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

for M1, M2 being Matrix of n,K st M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 holds

M1 * M2 is Idempotent

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 holds

M1 * M2 is Idempotent

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 implies M1 * M2 is Idempotent )

assume that

A1: M1 is Idempotent and

A2: M2 is Idempotent and

A3: M1 commutes_with M2 ; :: thesis: M1 * M2 is Idempotent

A4: len M1 = n by MATRIX_0:24;

A5: width M2 = n by MATRIX_0:24;

A6: ( width M1 = n & 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 A4, A6, MATRIX_3:33

.= (M1 * (M2 * M1)) * M2 by A4, A6, A5, MATRIX_3:33

.= (M1 * (M1 * M2)) * M2 by A3, MATRIX_6:def 1

.= ((M1 * M1) * M2) * M2 by A4, A6, MATRIX_3:33

.= (M1 * M2) * M2 by A1

.= M1 * (M2 * M2) by A6, A5, MATRIX_3:33

.= M1 * M2 by A2 ;

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

for M1, M2 being Matrix of n,K st M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 holds

M1 * M2 is Idempotent

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 holds

M1 * M2 is Idempotent

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 implies M1 * M2 is Idempotent )

assume that

A1: M1 is Idempotent and

A2: M2 is Idempotent and

A3: M1 commutes_with M2 ; :: thesis: M1 * M2 is Idempotent

A4: len M1 = n by MATRIX_0:24;

A5: width M2 = n by MATRIX_0:24;

A6: ( width M1 = n & 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 A4, A6, MATRIX_3:33

.= (M1 * (M2 * M1)) * M2 by A4, A6, A5, MATRIX_3:33

.= (M1 * (M1 * M2)) * M2 by A3, MATRIX_6:def 1

.= ((M1 * M1) * M2) * M2 by A4, A6, MATRIX_3:33

.= (M1 * M2) * M2 by A1

.= M1 * (M2 * M2) by A6, A5, MATRIX_3:33

.= M1 * M2 by A2 ;

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