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) )

A1: ( len M1 = n & width M1 = n ) by MATRIX_0:24;

A2: width (M1 ~) = n by MATRIX_0:24;

assume A3: ( M1 is invertible & M1 is Idempotent ) ; :: thesis: M1 = 1. (K,n)

then A4: M1 ~ is_reverse_of M1 by MATRIX_6:def 4;

M1 * M1 = M1 by A3;

then 1. (K,n) = (M1 ~) * (M1 * M1) by A4, MATRIX_6:def 2

.= ((M1 ~) * M1) * M1 by A1, A2, MATRIX_3:33

.= (1. (K,n)) * M1 by A4, MATRIX_6:def 2

.= M1 by MATRIX_3:18 ;

hence M1 = 1. (K,n) ; :: thesis: verum

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) )

A1: ( len M1 = n & width M1 = n ) by MATRIX_0:24;

A2: width (M1 ~) = n by MATRIX_0:24;

assume A3: ( M1 is invertible & M1 is Idempotent ) ; :: thesis: M1 = 1. (K,n)

then A4: M1 ~ is_reverse_of M1 by MATRIX_6:def 4;

M1 * M1 = M1 by A3;

then 1. (K,n) = (M1 ~) * (M1 * M1) by A4, MATRIX_6:def 2

.= ((M1 ~) * M1) * M1 by A1, A2, MATRIX_3:33

.= (1. (K,n)) * M1 by A4, MATRIX_6:def 2

.= M1 by MATRIX_3:18 ;

hence M1 = 1. (K,n) ; :: thesis: verum