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Theorem coe1mul2 18097
Description: The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Hypotheses
Ref Expression
coe1mul2.s  |-  S  =  (PwSer1 `  R )
coe1mul2.t  |-  .xb  =  ( .r `  S )
coe1mul2.u  |-  .x.  =  ( .r `  R )
coe1mul2.b  |-  B  =  ( Base `  S
)
Assertion
Ref Expression
coe1mul2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( k  e.  NN0  |->  ( R 
gsumg  ( x  e.  (
0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) ) ) )
Distinct variable groups:    x, k, B    k, F, x    .x. , k, x    k, G, x    R, k, x    .xb , k
Allowed substitution hints:    S( x, k)    .xb (
x)

Proof of Theorem coe1mul2
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fconst6g 5773 . . . . 5  |-  ( k  e.  NN0  ->  ( 1o 
X.  { k } ) : 1o --> NN0 )
2 nn0ex 10800 . . . . . 6  |-  NN0  e.  _V
3 1on 7137 . . . . . . 7  |-  1o  e.  On
43elexi 3123 . . . . . 6  |-  1o  e.  _V
52, 4elmap 7447 . . . . 5  |-  ( ( 1o  X.  { k } )  e.  ( NN0  ^m  1o )  <-> 
( 1o  X.  {
k } ) : 1o --> NN0 )
61, 5sylibr 212 . . . 4  |-  ( k  e.  NN0  ->  ( 1o 
X.  { k } )  e.  ( NN0 
^m  1o ) )
76adantl 466 . . 3  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( 1o  X.  { k } )  e.  ( NN0  ^m  1o ) )
8 eqidd 2468 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (
k  e.  NN0  |->  ( 1o 
X.  { k } ) )  =  ( k  e.  NN0  |->  ( 1o 
X.  { k } ) ) )
9 eqid 2467 . . . 4  |-  ( 1o mPwSer  R )  =  ( 1o mPwSer  R )
10 coe1mul2.s . . . . 5  |-  S  =  (PwSer1 `  R )
11 coe1mul2.b . . . . 5  |-  B  =  ( Base `  S
)
1210, 11, 9psr1bas2 18016 . . . 4  |-  B  =  ( Base `  ( 1o mPwSer  R ) )
13 coe1mul2.u . . . 4  |-  .x.  =  ( .r `  R )
14 coe1mul2.t . . . . 5  |-  .xb  =  ( .r `  S )
1510, 9, 14psr1mulr 18052 . . . 4  |-  .xb  =  ( .r `  ( 1o mPwSer  R ) )
16 psr1baslem 18011 . . . 4  |-  ( NN0 
^m  1o )  =  { a  e.  ( NN0  ^m  1o )  |  ( `' a
" NN )  e. 
Fin }
17 simp2 997 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  F  e.  B )
18 simp3 998 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  G  e.  B )
199, 12, 13, 15, 16, 17, 18psrmulfval 17825 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .xb  G )  =  ( b  e.  ( NN0  ^m  1o ) 
|->  ( R  gsumg  ( c  e.  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
b }  |->  ( ( F `  c ) 
.x.  ( G `  ( b  oF  -  c ) ) ) ) ) ) )
20 breq2 4451 . . . . . 6  |-  ( b  =  ( 1o  X.  { k } )  ->  ( d  oR  <_  b  <->  d  oR  <_  ( 1o  X.  { k } ) ) )
2120rabbidv 3105 . . . . 5  |-  ( b  =  ( 1o  X.  { k } )  ->  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  b }  =  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )
22 oveq1 6290 . . . . . . 7  |-  ( b  =  ( 1o  X.  { k } )  ->  ( b  oF  -  c )  =  ( ( 1o 
X.  { k } )  oF  -  c ) )
2322fveq2d 5869 . . . . . 6  |-  ( b  =  ( 1o  X.  { k } )  ->  ( G `  ( b  oF  -  c ) )  =  ( G `  ( ( 1o  X.  { k } )  oF  -  c
) ) )
2423oveq2d 6299 . . . . 5  |-  ( b  =  ( 1o  X.  { k } )  ->  ( ( F `
 c )  .x.  ( G `  ( b  oF  -  c
) ) )  =  ( ( F `  c )  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) ) ) )
2521, 24mpteq12dv 4525 . . . 4  |-  ( b  =  ( 1o  X.  { k } )  ->  ( c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  b }  |->  ( ( F `  c
)  .x.  ( G `  ( b  oF  -  c ) ) ) )  =  ( c  e.  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o 
X.  { k } ) }  |->  ( ( F `  c ) 
.x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c
) ) ) ) )
2625oveq2d 6299 . . 3  |-  ( b  =  ( 1o  X.  { k } )  ->  ( R  gsumg  ( c  e.  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  b }  |->  ( ( F `  c
)  .x.  ( G `  ( b  oF  -  c ) ) ) ) )  =  ( R  gsumg  ( c  e.  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) } 
|->  ( ( F `  c )  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) ) ) ) ) )
277, 8, 19, 26fmptco 6053 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (
( F  .xb  G
)  o.  ( k  e.  NN0  |->  ( 1o 
X.  { k } ) ) )  =  ( k  e.  NN0  |->  ( R  gsumg  ( c  e.  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) } 
|->  ( ( F `  c )  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) ) ) ) ) ) )
2810psr1rng 18075 . . . 4  |-  ( R  e.  Ring  ->  S  e. 
