let n be Nat; for K being Field
for p, q being FinSequence of K
for M1, M2 being Matrix of n,K st M1 is_symmetry_circulant_about p & M2 is_symmetry_circulant_about q holds
M1 + M2 is_symmetry_circulant_about p + q
let K be Field; for p, q being FinSequence of K
for M1, M2 being Matrix of n,K st M1 is_symmetry_circulant_about p & M2 is_symmetry_circulant_about q holds
M1 + M2 is_symmetry_circulant_about p + q
let p, q be FinSequence of K; for M1, M2 being Matrix of n,K st M1 is_symmetry_circulant_about p & M2 is_symmetry_circulant_about q holds
M1 + M2 is_symmetry_circulant_about p + q
let M1, M2 be Matrix of n,K; ( M1 is_symmetry_circulant_about p & M2 is_symmetry_circulant_about q implies M1 + M2 is_symmetry_circulant_about p + q )
assume A1:
M1 is_symmetry_circulant_about p
; ( not M2 is_symmetry_circulant_about q or M1 + M2 is_symmetry_circulant_about p + q )
A2:
len p = width M1
by A1;
A3:
Indices M2 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A4:
Indices (M1 + M2) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A5:
width M1 = n
by MATRIX_0:24;
then A6:
dom p = Seg n
by A2, FINSEQ_1:def 3;
assume A7:
M2 is_symmetry_circulant_about q
; M1 + M2 is_symmetry_circulant_about p + q
then A8:
len q = width M2
;
A9:
Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A10:
n in NAT
by ORDINAL1:def 12;
A11:
width M2 = n
by MATRIX_0:24;
then
dom q = Seg n
by A8, FINSEQ_1:def 3;
then A12:
dom (p + q) = dom p
by A6, POLYNOM1:1;
then A13:
len (p + q) = n
by A6, A10, FINSEQ_1:def 3;
A14:
width (M1 + M2) = n
by MATRIX_0:24;
A15:
dom (p + q) = Seg (len (p + q))
by FINSEQ_1:def 3;
A16:
for i, j being Nat st [i,j] in Indices (M1 + M2) & i + j <> (len (p + q)) + 1 holds
(M1 + M2) * (i,j) = (p + q) . (((i + j) - 1) mod (len (p + q)))
proof
let i,
j be
Nat;
( [i,j] in Indices (M1 + M2) & i + j <> (len (p + q)) + 1 implies (M1 + M2) * (i,j) = (p + q) . (((i + j) - 1) mod (len (p + q))) )
assume that A17:
[i,j] in Indices (M1 + M2)
and A18:
i + j <> (len (p + q)) + 1
;
(M1 + M2) * (i,j) = (p + q) . (((i + j) - 1) mod (len (p + q)))
A19:
((i + j) - 1) mod (len (p + q)) in dom (p + q)
by A4, A15, A18, A17, A12, A6, Lm4;
(M1 + M2) * (
i,
j) =
(M1 * (i,j)) + (M2 * (i,j))
by A9, A4, A17, MATRIX_3:def 3
.=
the
addF of
K . (
(M1 * (i,j)),
(q . (((i + j) - 1) mod (len (p + q)))))
by A7, A3, A4, A11, A17, A18, A13
.=
the
addF of
K . (
(p . (((i + j) - 1) mod (len (p + q)))),
(q . (((i + j) - 1) mod (len (p + q)))))
by A1, A5, A9, A4, A13, A17, A18
.=
(p + q) . (((i + j) - 1) mod (len (p + q)))
by A19, FUNCOP_1:22
;
hence
(M1 + M2) * (
i,
j)
= (p + q) . (((i + j) - 1) mod (len (p + q)))
;
verum
end;
for i, j being Nat st [i,j] in Indices (M1 + M2) & i + j = (len (p + q)) + 1 holds
(M1 + M2) * (i,j) = (p + q) . (len (p + q))
proof
let i,
j be
Nat;
( [i,j] in Indices (M1 + M2) & i + j = (len (p + q)) + 1 implies (M1 + M2) * (i,j) = (p + q) . (len (p + q)) )
assume that A20:
[i,j] in Indices (M1 + M2)
and A21:
i + j = (len (p + q)) + 1
;
(M1 + M2) * (i,j) = (p + q) . (len (p + q))
(
i in Seg n &
j in Seg n )
by A4, A20, ZFMISC_1:87;
then
( 1
<= i & 1
<= j )
by FINSEQ_1:1;
then
1
+ 1
<= i + j
by XREAL_1:7;
then
((len (p + q)) + 1) - 1
>= (1 + 1) - 1
by A21, XREAL_1:9;
then
len (p + q) in Seg (len (p + q))
;
then A22:
len (p + q) in dom (p + q)
by FINSEQ_1:def 3;
(M1 + M2) * (
i,
j) =
(M1 * (i,j)) + (M2 * (i,j))
by A9, A4, A20, MATRIX_3:def 3
.=
the
addF of
K . (
(M1 * (i,j)),
(q . (len (p + q))))
by A7, A11, A13, A3, A4, A20, A21
.=
the
addF of
K . (
(p . (len (p + q))),
(q . (len (p + q))))
by A1, A9, A5, A4, A13, A20, A21
.=
(p + q) . (len (p + q))
by A22, FUNCOP_1:22
;
hence
(M1 + M2) * (
i,
j)
= (p + q) . (len (p + q))
;
verum
end;
hence
M1 + M2 is_symmetry_circulant_about p + q
by A14, A13, A16; verum