let j be Nat; :: thesis: for K being Field
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT
for i being Nat st j in Seg (card P) & Det (EqSegm M,P,Q) <> 0. K holds
ex i being Nat st
( i in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K )
let K be Field; :: thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT
for i being Nat st j in Seg (card P) & Det (EqSegm M,P,Q) <> 0. K holds
ex i being Nat st
( i in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K )
let M be Matrix of K; :: thesis: for P, Q being finite without_zero Subset of NAT
for i being Nat st j in Seg (card P) & Det (EqSegm M,P,Q) <> 0. K holds
ex i being Nat st
( i in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K )
let P, Q be finite without_zero Subset of NAT ; :: thesis: for i being Nat st j in Seg (card P) & Det (EqSegm M,P,Q) <> 0. K holds
ex i being Nat st
( i in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K )
let i be Nat; :: thesis: ( j in Seg (card P) & Det (EqSegm M,P,Q) <> 0. K implies ex i being Nat st
( i in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K ) )
assume that
A1: j in Seg (card P) and
A2: Det (EqSegm M,P,Q) <> 0. K ; :: thesis: ex i being Nat st
( i in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K )
set C = card P;
set E = EqSegm M,P,Q;
set LC = LaplaceExpC (EqSegm M,P,Q),j;
set CC = (card P) |-> (0. K);
Sum ((card P) |-> (0. K)) = 0. K by MATRIX_3:13;
then A3: LaplaceExpC (EqSegm M,P,Q),j <> (card P) |-> (0. K) by A1, A2, LAPLACE:27;
len (LaplaceExpC (EqSegm M,P,Q),j) = card P by LAPLACE:def 8;
then A4: dom (LaplaceExpC (EqSegm M,P,Q),j) = Seg (card P) by FINSEQ_1:def 3;
dom ((card P) |-> (0. K)) = Seg (card P) by FUNCOP_1:19;
then consider i being Nat such that
A5: i in dom (LaplaceExpC (EqSegm M,P,Q),j) and
A6: (LaplaceExpC (EqSegm M,P,Q),j) . i <> ((card P) |-> (0. K)) . i by A3, A4, FINSEQ_1:17;
A7: (LaplaceExpC (EqSegm M,P,Q),j) . i = ((EqSegm M,P,Q) * i,j) * (Cofactor (EqSegm M,P,Q),i,j) by A5, LAPLACE:def 8;
((card P) |-> (0. K)) . i = 0. K by A4, A5, FINSEQ_2:71;
then Cofactor (EqSegm M,P,Q),i,j <> 0. K by A6, A7, VECTSP_1:44;
then Minor (EqSegm M,P,Q),i,j <> 0. K by VECTSP_1:44;
hence ex i being Nat st
( i in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K ) by A4, A5; :: thesis: verum