let n be Ordinal; for L being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for p being Series of n,L
for a being Element of L holds p * a = p *' (a | n,L)
let L be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr ; for p being Series of n,L
for a being Element of L holds p * a = p *' (a | n,L)
let p be Series of n,L; for a being Element of L holds p * a = p *' (a | n,L)
let a be Element of L; p * a = p *' (a | n,L)
A1:
for x being set st x in Bags n holds
(p * a) . x = (p *' (a | n,L)) . x
proof
set O =
a | n,
L;
set cL = the
carrier of
L;
let x be
set ;
( x in Bags n implies (p * a) . x = (p *' (a | n,L)) . x )
assume
x in Bags n
;
(p * a) . x = (p *' (a | n,L)) . x
then reconsider b =
x as
bag of
n ;
A2:
for
b being
Element of
Bags n holds
(p *' (a | n,L)) . b = (p . b) * a
proof
let b be
Element of
Bags n;
(p *' (a | n,L)) . b = (p . b) * a
consider s being
FinSequence of the
carrier of
L such that A3:
(p *' (a | n,L)) . b = Sum s
and A4:
len s = len (decomp b)
and A5:
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 = (p . b1) * ((a | n,L) . b2) )
by POLYNOM1:def 26;
consider t being
FinSequence of the
carrier of
L,
s1 being
Element of the
carrier of
L such that A6:
s = t ^ <*s1*>
by A4, FINSEQ_2:22;
A7:
now per cases
( t = <*> the carrier of L or t <> <*> the carrier of L )
;
suppose A8:
t <> <*> the
carrier of
L
;
Sum t = 0. Lnow let k be
Nat;
( k in dom t implies t /. b1 = 0. L )A9:
len s =
(len t) + (len <*s1*>)
by A6, FINSEQ_1:35
.=
(len t) + 1
by FINSEQ_1:56
;
assume A10:
k in dom t
;
t /. b1 = 0. Lthen A11:
t /. k =
t . k
by PARTFUN1:def 8
.=
s . k
by A6, A10, FINSEQ_1:def 7
;
k <= len t
by A10, FINSEQ_3:27;
then A12:
k < len s
by A9, NAT_1:13;
A13:
1
<= k
by A10, FINSEQ_3:27;
then A14:
k in dom (decomp b)
by A4, A12, FINSEQ_3:27;
A15:
dom s = dom (decomp b)
by A4, FINSEQ_3:31;
then A16:
s /. k = s . k
by A14, PARTFUN1:def 8;
per cases
( 1 < k or k = 1 )
by A13, XXREAL_0:1;
suppose A17:
1
< k
;
t /. b1 = 0. Lreconsider k1 =
k as
Element of
NAT by ORDINAL1:def 13;
consider b1,
b2 being
bag of
n such that A18:
(decomp b) /. k1 = <*b1,b2*>
and A19:
s /. k1 = (p . b1) * ((a | n,L) . b2)
by A5, A15, A14;
b2 <> EmptyBag n
by A4, A12, A17, A18, PRE_POLY:72;
hence t /. k =
(p . b1) * (0. L)
by A11, A16, A19, Th18
.=
0. L
by VECTSP_1:36
;
verum end; suppose A20:
k = 1
;
t /. b1 = 0. Lconsider b1,
b2 being
bag of
n such that A22:
(decomp b) /. k = <*b1,b2*>
and A23:
s /. k = (p . b1) * ((a | n,L) . b2)
by A5, A15, A14;
(decomp b) /. 1
= <*(EmptyBag n),b*>
by PRE_POLY:71;
then
(
b1 = EmptyBag n &
b2 = b )
by A20, A22, FINSEQ_1:98;
then s . k =
(p . (EmptyBag n)) * (0. L)
by A16, A23, A21, Th18
.=
0. L
by VECTSP_1:36
;
hence
t /. k = 0. L
by A11;
verum end; end; hence
Sum t = 0. L
by MATRLIN:15;
verum
A24:
s . (len s) =
(t ^ <*s1*>) . ((len t) + 1)
by A6, FINSEQ_2:19
.=
s1
by FINSEQ_1:59
;
A25:
Sum s = (Sum t) + (Sum <*s1*>)
by A6, RLVECT_1:58;
not
s is
empty
by A4;
then A26:
len s in dom s
by FINSEQ_5:6;
then consider b1,
b2 being
bag of
n such that A27:
(decomp b) /. (len s) = <*b1,b2*>
and A28:
s /. (len s) = (p . b1) * ((a | n,L) . b2)
by A5;
A29:
s /. (len s) = s . (len s)
by A26, PARTFUN1:def 8;
(decomp b) /. (len s) = <*b,(EmptyBag n)*>
by A4, PRE_POLY:71;
then A30:
(
b1 = b &
b2 = EmptyBag n )
by A27, FINSEQ_1:98;
Sum <*s1*> =
s1
by RLVECT_1:61
.=
(p . b) * a
by A28, A30, A29, A24, Th18
;
hence
(p *' (a | n,L)) . b = (p . b) * a
by A3, A25, A7, RLVECT_1:10;
verum
end;
b is
Element of
Bags n
by PRE_POLY:def 12;
then (p *' (a | n,L)) . b =
(p . b) * a
by A2
.=
(p * a) . b
by Def11
;
hence
(p * a) . x = (p *' (a | n,L)) . x
;
verum
end;
( Bags n = dom (p * a) & Bags n = dom (p *' (a | n,L)) )
by FUNCT_2:def 1;
hence
p * a = p *' (a | n,L)
by A1, FUNCT_1:9; verum