let K be Field; :: thesis: for a being Element of K
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
Det (EqSegm (a * M),P,Q) = ((power K) . a,(card P)) * (Det (EqSegm M,P,Q))
let a be Element of K; :: thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
Det (EqSegm (a * M),P,Q) = ((power K) . a,(card P)) * (Det (EqSegm M,P,Q))
let M be Matrix of K; :: thesis: for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
Det (EqSegm (a * M),P,Q) = ((power K) . a,(card P)) * (Det (EqSegm M,P,Q))
let P, Q be finite without_zero Subset of NAT ; :: thesis: ( [:P,Q:] c= Indices M & card P = card Q implies Det (EqSegm (a * M),P,Q) = ((power K) . a,(card P)) * (Det (EqSegm M,P,Q)) )
assume that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q ; :: thesis: Det (EqSegm (a * M),P,Q) = ((power K) . a,(card P)) * (Det (EqSegm M,P,Q))
EqSegm (a * M),P,Q = Segm (a * M),P,Q by A2, Def3
.= a * (Segm M,P,Q) by A1, Th63
.= a * (EqSegm M,P,Q) by A2, Def3 ;
hence Det (EqSegm (a * M),P,Q) = ((power K) . a,(card P)) * (Det (EqSegm M,P,Q)) by Th71; :: thesis: verum