let l, k, n, m be Nat; for K being Field
for M, M1 being Matrix of n,m,K st l in Seg (width M) & k in Seg (width M) & n > 0 & m > 0 & M1 = M @ holds
(ILine M1,l,k) @ = ICol M,l,k
let K be Field; for M, M1 being Matrix of n,m,K st l in Seg (width M) & k in Seg (width M) & n > 0 & m > 0 & M1 = M @ holds
(ILine M1,l,k) @ = ICol M,l,k
let M, M1 be Matrix of n,m,K; ( l in Seg (width M) & k in Seg (width M) & n > 0 & m > 0 & M1 = M @ implies (ILine M1,l,k) @ = ICol M,l,k )
assume that
A1:
l in Seg (width M)
and
A2:
k in Seg (width M)
and
A3:
n > 0
and
A4:
m > 0
and
A5:
M1 = M @
; (ILine M1,l,k) @ = ICol M,l,k
A6:
width M = m
by A3, MATRIX_1:24;
A7:
width (ILine M1,l,k) = width M1
by Th1;
len M = n
by A3, MATRIX_1:24;
then A8:
width M1 = n
by A4, A5, A6, MATRIX_2:12;
then A9:
len ((ILine M1,l,k) @ ) = n
by A3, A7, MATRIX_2:12;
A10:
len (ILine M1,l,k) = len M1
by Def1;
len M1 = m
by A4, A5, A6, MATRIX_2:12;
then
width ((ILine M1,l,k) @ ) = m
by A3, A8, A10, A7, MATRIX_2:12;
then A11:
(ILine M1,l,k) @ is Matrix of n,m,K
by A3, A9, MATRIX_1:20;
then consider M2 being Matrix of n,m,K such that
A12:
M2 = (ILine M1,l,k) @
;
A13:
for i, j being Nat st i in dom M & j in Seg (width M) holds
( ( j = l implies M2 * i,j = M * i,k ) & ( j = k implies M2 * i,j = M * i,l ) & ( j <> l & j <> k implies M2 * i,j = M * i,j ) )
proof
let i,
j be
Nat;
( i in dom M & j in Seg (width M) implies ( ( j = l implies M2 * i,j = M * i,k ) & ( j = k implies M2 * i,j = M * i,l ) & ( j <> l & j <> k implies M2 * i,j = M * i,j ) ) )
assume that A14:
i in dom M
and A15:
j in Seg (width M)
;
( ( j = l implies M2 * i,j = M * i,k ) & ( j = k implies M2 * i,j = M * i,l ) & ( j <> l & j <> k implies M2 * i,j = M * i,j ) )
A16:
[i,j] in Indices M
by A14, A15, ZFMISC_1:106;
then A17:
[j,i] in Indices M1
by A5, MATRIX_1:def 7;
then A18:
(
j in dom M1 &
i in Seg (width M1) )
by ZFMISC_1:106;
dom (ILine M1,l,k) =
Seg (len (ILine M1,l,k))
by FINSEQ_1:def 3
.=
Seg (len M1)
by Def1
.=
dom M1
by FINSEQ_1:def 3
;
then A19:
[j,i] in Indices (ILine M1,l,k)
by A17, Th1;
thus
(
j = l implies
M2 * i,
j = M * i,
k )
( ( j = k implies M2 * i,j = M * i,l ) & ( j <> l & j <> k implies M2 * i,j = M * i,j ) )proof
A20:
[i,k] in Indices M
by A2, A14, ZFMISC_1:106;
assume A21:
j = l
;
M2 * i,j = M * i,k
M2 * i,
j =
(ILine M1,l,k) * j,
i
by A12, A19, MATRIX_1:def 7
.=
M1 * k,
i
by A18, A21, Def1
.=
M * i,
k
by A5, A20, MATRIX_1:def 7
;
hence
M2 * i,
j = M * i,
k
;
verum
end;
thus
(
j = k implies
M2 * i,
j = M * i,
l )
( j <> l & j <> k implies M2 * i,j = M * i,j )proof
A22:
[i,l] in Indices M
by A1, A14, ZFMISC_1:106;
assume A23:
j = k
;
M2 * i,j = M * i,l
M2 * i,
j =
(ILine M1,l,k) * j,
i
by A12, A19, MATRIX_1:def 7
.=
M1 * l,
i
by A18, A23, Def1
.=
M * i,
l
by A5, A22, MATRIX_1:def 7
;
hence
M2 * i,
j = M * i,
l
;
verum
end;
assume A24:
(
j <> l &
j <> k )
;
M2 * i,j = M * i,j
M2 * i,
j =
(ILine M1,l,k) * j,
i
by A12, A19, MATRIX_1:def 7
.=
M1 * j,
i
by A18, A24, Def1
.=
M * i,
j
by A5, A16, MATRIX_1:def 7
;
hence
M2 * i,
j = M * i,
j
;
verum
end;
for i, j being Nat st [i,j] in Indices (ICol M,l,k) holds
(ICol M,l,k) * i,j = ((ILine M1,l,k) @ ) * i,j
proof
A25:
Indices M = Indices (ICol M,l,k)
by MATRIX_1:27;
let i,
j be
Nat;
( [i,j] in Indices (ICol M,l,k) implies (ICol M,l,k) * i,j = ((ILine M1,l,k) @ ) * i,j )
assume
[i,j] in Indices (ICol M,l,k)
;
(ICol M,l,k) * i,j = ((ILine M1,l,k) @ ) * i,j
then A26:
(
i in dom M &
j in Seg (width M) )
by A25, ZFMISC_1:106;
then A27:
(
j = l implies
((ILine M1,l,k) @ ) * i,
j = M * i,
k )
by A12, A13;
A28:
(
j = k implies
(ICol M,l,k) * i,
j = M * i,
l )
by A1, A3, A4, A26, Def4;
A29:
(
j = k implies
((ILine M1,l,k) @ ) * i,
j = M * i,
l )
by A12, A13, A26;
A30:
(
j <> l &
j <> k implies
((ILine M1,l,k) @ ) * i,
j = M * i,
j )
by A12, A13, A26;
(
j = l implies
(ICol M,l,k) * i,
j = M * i,
k )
by A2, A3, A4, A26, Def4;
hence
(ICol M,l,k) * i,
j = ((ILine M1,l,k) @ ) * i,
j
by A1, A2, A3, A4, A26, A27, A29, A30, A28, Def4;
verum
end;
hence
(ILine M1,l,k) @ = ICol M,l,k
by A11, MATRIX_1:28; verum