let n be Nat; :: thesis: for K being Field
for M2, M1 being Matrix of n,K st M2 is invertible & M2 * M1 = 1. K,n holds
M1 is_reverse_of M2
let K be Field; :: thesis: for M2, M1 being Matrix of n,K st M2 is invertible & M2 * M1 = 1. K,n holds
M1 is_reverse_of M2
let M2, M1 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_1:25;
A2: ( len M2 = n & width (M2 ~ ) = n ) by MATRIX_1:25;
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_3:21;
then ((M2 ~ ) * M2) * M1 = M2 ~ by A1, A2, MATRIX_3:35;
then (1. K,n) * M1 = M2 ~ by A5, Def2;
then M1 = M2 ~ by MATRIX_3:20;
hence M1 is_reverse_of M2 by A3, Def4; :: thesis: verum