let n be Nat; for K being Field
for f being FinSequence of K
for M being Matrix of n,K
for p being Element of Permutations n
for i being Nat st len f = n & i in Seg n holds
mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)
let K be Field; for f being FinSequence of K
for M being Matrix of n,K
for p being Element of Permutations n
for i being Nat st len f = n & i in Seg n holds
mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)
let f be FinSequence of K; for M being Matrix of n,K
for p being Element of Permutations n
for i being Nat st len f = n & i in Seg n holds
mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)
let M be Matrix of n,K; for p being Element of Permutations n
for i being Nat st len f = n & i in Seg n holds
mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)
let p be Element of Permutations n; for i being Nat st len f = n & i in Seg n holds
mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)
let i be Nat; ( len f = n & i in Seg n implies mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i) )
assume that
A1:
len f = n
and
A2:
i in Seg n
; mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)
reconsider N = n as Element of NAT by ORDINAL1:def 12;
set KK = the carrier of K;
set C = Matrix_of_Cofactor M;
reconsider Tp = f, TL = Line ((Matrix_of_Cofactor M),i) as Element of N -tuples_on the carrier of K by A1, FINSEQ_2:92, MATRIX_0:24;
set R = RLine (M,i,f);
set LL = LaplaceExpL ((RLine (M,i,f)),i);
set MLT = mlt (TL,Tp);
A3:
len (LaplaceExpL ((RLine (M,i,f)),i)) = n
by Def7;
A4:
now for j being Nat st 1 <= j & j <= n holds
(mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . jA5:
dom (LaplaceExpL ((RLine (M,i,f)),i)) = Seg n
by A3, FINSEQ_1:def 3;
A6:
n = width M
by MATRIX_0:24;
let j be
Nat;
( 1 <= j & j <= n implies (mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . j )assume that A7:
1
<= j
and A8:
j <= n
;
(mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . jA9:
j in Seg n
by A7, A8;
n = width (Matrix_of_Cofactor M)
by MATRIX_0:24;
then A10:
(Line ((Matrix_of_Cofactor M),i)) . j = (Matrix_of_Cofactor M) * (
i,
j)
by A9, MATRIX_0:def 7;
Indices M = [:(Seg n),(Seg n):]
by MATRIX_0:24;
then
[i,j] in Indices M
by A2, A9, ZFMISC_1:87;
then A11:
(RLine (M,i,f)) * (
i,
j)
= f . j
by A1, A6, MATRIX11:def 3;
Indices (Matrix_of_Cofactor M) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
then
[i,j] in Indices (Matrix_of_Cofactor M)
by A2, A9, ZFMISC_1:87;
then
(Line ((Matrix_of_Cofactor M),i)) . j = Cofactor (
M,
i,
j)
by A10, Def6;
then A12:
(mlt (TL,Tp)) . j = (Cofactor (M,i,j)) * ((RLine (M,i,f)) * (i,j))
by A9, A11, FVSUM_1:61;
Cofactor (
M,
i,
j)
= Cofactor (
(RLine (M,i,f)),
i,
j)
by A2, A9, Th15;
hence
(mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . j
by A9, A5, A12, Def7;
verum end;
len (mlt (TL,Tp)) = n
by CARD_1:def 7;
hence
mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)
by A3, A4; verum