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

for M1, M2 being Matrix of n,K st n > 0 & M1 commutes_with M2 holds

M1 @ commutes_with M2 @

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st n > 0 & M1 commutes_with M2 holds

M1 @ commutes_with M2 @

let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 commutes_with M2 implies M1 @ commutes_with M2 @ )

A1: ( width M1 = n & width M2 = n ) by MATRIX_0:24;

set M3 = M1 @ ;

set M4 = M2 @ ;

A2: len M2 = n by MATRIX_0:24;

assume that

A3: n > 0 and

A4: M1 commutes_with M2 ; :: thesis: M1 @ commutes_with M2 @

len M1 = n by MATRIX_0:24;

then (M1 @) * (M2 @) = (M2 * M1) @ by A1, A3, MATRIX_3:22

.= (M1 * M2) @ by A4

.= (M2 @) * (M1 @) by A1, A2, A3, MATRIX_3:22 ;

hence M1 @ commutes_with M2 @ ; :: thesis: verum

for M1, M2 being Matrix of n,K st n > 0 & M1 commutes_with M2 holds

M1 @ commutes_with M2 @

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st n > 0 & M1 commutes_with M2 holds

M1 @ commutes_with M2 @

let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 commutes_with M2 implies M1 @ commutes_with M2 @ )

A1: ( width M1 = n & width M2 = n ) by MATRIX_0:24;

set M3 = M1 @ ;

set M4 = M2 @ ;

A2: len M2 = n by MATRIX_0:24;

assume that

A3: n > 0 and

A4: M1 commutes_with M2 ; :: thesis: M1 @ commutes_with M2 @

len M1 = n by MATRIX_0:24;

then (M1 @) * (M2 @) = (M2 * M1) @ by A1, A3, MATRIX_3:22

.= (M1 * M2) @ by A4

.= (M2 @) * (M1 @) by A1, A2, A3, MATRIX_3:22 ;

hence M1 @ commutes_with M2 @ ; :: thesis: verum