let n be Nat; :: thesis: for K being Field
for A being Matrix of n,K st Det A <> 0. K holds
for x, b being FinSequence of K st len x = n & A * x = <*b*> @ holds
( <*x*> @ = (A ~) * b & ( for i being Nat st i in Seg n holds
x . i = ((Det A) ") * (Det (ReplaceCol (A,i,b))) ) )
let K be Field; :: thesis: for A being Matrix of n,K st Det A <> 0. K holds
for x, b being FinSequence of K st len x = n & A * x = <*b*> @ holds
( <*x*> @ = (A ~) * b & ( for i being Nat st i in Seg n holds
x . i = ((Det A) ") * (Det (ReplaceCol (A,i,b))) ) )
let A be Matrix of n,K; :: thesis: ( Det A <> 0. K implies for x, b being FinSequence of K st len x = n & A * x = <*b*> @ holds
( <*x*> @ = (A ~) * b & ( for i being Nat st i in Seg n holds
x . i = ((Det A) ") * (Det (ReplaceCol (A,i,b))) ) ) )
assume A1: Det A <> 0. K ; :: thesis: for x, b being FinSequence of K st len x = n & A * x = <*b*> @ holds
( <*x*> @ = (A ~) * b & ( for i being Nat st i in Seg n holds
x . i = ((Det A) ") * (Det (ReplaceCol (A,i,b))) ) )
let x, b be FinSequence of K; :: thesis: ( len x = n & A * x = <*b*> @ implies ( <*x*> @ = (A ~) * b & ( for i being Nat st i in Seg n holds
x . i = ((Det A) ") * (Det (ReplaceCol (A,i,b))) ) ) )
assume that
A2: len x = n and
A3: A * x = <*b*> @ ; :: thesis: ( <*x*> @ = (A ~) * b & ( for i being Nat st i in Seg n holds
x . i = ((Det A) ") * (Det (ReplaceCol (A,i,b))) ) )
set X = <*x*>;
len <*x*> = 1 by MATRIX_0:def 2;
then A4: len x = width <*x*> by MATRIX_0:20;
then A5: len (<*x*> @) = len x by MATRIX_0:def 6;
hence <*x*> @ = (A ~) * b by A1, A2, A3, Th37; :: thesis: for i being Nat st i in Seg n holds
x . i = ((Det A) ") * (Det (ReplaceCol (A,i,b)))
set B = <*b*>;
A6: 1 in Seg 1 ;
then A7: Line (<*x*>,1) = <*x*> . 1 by MATRIX_0:52;
len <*b*> = 1 by MATRIX_0:def 2;
then A8: 1 in dom <*b*> by A6, FINSEQ_1:def 3;
A9: Line (<*b*>,1) = <*b*> . 1 by A6, MATRIX_0:52;
let i be Nat; :: thesis: ( i in Seg n implies x . i = ((Det A) ") * (Det (ReplaceCol (A,i,b))) )
assume A10: i in Seg n ; :: thesis: x . i = ((Det A) ") * (Det (ReplaceCol (A,i,b)))
n > 0 by A10;
then width (<*x*> @) = len <*x*> by A2, A4, MATRIX_0:54
.= 1 by MATRIX_0:def 2 ;
then Indices (<*x*> @) = [:(Seg n),(Seg 1):] by A2, A5, FINSEQ_1:def 3;
then A11: [i,1] in Indices (<*x*> @) by A10, A6, ZFMISC_1:87;
then [1,i] in Indices <*x*> by MATRIX_0:def 6;
then (<*x*> @) * (i,1) = <*x*> * (1,i) by MATRIX_0:def 6
.= (Line (<*x*>,1)) . i by A2, A4, A10, MATRIX_0:def 7
.= x . i by A7 ;
hence x . i = ((Det A) ") * (Det (ReplaceCol (A,i,(Col ((<*b*> @),1))))) by A1, A2, A3, A5, A11, Th37
.= ((Det A) ") * (Det (ReplaceCol (A,i,(Line (<*b*>,1))))) by A8, MATRIX_0:58
.= ((Det A) ") * (Det (ReplaceCol (A,i,b))) by A9 ;
:: thesis: verum