ex b₁ being Element of D ex p being FinSequence of D st
( p = M . i & b₁ = p . j )

proof end;

uniqueness
for b₁, b₂ being Element of D st ex p being FinSequence of D st
( p = M . i & b₁ = p . j ) & ex p being FinSequence of D st
( p = M . i & b₂ = p . j ) holds
b₁ = b₂ ;

end;

:: deftheorem Def6 defines * MATRIX_1:def 6 :
for D being set
for M being Matrix of D
for i, j being Nat st [i,j] in Indices M holds
for b₅ being Element of D holds
( b₅ = M * (i,j) iff ex p being FinSequence of D st
( p = M . i & b₅ = p . j ) );

theorem Th21: :: MATRIX_1:21

for D being non empty set
for M1, M2 being Matrix of D st len M1 = len M2 & width M1 = width M2 & ( for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j) ) holds
M1 = M2

proof end;

scheme :: MATRIX_1:sch 1
MatrixLambda{ F₁() -> non empty set , F₂() -> Element of NAT , F₃() -> Element of NAT , F₄( set , set ) -> Element of F₁() } :

ex M being Matrix of F₂(),F₃(),F₁() st
for i, j being Nat st [i,j] in Indices M holds
M * (i,j) = F₄(i,j)

proof end;

scheme :: MATRIX_1:sch 2
MatrixEx{ F₁() -> non empty set , F₂() -> Element of NAT , F₃() -> Element of NAT , P₁[ set , set , set ] } :

ex M being Matrix of F₂(),F₃(),F₁() st
for i, j being Nat st [i,j] in Indices M holds
P₁[i,j,M * (i,j)]

provided

for i, j being Nat st [i,j] in [:(Seg F₂()),(Seg F₃()):] holds
for x1, x2 being Element of F₁() st P₁[i,j,x1] & P₁[i,j,x2] holds
x1 = x2 and
A1: for i, j being Nat st [i,j] in [:(Seg F₂()),(Seg F₃()):] holds
ex x being Element of F₁() st P₁[i,j,x]

proof end;

theorem :: MATRIX_1:22

canceled;

theorem :: MATRIX_1:23

for m being Nat
for D being non empty set
for M being Matrix of 0 ,m,D holds
( len M = 0 & width M = 0 & Indices M = {} )

proof end;

theorem Th24: :: MATRIX_1:24

for n, m being Nat
for D being non empty set st n > 0 holds
for M being Matrix of n,m,D holds
( len M = n & width M = m & Indices M = [:(Seg n),(Seg m):] )

proof end;

theorem Th25: :: MATRIX_1:25

for n being Nat
for D being non empty set
for M being Matrix of n,D holds
( len M = n & width M = n & Indices M = [:(Seg n),(Seg n):] )

proof end;

theorem Th26: :: MATRIX_1:26

for n, m being Nat
for D being non empty set
for M being Matrix of n,m,D holds
( len M = n & Indices M = [:(Seg n),(Seg (width M)):] )

proof end;

theorem Th27: :: MATRIX_1:27

for n, m being Nat
for D being non empty set
for M1, M2 being Matrix of n,m,D holds Indices M1 = Indices M2

proof end;

theorem Th28: :: MATRIX_1:28

for n, m being Nat
for D being non empty set
for M1, M2 being Matrix of n,m,D st ( for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j) ) holds
M1 = M2

proof end;

theorem :: MATRIX_1:29

for n being Nat
for D being non empty set
for M1 being Matrix of n,D
for i, j being Nat st [i,j] in Indices M1 holds
[j,i] in Indices M1

proof end;

definition

let D be non empty set ;
let M be Matrix of D;
func M @ -> Matrix of D means :Def7: :: MATRIX_1:def 7
( len it = width M & ( for i, j being Nat holds
( [i,j] in Indices it iff [j,i] in Indices M ) ) & ( for i, j being Nat st [j,i] in Indices M holds
it * (i,j) = M * (j,i) ) );
existence
ex b₁ being Matrix of D st
( len b₁ = width M & ( for i, j being Nat holds
( [i,j] in Indices b₁ iff [j,i] in Indices M ) ) & ( for i, j being Nat st [j,i] in Indices M holds
b₁ * (i,j) = M * (j,i) ) )

proof end;

