let n be Nat; :: thesis: for K being Field

for M1, M2 being Matrix of n,K st M2 * M1 = 1. (K,n) holds

M1 * M2 is Idempotent

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M2 * M1 = 1. (K,n) holds

M1 * M2 is Idempotent

let M1, M2 be Matrix of n,K; :: thesis: ( M2 * M1 = 1. (K,n) implies M1 * M2 is Idempotent )

assume A1: M2 * M1 = 1. (K,n) ; :: thesis: M1 * M2 is Idempotent

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

A3: width M2 = n by MATRIX_0:24;

A4: len M2 = n by MATRIX_0:24;

width (M1 * M2) = n by MATRIX_0:24;

then (M1 * M2) * (M1 * M2) = ((M1 * M2) * M1) * M2 by A2, A4, MATRIX_3:33

.= (M1 * (1. (K,n))) * M2 by A1, A2, A4, A3, MATRIX_3:33

.= M1 * M2 by MATRIX_3:19 ;

hence M1 * M2 is Idempotent ; :: thesis: verum

for M1, M2 being Matrix of n,K st M2 * M1 = 1. (K,n) holds

M1 * M2 is Idempotent

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M2 * M1 = 1. (K,n) holds

M1 * M2 is Idempotent

let M1, M2 be Matrix of n,K; :: thesis: ( M2 * M1 = 1. (K,n) implies M1 * M2 is Idempotent )

assume A1: M2 * M1 = 1. (K,n) ; :: thesis: M1 * M2 is Idempotent

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

A3: width M2 = n by MATRIX_0:24;

A4: len M2 = n by MATRIX_0:24;

width (M1 * M2) = n by MATRIX_0:24;

then (M1 * M2) * (M1 * M2) = ((M1 * M2) * M1) * M2 by A2, A4, MATRIX_3:33

.= (M1 * (1. (K,n))) * M2 by A1, A2, A4, A3, MATRIX_3:33

.= M1 * M2 by MATRIX_3:19 ;

hence M1 * M2 is Idempotent ; :: thesis: verum