Ring )
2911, 14rngcl 17008 . . . 4  |-  ( ( S  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .xb  G )  e.  B )
3028, 29syl3an1 1261 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .xb  G )  e.  B )
31 eqid 2467 . . . 4  |-  (coe1 `  ( F  .xb  G ) )  =  (coe1 `  ( F  .xb  G ) )
32 eqid 2467 . . . 4  |-  ( k  e.  NN0  |->  ( 1o 
X.  { k } ) )  =  ( k  e.  NN0  |->  ( 1o 
X.  { k } ) )
3331, 11, 10, 32coe1fval3 18034 . . 3  |-  ( ( F  .xb  G )  e.  B  ->  (coe1 `  ( F  .xb  G ) )  =  ( ( F 
.xb  G )  o.  ( k  e.  NN0  |->  ( 1o  X.  { k } ) ) ) )
3430, 33syl 16 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( ( F  .xb  G )  o.  ( k  e.  NN0  |->  ( 1o  X.  { k } ) ) ) )
35 eqid 2467 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
36 eqid 2467 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
37 simpl1 999 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  R  e.  Ring )
38 rngcmn 17025 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. CMnd
)
3937, 38syl 16 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  R  e. CMnd )
40 fzfi 12049 . . . . . 6  |-  ( 0 ... k )  e. 
Fin
4140a1i 11 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( 0 ... k )  e.  Fin )
42 simpll1 1035 . . . . . . 7  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  R  e.  Ring )
43 simpll2 1036 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  F  e.  B )
44 eqid 2467 . . . . . . . . . 10  |-  (coe1 `  F
)  =  (coe1 `  F
)
4544, 11, 10, 35coe1f2 18035 . . . . . . . . 9  |-  ( F  e.  B  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
4643, 45syl 16 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
47 elfznn0 11769 . . . . . . . . 9  |-  ( x  e.  ( 0 ... k )  ->  x  e.  NN0 )
4847adantl 466 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  x  e.  NN0 )
4946, 48ffvelrnd 6021 . . . . . . 7  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  (
(coe1 `  F ) `  x )  e.  (
Base `  R )
)
50 simpll3 1037 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  G  e.  B )
51 eqid 2467 . . . . . . . . . 10  |-  (coe1 `  G
)  =  (coe1 `  G
)
5251, 11, 10, 35coe1f2 18035 . . . . . . . . 9  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
5350, 52syl 16 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
54 fznn0sub 11715 . . . . . . . . 9  |-  ( x  e.  ( 0 ... k )  ->  (
k  -  x )  e.  NN0 )
5554adantl 466 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  (
k  -  x )  e.  NN0 )
5653, 55ffvelrnd 6021 . . . . . . 7  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  (
(coe1 `  G ) `  ( k  -  x
) )  e.  (
Base `  R )
)
5735, 13rngcl 17008 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
(coe1 `  F ) `  x )  e.  (
Base `  R )  /\  ( (coe1 `  G ) `  ( k  -  x
) )  e.  (
Base `  R )
)  ->  ( (
(coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) )  e.  ( Base `  R
) )
5842, 49, 56, 57syl3anc 1228 . . . . . 6  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  (
( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) )  e.  ( Base `  R
) )
59 eqid 2467 . . . . . 6  |-  ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )  =  ( x  e.  ( 0 ... k
)  |->  ( ( (coe1 `  F ) `  x
)  .x.  ( (coe1 `  G ) `  (
k  -  x ) ) ) )
6058, 59fmptd 6044 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( x  e.  ( 0 ... k
)  |->  ( ( (coe1 `  F ) `  x
)  .x.  ( (coe1 `  G ) `  (
k  -  x ) ) ) ) : ( 0 ... k
) --> ( Base `  R
) )
6140elexi 3123 . . . . . . . . 9  |-  ( 0 ... k )  e. 