uniqueness
for b₁, b₂ being Matrix of D st len b₁ = width M & ( for i, j being Nat holds
( [i,j] in Indices b₁ iff [j,i] in Indices M ) ) & ( for i, j being Nat st [j,i] in Indices M holds
b₁ * (i,j) = M * (j,i) ) & len b₂ = width M & ( for i, j being Nat holds
( [i,j] in Indices b₂ iff [j,i] in Indices M ) ) & ( for i, j being Nat st [j,i] in Indices M holds
b₂ * (i,j) = M * (j,i) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines @ MATRIX_1:def 7 :
for D being non empty set
for M, b₃ being Matrix of D holds
( b₃ = M @ iff ( len b₃ = width M & ( for i, j being Nat holds
( [i,j] in Indices b₃ iff [j,i] in Indices M ) ) & ( for i, j being Nat st [j,i] in Indices M holds
b₃ * (i,j) = M * (j,i) ) ) );

Lm1: for D being non empty set
for M being Matrix of D st width M > 0 holds
( len (M @) = width M & width (M @) = len M )

proof end;

Lm2: for n being Element of NAT
for K being non empty set
for A being Matrix of n,K holds A @ is Matrix of n,K

proof end;

definition

let n be Nat;
let D be non empty set ;
let M be Matrix of n,D;
:: original: @
redefine func M @ -> Matrix of n,D;
coherence
M @ is Matrix of n,D

proof end;

end;

definition

let D be non empty set ;
let M be Matrix of D;
let i be Nat;
func Line (M,i) -> FinSequence of D means :Def8: :: MATRIX_1:def 8
( len it = width M & ( for j being Nat st j in Seg (width M) holds
it . j = M * (i,j) ) );
existence
ex b₁ being FinSequence of D st
( len b₁ = width M & ( for j being Nat st j in Seg (width M) holds
b₁ . j = M * (i,j) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of D st len b₁ = width M & ( for j being Nat st j in Seg (width M) holds
b₁ . j = M * (i,j) ) & len b₂ = width M & ( for j being Nat st j in Seg (width M) holds
b₂ . j = M * (i,j) ) holds
b₁ = b₂

proof end;

func Col (M,i) -> FinSequence of D means :Def9: :: MATRIX_1:def 9
( len it = len M & ( for j being Nat st j in dom M holds
it . j = M * (j,i) ) );
existence
ex b₁ being FinSequence of D st
( len b₁ = len M & ( for j being Nat st j in dom M holds
b₁ . j = M * (j,i) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of D st len b₁ = len M & ( for j being Nat st j in dom M holds
b₁ . j = M * (j,i) ) & len b₂ = len M & ( for j being Nat st j in dom M holds
b₂ . j = M * (j,i) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines Line MATRIX_1:def 8 :
for D being non empty set
for M being Matrix of D
for i being Nat
for b₄ being FinSequence of D holds
( b₄ = Line (M,i) iff ( len b₄ = width M & ( for j being Nat st j in Seg (width M) holds
b₄ . j = M * (i,j) ) ) );

:: deftheorem Def9 defines Col MATRIX_1:def 9 :
for D being non empty set
for M being Matrix of D
for i being Nat
for b₄ being FinSequence of D holds
( b₄ = Col (M,i) iff ( len b₄ = len M & ( for j being Nat st j in dom M holds
b₄ . j = M * (j,i) ) ) );

definition

let D be non empty set ;
let M be Matrix of D;
let i be Nat;
:: original: Line
redefine func Line (M,i) -> Element of (width M) -tuples_on D;
coherence
Line (M,i) is Element of (width M) -tuples_on D

proof end;

:: original: Col
redefine func Col (M,i) -> Element of (len M) -tuples_on D;
coherence
Col (M,i) is Element of (len M) -tuples_on D

proof end;

end;

definition

let K be non empty doubleLoopStr ;
let n be Nat;
func n -Matrices_over K -> set equals :: MATRIX_1:def 10
n -tuples_on (n -tuples_on the carrier of K);
coherence
n -tuples_on (n -tuples_on the carrier of K) is set ;
func 0. (K,n) -> Matrix of n,K equals :: MATRIX_1:def 11
n |-> (n |-> (0. K));
coherence
n |-> (n |-> (0. K)) is Matrix of n,K by Th10;
func 1. (K,n) -> Matrix of n,K means :Def12: :: MATRIX_1:def 12
( ( for i being Nat st [i,i] in Indices it holds
it * (i,i) = 1. K ) & ( for i, j being Nat st [i,j] in Indices it & i <> j holds
it * (i,j) = 0. K ) );
existence
ex b₁ being Matrix of n,K st
( ( for i being Nat st [i,i] in Indices b₁ holds
b₁ * (i,i) = 1. K ) & ( for i, j being Nat st [i,j] in Indices b₁ & i <> j holds
b₁ * (i,j) = 0. K ) )

