let n be Nat; :: thesis: for K being Field
for M1 being Matrix of n,K st M1 is invertible & M1 is Idempotent holds
M1 = 1. K,n
let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is invertible & M1 is Idempotent holds
M1 = 1. K,n
let M1 be Matrix of n,K; :: thesis: ( M1 is invertible & M1 is Idempotent implies M1 = 1. K,n )
assume ( M1 is invertible & M1 is Idempotent ) ; :: thesis: M1 = 1. K,n
then A1: ( M1 * M1 = M1 & M1 ~ is_reverse_of M1 ) by Def1, MATRIX_6:def 4;
A2: ( len M1 = n & width M1 = n & len (M1 ~ ) = n & width (M1 ~ ) = n ) by MATRIX_1:25;
1. K,n = (M1 ~ ) * (M1 * M1) by A1, MATRIX_6:def 2
.= ((M1 ~ ) * M1) * M1 by A2, MATRIX_3:35
.= (1. K,n) * M1 by A1, MATRIX_6:def 2
.= M1 by MATRIX_3:20 ;
hence M1 = 1. K,n ; :: thesis: verum