_V
6261mptex 6130 . . . . . . . 8  |-  ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )  e.  _V
63 funmpt 5623 . . . . . . . 8  |-  Fun  (
x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )
64 fvex 5875 . . . . . . . 8  |-  ( 0g
`  R )  e. 
_V
6562, 63, 643pm3.2i 1174 . . . . . . 7  |-  ( ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )  e.  _V  /\  Fun  ( x  e.  (
0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )  /\  ( 0g `  R )  e.  _V )
66 suppssdm 6914 . . . . . . . . 9  |-  ( ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) supp  ( 0g `  R
) )  C_  dom  ( x  e.  (
0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )
6759dmmptss 5502 . . . . . . . . 9  |-  dom  (
x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) 
C_  ( 0 ... k )
6866, 67sstri 3513 . . . . . . . 8  |-  ( ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) supp  ( 0g `  R
) )  C_  (
0 ... k )
6940, 68pm3.2i 455 . . . . . . 7  |-  ( ( 0 ... k )  e.  Fin  /\  (
( x  e.  ( 0 ... k ) 
|->  ( ( (coe1 `  F
) `  x )  .x.  ( (coe1 `  G ) `  ( k  -  x
) ) ) ) supp  ( 0g `  R
) )  C_  (
0 ... k ) )
70 suppssfifsupp 7843 . . . . . . 7  |-  ( ( ( ( x  e.  ( 0 ... k
)  |->  ( ( (coe1 `  F ) `  x
)  .x.  ( (coe1 `  G ) `  (
k  -  x ) ) ) )  e. 
_V  /\  Fun  ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )  /\  ( 0g `  R )  e.  _V )  /\  ( ( 0 ... k )  e. 
Fin  /\  ( (
x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) supp  ( 0g `  R
) )  C_  (
0 ... k ) ) )  ->  ( x  e.  ( 0 ... k
)  |->  ( ( (coe1 `  F ) `  x
)  .x.  ( (coe1 `  G ) `  (
k  -  x ) ) ) ) finSupp  ( 0g `  R ) )
7165, 69, 70mp2an 672 . . . . . 6  |-  ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) finSupp 
( 0g `  R
)
7271a1i 11 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( x  e.  ( 0 ... k
)  |->  ( ( (coe1 `  F ) `  x
)  .x.  ( (coe1 `  G ) `  (
k  -  x ) ) ) ) finSupp  ( 0g `  R ) )
73 eqid 2467 . . . . . . 7  |-  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o 
X.  { k } ) }  =  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) }
7473coe1mul2lem2 18096 . . . . . 6  |-  ( k  e.  NN0  ->  ( c  e.  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  |->  ( c `  (/) ) ) : {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) } -1-1-onto-> ( 0 ... k ) )
7574adantl 466 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  |->  ( c `  (/) ) ) : {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) } -1-1-onto-> ( 0 ... k ) )
7635, 36, 39, 41, 60, 72, 75gsumf1o 16724 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( R  gsumg  ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) )  =  ( R 
gsumg  ( ( x  e.  ( 0 ... k
)  |->  ( ( (coe1 `  F ) `  x
)  .x.  ( (coe1 `  G ) `  (
k  -  x ) ) ) )  o.  ( c  e.  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) } 
|->  ( c `  (/) ) ) ) ) )
77 breq1 4450 . . . . . . . . . . 11  |-  ( d  =  c  ->  (
d  oR  <_ 
( 1o  X.  {
k } )  <->  c  oR  <_  ( 1o  X.  { k } ) ) )
7877elrab 3261 . . . . . . . . . 10  |-  ( c  e.  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  <->  ( c  e.  ( NN0  ^m  1o )  /\  c  oR  <_  ( 1o  X.  { k } ) ) )
7978simprbi 464 . . . . . . . . 9  |-  ( c  e.  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  ->  c  oR  <_  ( 1o  X.  { k } ) )
8079adantl 466 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  c  oR  <_  ( 1o 