proof end;

uniqueness
for b₁, b₂ being Matrix of n,K st ( for i being Nat st [i,i] in Indices b₁ holds
b₁ * (i,i) = 1. K ) & ( for i, j being Nat st [i,j] in Indices b₁ & i <> j holds
b₁ * (i,j) = 0. K ) & ( for i being Nat st [i,i] in Indices b₂ holds
b₂ * (i,i) = 1. K ) & ( for i, j being Nat st [i,j] in Indices b₂ & i <> j holds
b₂ * (i,j) = 0. K ) holds
b₁ = b₂

proof end;

let A be Matrix of n,K;
func - A -> Matrix of n,K means :Def13: :: MATRIX_1:def 13
for i, j being Nat st [i,j] in Indices A holds
it * (i,j) = - (A * (i,j));
existence
ex b₁ being Matrix of n,K st
for i, j being Nat st [i,j] in Indices A holds
b₁ * (i,j) = - (A * (i,j))

proof end;

uniqueness
for b₁, b₂ being Matrix of n,K st ( for i, j being Nat st [i,j] in Indices A holds
b₁ * (i,j) = - (A * (i,j)) ) & ( for i, j being Nat st [i,j] in Indices A holds
b₂ * (i,j) = - (A * (i,j)) ) holds
b₁ = b₂

proof end;

let B be Matrix of n,K;
func A + B -> Matrix of n,K means :Def14: :: MATRIX_1:def 14
for i, j being Nat st [i,j] in Indices A holds
it * (i,j) = (A * (i,j)) + (B * (i,j));
existence
ex b₁ being Matrix of n,K st
for i, j being Nat st [i,j] in Indices A holds
b₁ * (i,j) = (A * (i,j)) + (B * (i,j))

proof end;

uniqueness
for b₁, b₂ being Matrix of n,K st ( for i, j being Nat st [i,j] in Indices A holds
b₁ * (i,j) = (A * (i,j)) + (B * (i,j)) ) & ( for i, j being Nat st [i,j] in Indices A holds
b₂ * (i,j) = (A * (i,j)) + (B * (i,j)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem defines -Matrices_over MATRIX_1:def 10 :
for K being non empty doubleLoopStr
for n being Nat holds n -Matrices_over K = n -tuples_on (n -tuples_on the carrier of K);

:: deftheorem defines 0. MATRIX_1:def 11 :
for K being non empty doubleLoopStr
for n being Nat holds 0. (K,n) = n |-> (n |-> (0. K));

:: deftheorem Def12 defines 1. MATRIX_1:def 12 :
for K being non empty doubleLoopStr
for n being Nat
for b₃ being Matrix of n,K holds
( b₃ = 1. (K,n) iff ( ( for i being Nat st [i,i] in Indices b₃ holds
b₃ * (i,i) = 1. K ) & ( for i, j being Nat st [i,j] in Indices b₃ & i <> j holds
b₃ * (i,j) = 0. K ) ) );

:: deftheorem Def13 defines - MATRIX_1:def 13 :
for K being non empty doubleLoopStr
for n being Nat
for A, b₄ being Matrix of n,K holds
( b₄ = - A iff for i, j being Nat st [i,j] in Indices A holds
b₄ * (i,j) = - (A * (i,j)) );

:: deftheorem Def14 defines + MATRIX_1:def 14 :
for K being non empty doubleLoopStr
for n being Nat
for A, B, b₅ being Matrix of n,K holds
( b₅ = A + B iff for i, j being Nat st [i,j] in Indices A holds
b₅ * (i,j) = (A * (i,j)) + (B * (i,j)) );

registration

let K be non empty doubleLoopStr ;
let n be Nat;
cluster n -Matrices_over K -> non empty ;
coherence
not n -Matrices_over K is empty ;

end;

theorem Th30: :: MATRIX_1:30

for i, j, n being Nat
for K being non empty doubleLoopStr st [i,j] in Indices (0. (K,n)) holds
(0. (K,n)) * (i,j) = 0. K

proof end;

theorem Th31: :: MATRIX_1:31

for x being set
for n being Nat
for K being non empty doubleLoopStr holds
( x is Element of n -Matrices_over K iff x is Matrix of n,K )

proof end;

definition

let K be non empty ZeroStr ;
let M be Matrix of K;
attr M is diagonal means :: MATRIX_1:def 15
for i, j being Nat st [i,j] in Indices M & M * (i,j) <> 0. K holds
i = j;

end;

