let n be Element of NAT ; :: thesis: for K being Field
for A being Matrix of n,K holds
( A = 0. K,n iff for i, j being Element of NAT st 1 <= i & i <= n & 1 <= j & j <= n holds
A * i,j = 0. K )
let K be Field; :: thesis: for A being Matrix of n,K holds
( A = 0. K,n iff for i, j being Element of NAT st 1 <= i & i <= n & 1 <= j & j <= n holds
A * i,j = 0. K )
let A be Matrix of n,K; :: thesis: ( A = 0. K,n iff for i, j being Element of NAT st 1 <= i & i <= n & 1 <= j & j <= n holds
A * i,j = 0. K )
thus ( A = 0. K,n implies for i, j being Element of NAT st 1 <= i & i <= n & 1 <= j & j <= n holds
A * i,j = 0. K ) :: thesis: ( ( for i, j being Element of NAT st 1 <= i & i <= n & 1 <= j & j <= n holds
A * i,j = 0. K ) implies A = 0. K,n )

proof

set A2 = 0. K,n;
set B2 = 0. K,n;
reconsider B3 = 0. K,n,n as Matrix of n,K ;
set A3 = 0. K,n,n;
assume A1: A = 0. K,n ; :: thesis: for i, j being Element of NAT st 1 <= i & i <= n & 1 <= j & j <= n holds
A * i,j = 0. K
let i, j be Element of NAT ; :: thesis: ( 1 <= i & i <= n & 1 <= j & j <= n implies A * i,j = 0. K )
assume ( 1 <= i & i <= n & 1 <= j & j <= n ) ; :: thesis: A * i,j = 0. K
then [i,j] in Indices (0. K,n) by MATRIX_1:38;
then ( 0. K,n,n = 0. K,n & ((0. K,n) + (0. K,n)) * i,j = ((0. K,n) * i,j) + ((0. K,n) * i,j) ) by MATRIX_3:def 1, MATRIX_3:def 3;
then (0. K,n) * i,j = ((0. K,n) * i,j) + ((0. K,n) * i,j) by MATRIX_3:6;
then ((0. K,n) * i,j) - ((0. K,n) * i,j) = ((0. K,n) * i,j) + (((0. K,n) * i,j) - ((0. K,n) * i,j)) by RLVECT_1:42
.= ((0. K,n) * i,j) + (0. K) by RLVECT_1:28
.= (0. K,n) * i,j by RLVECT_1:10 ;
hence A * i,j = 0. K by A1, RLVECT_1:28; :: thesis: verum

end;

A2: len A = n by MATRIX_1:25;
A3: Indices A = [:(Seg n),(Seg n):] by MATRIX_1:25;
A4: width A = n by MATRIX_1:25;
assume A5: for i, j being Element of NAT st 1 <= i & i <= n & 1 <= j & j <= n holds
A * i,j = 0. K ; :: thesis: A = 0. K,n
for i, j being Nat st [i,j] in Indices A holds
A * i,j = (A * i,j) + (A * i,j)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices A implies A * i,j = (A * i,j) + (A * i,j) )
reconsider i0 = i, j0 = j as Element of NAT by ORDINAL1:def 13;
assume A6: [i,j] in Indices A ; :: thesis: A * i,j = (A * i,j) + (A * i,j)
then j in Seg n by A4, ZFMISC_1:106;
then A7: ( 1 <= j & j <= n ) by FINSEQ_1:3;
i in Seg n by A3, A6, ZFMISC_1:106;
then ( 1 <= i & i <= n ) by FINSEQ_1:3;
then A * i0,j0 = 0. K by A5, A7;
hence A * i,j = (A * i,j) + (A * i,j) by RLVECT_1:10; :: thesis: verum

end;

then A = A + A by A2, MATRIX_3:def 3;
then A = 0. K,(len A),(width A) by MATRIX_4:6;
hence A = 0. K,n by A2, A4, MATRIX_3:def 1; :: thesis: verum