X.  { k } ) )
81 simplr 754 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  k  e.  NN0 )
82 elrabi 3258 . . . . . . . . . 10  |-  ( c  e.  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  ->  c  e.  ( NN0  ^m  1o ) )
8382adantl 466 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  c  e.  ( NN0  ^m  1o ) )
84 coe1mul2lem1 18095 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  c  e.  ( NN0  ^m  1o ) )  -> 
( c  oR  <_  ( 1o  X.  { k } )  <-> 
( c `  (/) )  e.  ( 0 ... k
) ) )
8581, 83, 84syl2anc 661 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  (
c  oR  <_ 
( 1o  X.  {
k } )  <->  ( c `  (/) )  e.  ( 0 ... k ) ) )
8680, 85mpbid 210 . . . . . . 7  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  (
c `  (/) )  e.  ( 0 ... k
) )
87 eqidd 2468 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  |->  ( c `  (/) ) )  =  ( c  e.  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o 
X.  { k } ) }  |->  ( c `
 (/) ) ) )
88 eqidd 2468 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( x  e.  ( 0 ... k
)  |->  ( ( (coe1 `  F ) `  x
)  .x.  ( (coe1 `  G ) `  (
k  -  x ) ) ) )  =  ( x  e.  ( 0 ... k ) 
|->  ( ( (coe1 `  F
) `  x )  .x.  ( (coe1 `  G ) `  ( k  -  x
) ) ) ) )
89 fveq2 5865 . . . . . . . 8  |-  ( x  =  ( c `  (/) )  ->  ( (coe1 `  F ) `  x
)  =  ( (coe1 `  F ) `  (
c `  (/) ) ) )
90 oveq2 6291 . . . . . . . . 9  |-  ( x  =  ( c `  (/) )  ->  ( k  -  x )  =  ( k  -  ( c `
 (/) ) ) )
9190fveq2d 5869 . . . . . . . 8  |-  ( x  =  ( c `  (/) )  ->  ( (coe1 `  G ) `  (
k  -  x ) )  =  ( (coe1 `  G ) `  (
k  -  ( c `
 (/) ) ) ) )
9289, 91oveq12d 6301 . . . . . . 7  |-  ( x  =  ( c `  (/) )  ->  ( (
(coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) )  =  ( ( (coe1 `  F
) `  ( c `  (/) ) )  .x.  ( (coe1 `  G ) `  ( k  -  (
c `  (/) ) ) ) ) )
9386, 87, 88, 92fmptco 6053 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )  o.  ( c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  |->  ( c `  (/) ) ) )  =  ( c  e.  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) } 
|->  ( ( (coe1 `  F
) `  ( c `  (/) ) )  .x.  ( (coe1 `  G ) `  ( k  -  (
c `  (/) ) ) ) ) ) )
94 simpll2 1036 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  F  e.  B )
9544fvcoe1 18033 . . . . . . . . 9  |-  ( ( F  e.  B  /\  c  e.  ( NN0  ^m  1o ) )  -> 
( F `  c
)  =  ( (coe1 `  F ) `  (
c `  (/) ) ) )
9694, 83, 95syl2anc 661 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  ( F `  c )  =  ( (coe1 `  F
) `  ( c `  (/) ) ) )
97 df1o2 7142 . . . . . . . . . . . . . 14  |-  1o  =  { (/) }
98 0ex 4577 . . . . . . . . . . . . . 14  |-  (/)  e.  _V
9997, 2, 98mapsnconst 7464 . . . . . . . . . . . . 13  |-  ( c  e.  ( NN0  ^m  1o )  ->  c  =  ( 1o  X.  {
( c `  (/) ) } ) )
10083, 99syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  c  =  ( 1o  X.  { ( c `  (/) ) } ) )
101100oveq2d 6299 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  (
( 1o  X.  {
k } )  oF  -  c )  =  ( ( 1o 
X.  { k } )  oF  -  ( 1o  X.  { ( c `  (/) ) } ) ) )
1023a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  1o  e.  On )
103 vex 3116 . . . . . . . . . . . . 13  |-  k  e. 
_V
104103a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  k  e.  _V )
105 fvex 5875 . . . . . . . . . . . . 13  |-  ( c `
 (/) )  e.  _V
106105a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  (
c `  (/) )  e. 