:: deftheorem defines diagonal MATRIX_1:def 15 :
for K being non empty ZeroStr
for M being Matrix of K holds
( M is diagonal iff for i, j being Nat st [i,j] in Indices M & M * (i,j) <> 0. K holds
i = j );

registration

let n be Nat;
let K be non empty doubleLoopStr ;
cluster V1() V5( the carrier of K * ) Function-like V32() FinSequence-like FinSubsequence-like tabular diagonal Matrix of n,n, the carrier of K;
existence
ex b₁ being Matrix of n,K st b₁ is diagonal

proof end;

end;

definition

let K be non empty doubleLoopStr ;
let n be Nat;
mode Diagonal of n,K is diagonal Matrix of n,K;

end;

theorem Th32: :: MATRIX_1:32

for n being Nat
for F being non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr
for A, B being Matrix of n,F holds A + B = B + A

proof end;

theorem Th33: :: MATRIX_1:33

for n being Nat
for F being non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr
for A, B, C being Matrix of n,F holds (A + B) + C = A + (B + C)

proof end;

theorem Th34: :: MATRIX_1:34

for n being Nat
for F being non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr
for A being Matrix of n,F holds A + (0. (F,n)) = A

proof end;

theorem Th35: :: MATRIX_1:35

for n being Nat
for F being non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr
for A being Matrix of n,F holds A + (- A) = 0. (F,n)

proof end;

definition

let F be non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr ;
let n be Nat;
func n -G_Matrix_over F -> strict AbGroup means :: MATRIX_1:def 16
( the carrier of it = n -Matrices_over F & ( for A, B being Matrix of n,F holds the addF of it . (A,B) = A + B ) & 0. it = 0. (F,n) );
existence
ex b₁ being strict AbGroup st
( the carrier of b₁ = n -Matrices_over F & ( for A, B being Matrix of n,F holds the addF of b₁ . (A,B) = A + B ) & 0. b₁ = 0. (F,n) )

proof end;

uniqueness
for b₁, b₂ being strict AbGroup st the carrier of b₁ = n -Matrices_over F & ( for A, B being Matrix of n,F holds the addF of b₁ . (A,B) = A + B ) & 0. b₁ = 0. (F,n) & the carrier of b₂ = n -Matrices_over F & ( for A, B being Matrix of n,F holds the addF of b₂ . (A,B) = A + B ) & 0. b₂ = 0. (F,n) holds
b₁ = b₂

proof end;

end;

:: deftheorem defines -G_Matrix_over MATRIX_1:def 16 :
for F being non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr
for n being Nat
for b₃ being strict AbGroup holds
( b₃ = n -G_Matrix_over F iff ( the carrier of b₃ = n -Matrices_over F & ( for A, B being Matrix of n,F holds the addF of b₃ . (A,B) = A + B ) & 0. b₃ = 0. (F,n) ) );

theorem :: MATRIX_1:36

for n, m being Nat
for K being non empty doubleLoopStr
for M1 being Matrix of 0 ,n,K
for M2 being Matrix of 0 ,m,K holds M1 = M2

proof end;

begin

theorem Th37: :: MATRIX_1:37

for D being set
for i, j being Nat
for M being Matrix of D st 1 <= i & i <= len M & 1 <= j & j <= width M holds
[i,j] in Indices M

proof end;

theorem :: MATRIX_1:38

for n, m, i, j being Nat
for D being non empty set
for M being Matrix of n,m,D st 1 <= i & i <= n & 1 <= j & j <= m holds
[i,j] in Indices M

proof end;

theorem Th39: :: MATRIX_1:39

for M being tabular FinSequence
for i, j being Nat st [i,j] in Indices M holds
( 1 <= i & i <= len M & 1 <= j & j <= width M )

proof end;

theorem :: MATRIX_1:40

for n, m, i, j being Nat
for D being non empty set
for M being Matrix of n,m,D st [i,j] in Indices M holds
( 1 <= i & i <= n & 1 <= j & j <= m )

proof end;