let n be Nat; :: thesis: for K being Ring
for M1, M2 being Matrix of n,K st M2 is invertible & M2 * M1 = 1. (K,n) holds
M1 is_reverse_of M2
let K be Ring; :: thesis: for M1, M2 being Matrix of n,K st M2 is invertible & M2 * M1 = 1. (K,n) holds
M1 is_reverse_of M2
let M1, M2 be Matrix of n,K; :: thesis: ( M2 is invertible & M2 * M1 = 1. (K,n) implies M1 is_reverse_of M2 )
A1: ( len M1 = n & width M2 = n ) by MATRIX_0:24;
A2: ( len M2 = n & width (M2 ~) = n ) by MATRIX_0:24;
assume that
A3: M2 is invertible and
A4: M2 * M1 = 1. (K,n) ; :: thesis: M1 is_reverse_of M2
A5: M2 ~ is_reverse_of M2 by A3, Def4;
(M2 ~) * (M2 * M1) = M2 ~ by A4, MATRIX_3:19;
then ((M2 ~) * M2) * M1 = M2 ~ by A1, A2, MATRIX_3:33;
then (1. (K,n)) * M1 = M2 ~ by A5;
then M1 = M2 ~ by MATRIX_3:18;
hence M1 is_reverse_of M2 by A3, Def4; :: thesis: verum