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

for M1, M2 being Matrix of n,K st M1 is invertible & M2 * M1 = 0. (K,n) holds

M2 = 0. (K,n)

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is invertible & M2 * M1 = 0. (K,n) holds

M2 = 0. (K,n)

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is invertible & M2 * M1 = 0. (K,n) implies M2 = 0. (K,n) )

assume that

A1: M1 is invertible and

A2: M2 * M1 = 0. (K,n) ; :: thesis: M2 = 0. (K,n)

A3: M1 ~ is_reverse_of M1 by A1, MATRIX_6:def 4;

A4: width M2 = n by MATRIX_0:24;

A5: ( width M1 = n & len M1 = n ) by MATRIX_0:24;

A6: width (M1 ~) = n by MATRIX_0:24;

A7: len (M1 ~) = n by MATRIX_0:24;

M2 = M2 * (1. (K,n)) by MATRIX_3:19

.= M2 * (M1 * (M1 ~)) by A3, MATRIX_6:def 2

.= (M2 * M1) * (M1 ~) by A5, A4, A7, MATRIX_3:33

.= 0. (K,n,n) by A2, A6, A7, MATRIX_6:1

.= 0. (K,n) ;

hence M2 = 0. (K,n) ; :: thesis: verum

for M1, M2 being Matrix of n,K st M1 is invertible & M2 * M1 = 0. (K,n) holds

M2 = 0. (K,n)

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is invertible & M2 * M1 = 0. (K,n) holds

M2 = 0. (K,n)

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is invertible & M2 * M1 = 0. (K,n) implies M2 = 0. (K,n) )

assume that

A1: M1 is invertible and

A2: M2 * M1 = 0. (K,n) ; :: thesis: M2 = 0. (K,n)

A3: M1 ~ is_reverse_of M1 by A1, MATRIX_6:def 4;

A4: width M2 = n by MATRIX_0:24;

A5: ( width M1 = n & len M1 = n ) by MATRIX_0:24;

A6: width (M1 ~) = n by MATRIX_0:24;

A7: len (M1 ~) = n by MATRIX_0:24;

M2 = M2 * (1. (K,n)) by MATRIX_3:19

.= M2 * (M1 * (M1 ~)) by A3, MATRIX_6:def 2

.= (M2 * M1) * (M1 ~) by A5, A4, A7, MATRIX_3:33

.= 0. (K,n,n) by A2, A6, A7, MATRIX_6:1

.= 0. (K,n) ;

hence M2 = 0. (K,n) ; :: thesis: verum