_V )
107102, 104, 106ofc12 6548 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  (
( 1o  X.  {
k } )  oF  -  ( 1o 
X.  { ( c `
 (/) ) } ) )  =  ( 1o 
X.  { ( k  -  ( c `  (/) ) ) } ) )
108101, 107eqtrd 2508 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  (
( 1o  X.  {
k } )  oF  -  c )  =  ( 1o  X.  { ( k  -  ( c `  (/) ) ) } ) )
109108fveq2d 5869 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) )  =  ( G `  ( 1o  X.  { ( k  -  ( c `  (/) ) ) } ) ) )
110 simpll3 1037 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  G  e.  B )
111 fznn0sub 11715 . . . . . . . . . . 11  |-  ( ( c `  (/) )  e.  ( 0 ... k
)  ->  ( k  -  ( c `  (/) ) )  e.  NN0 )
11286, 111syl 16 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  (
k  -  ( c `
 (/) ) )  e. 
NN0 )
11351coe1fv 18032 . . . . . . . . . 10  |-  ( ( G  e.  B  /\  ( k  -  (
c `  (/) ) )  e.  NN0 )  -> 
( (coe1 `  G ) `  ( k  -  (
c `  (/) ) ) )  =  ( G `
 ( 1o  X.  { ( k  -  ( c `  (/) ) ) } ) ) )
114110, 112, 113syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  (
(coe1 `  G ) `  ( k  -  (
c `  (/) ) ) )  =  ( G `
 ( 1o  X.  { ( k  -  ( c `  (/) ) ) } ) ) )
115109, 114eqtr4d 2511 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) )  =  ( (coe1 `  G ) `  ( k  -  (
c `  (/) ) ) ) )
11696, 115oveq12d 6301 . . . . . . 7  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) } )  ->  (
( F `  c
)  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c
) ) )  =  ( ( (coe1 `  F
) `  ( c `  (/) ) )  .x.  ( (coe1 `  G ) `  ( k  -  (
c `  (/) ) ) ) ) )
117116mpteq2dva 4533 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  |->  ( ( F `
 c )  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) ) ) )  =  ( c  e.  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o 
X.  { k } ) }  |->  ( ( (coe1 `  F ) `  ( c `  (/) ) ) 
.x.  ( (coe1 `  G
) `  ( k  -  ( c `  (/) ) ) ) ) ) )
11893, 117eqtr4d 2511 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )  o.  ( c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  |->  ( c `  (/) ) ) )  =  ( c  e.  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) } 
|->  ( ( F `  c )  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) ) ) ) )
119118oveq2d 6299 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( R  gsumg  ( ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )  o.  ( c  e. 
{ d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  |->  ( c `  (/) ) ) ) )  =  ( R  gsumg  ( c  e.  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o  X.  { k } ) }  |->  ( ( F `
 c )  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) ) ) ) ) )
12076, 119eqtrd 2508 . . 3  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  ( R  gsumg  ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) )  =  ( R 
gsumg  ( c  e.  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) } 
|->  ( ( F `  c )  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) ) ) ) ) )
121120mpteq2dva 4533 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (
k  e.  NN0  |->  ( R 
gsumg  ( x  e.  (
0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) ) )  =  ( k  e.  NN0  |->  ( R 
gsumg  ( c  e.  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) } 
|->  ( ( F `  c )  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) ) ) ) ) ) )
12227, 34, 1213eqtr4d 2518 1  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( k  e.  NN0  |->  ( R 
gsumg  ( x  e.  (
0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447    |-> cmpt 4505   Oncon0 4878    X. cxp 4997   dom cdm 4999    o. ccom 5003   Fun wfun 5581   -->wf 5583   -1-1-onto->wf1o 5586   ` cfv 5587  (class class class)co 6283    oFcof 6521    oRcofr 6522   supp csupp 6901   1oc1o 7123    ^m cmap 7420   Fincfn 7516   finSupp cfsupp 7828   0cc0 9491    <_ cle 9628    - cmin 9804   NN0cn0 10794   ...cfz 11671   Basecbs 14489   .rcmulr 14555   0gc0g 14694    gsumg cgsu 14695  CMndccmn 16601   Ringcrg 16995   mPwSer cmps 17787  PwSer1cps1 18001  coe1cco1 18004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-ofr 6524  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-oi 7934  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-fzo 11792  df-seq 12075  df-hash 12373  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-sca 14570  df-vsca 14571  df-tset 14573  df-ple 14574  df-0g 14696  df-gsum 14697  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-mhm 15783  df-submnd 15784  df-grp 15864  df-minusg 15865  df-mulg 15867  df-ghm 16067  df-cntz 16157  df-cmn 16603  df-abl 16604  df-mgp 16941  df-ur 16953  df-rng 16997  df-psr 17792  df-opsr 17796  df-psr1 18006  df-coe1 18009
This theorem is referenced by:  coe1mul  18098
  Copyright terms: Public domain W3C validator