let n be Ordinal; for L being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr
for p being Series of n,L
for a being Element of L holds a * p = (a | (n,L)) *' p
let L be non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr ; for p being Series of n,L
for a being Element of L holds a * p = (a | (n,L)) *' p
let p be Series of n,L; for a being Element of L holds a * p = (a | (n,L)) *' p
let a be Element of L; a * p = (a | (n,L)) *' p
for x being object st x in Bags n holds
(a * p) . x = ((a | (n,L)) *' p) . x
proof
set O =
a | (
n,
L);
set cL = the
carrier of
L;
let x be
object ;
( x in Bags n implies (a * p) . x = ((a | (n,L)) *' p) . x )
assume
x in Bags n
;
(a * p) . x = ((a | (n,L)) *' p) . x
then reconsider b =
x as
bag of
n ;
A1:
for
b being
Element of
Bags n holds
((a | (n,L)) *' p) . b = a * (p . b)
proof
let b be
Element of
Bags n;
((a | (n,L)) *' p) . b = a * (p . b)
consider s being
FinSequence of the
carrier of
L such that A2:
((a | (n,L)) *' p) . b = Sum s
and A3:
len s = len (decomp b)
and A4:
for
k being
Element of
NAT st
k in dom s holds
ex
b1,
b2 being
bag of
n st
(
(decomp b) /. k = <*b1,b2*> &
s /. k = ((a | (n,L)) . b1) * (p . b2) )
by POLYNOM1:def 10;
not
s is
empty
by A3;
then consider s1 being
Element of the
carrier of
L,
t being
FinSequence of the
carrier of
L such that A5:
s1 = s . 1
and A6:
s = <*s1*> ^ t
by FINSEQ_3:102;
A7:
Sum s = (Sum <*s1*>) + (Sum t)
by A6, RLVECT_1:41;
A8:
now Sum t = 0. Lper cases
( t = <*> the carrier of L or t <> <*> the carrier of L )
;
suppose A9:
t <> <*> the
carrier of
L
;
Sum t = 0. Lnow for k being Nat st k in dom t holds
t /. k = 0. Llet k be
Nat;
( k in dom t implies t /. b1 = 0. L )A10:
len s =
(len t) + (len <*s1*>)
by A6, FINSEQ_1:22
.=
(len t) + 1
by FINSEQ_1:39
;
assume A11:
k in dom t
;
t /. b1 = 0. Lthen A12:
t /. k =
t . k
by PARTFUN1:def 6
.=
s . (k + 1)
by A6, A11, FINSEQ_3:103
;
1
<= k
by A11, FINSEQ_3:25;
then A13:
1
< k + 1
by NAT_1:13;
k <= len t
by A11, FINSEQ_3:25;
then A14:
k + 1
<= len s
by A10, XREAL_1:6;
then A15:
k + 1
in dom (decomp b)
by A3, A13, FINSEQ_3:25;
A16:
dom s = dom (decomp b)
by A3, FINSEQ_3:29;
then A17:
s /. (k + 1) = s . (k + 1)
by A15, PARTFUN1:def 6;
per cases
( k + 1 < len s or k + 1 = len s )
by A14, XXREAL_0:1;
suppose A18:
k + 1
< len s
;
t /. b1 = 0. Lreconsider k1 =
k as
Element of
NAT by ORDINAL1:def 12;
consider b1,
b2 being
bag of
n such that A19:
(decomp b) /. (k1 + 1) = <*b1,b2*>
and A20:
s /. (k1 + 1) = ((a | (n,L)) . b1) * (p . b2)
by A4, A16, A15;
b1 <> EmptyBag n
by A3, A13, A18, A19, PRE_POLY:72;
hence t /. k =
(0. L) * (p . b2)
by A12, A17, A20, Th18
.=
0. L
;
verum end; suppose A21:
k + 1
= len s
;
t /. b1 = 0. Lconsider b1,
b2 being
bag of
n such that A23:
(decomp b) /. (k + 1) = <*b1,b2*>
and A24:
s /. (k + 1) = ((a | (n,L)) . b1) * (p . b2)
by A4, A16, A15;
(decomp b) /. (len s) = <*b,(EmptyBag n)*>
by A3, PRE_POLY:71;
then
(
b2 = EmptyBag n &
b1 = b )
by A21, A23, FINSEQ_1:77;
then s . (k + 1) =
(0. L) * (p . (EmptyBag n))
by A17, A24, A22, Th18
.=
0. L
;
hence
t /. k = 0. L
by A12;
verum end; end; hence
Sum t = 0. L
by MATRLIN:11;
verum
A25:
not
s is
empty
by A3;
then consider b1,
b2 being
bag of
n such that A26:
(decomp b) /. 1
= <*b1,b2*>
and A27:
s /. 1
= ((a | (n,L)) . b1) * (p . b2)
by A4, FINSEQ_5:6;
1
in dom s
by A25, FINSEQ_5:6;
then A28:
s /. 1
= s . 1
by PARTFUN1:def 6;
(decomp b) /. 1
= <*(EmptyBag n),b*>
by PRE_POLY:71;
then A29:
(
b2 = b &
b1 = EmptyBag n )
by A26, FINSEQ_1:77;
Sum <*s1*> =
s1
by RLVECT_1:44
.=
a * (p . b)
by A5, A27, A29, A28, Th18
;
hence
((a | (n,L)) *' p) . b = a * (p . b)
by A2, A7, A8, RLVECT_1:4;
verum
end;
b is
Element of
Bags n
by PRE_POLY:def 12;
then ((a | (n,L)) *' p) . b =
a * (p . b)
by A1
.=
(a * p) . b
by Def9
;
hence
(a * p) . x = ((a | (n,L)) *' p) . x
;
verum
end;
hence
a * p = (a | (n,L)) *' p
; verum