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Theorem coe1mul2 17735
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 5611 . . . . 5  |-  ( k  e.  NN0  ->  ( 1o 
X.  { k } ) : 1o --> NN0 )
2 nn0ex 10597 . . . . . 6  |-  NN0  e.  _V
3 1on 6939 . . . . . . 7  |-  1o  e.  On
43elexi 2994 . . . . . 6  |-  1o  e.  _V
52, 4elmap 7253 . . . . 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 2444 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (
k  e.  NN0  |->  ( 1o 
X.  { k } ) )  =  ( k  e.  NN0  |->  ( 1o 
X.  { k } ) ) )
9 eqid 2443 . . . 4  |-  ( 1o mPwSer  R )  =  ( 1o mPwSer  R )
10 coe1mul2.s . . . . 5  |-  S  =  (PwSer1 `  R )
11 coe1mul2.b . . . . 5  |-  B  =  ( Base `  S
)
1210, 11, 9psr1bas2 17658 . . . 4  |-  B  =  ( Base `  ( 1o mPwSer  R ) )
13 coe1mul2.u . . . 4  |-  .x.  =  ( .r `  R )
14 coe1mul2.t . . . . 5  |-  .xb  =  ( .r `  S )
1510, 9, 14psr1mulr 17690 . . . 4  |-  .xb  =  ( .r `  ( 1o mPwSer  R ) )
16 psr1baslem 17653 . . . 4  |-  ( NN0 
^m  1o )  =  { a  e.  ( NN0  ^m  1o )  |  ( `' a
" NN )  e. 
Fin }
17 simp2 989 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  F  e.  B )
18 simp3 990 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  G  e.  B )
199, 12, 13, 15, 16, 17, 18psrmulfval 17468 . . 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 4308 . . . . . 6  |-  ( b  =  ( 1o  X.  { k } )  ->  ( d  oR  <_  b  <->  d  oR  <_  ( 1o  X.  { k } ) ) )
2120rabbidv 2976 . . . . 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 6110 . . . . . . 7  |-  ( b  =  ( 1o  X.  { k } )  ->  ( b  oF  -  c )  =  ( ( 1o 
X.  { k } )  oF  -  c ) )
2322fveq2d 5707 . . . . . 6  |-  ( b  =  ( 1o  X.  { k } )  ->  ( G `  ( b  oF  -  c ) )  =  ( G `  ( ( 1o  X.  { k } )  oF  -  c
) ) )
2423oveq2d 6119 . . . . 5  |-  ( b  =  ( 1o  X.  { k } )  ->  ( ( F `
 c )  .x.  ( G `  ( b  oF  -  c
) ) )  =  ( ( F `  c )  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) ) ) )
2521, 24mpteq12dv 4382 . . . 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 6119 . . 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 5888 . 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 17714 . . . 4  |-  ( R  e.  Ring  ->  S  e. 
Ring )
2911, 14rngcl 16670 . . . 4  |-  ( ( S  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .xb  G )  e.  B )
3028, 29syl3an1 1251 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .xb  G )  e.  B )
31 eqid 2443 . . . 4  |-  (coe1 `  ( F  .xb  G ) )  =  (coe1 `  ( F  .xb  G ) )
32 eqid 2443 . . . 4  |-  ( k  e.  NN0  |->  ( 1o 
X.  { k } ) )  =  ( k  e.  NN0  |->  ( 1o 
X.  { k } ) )
3331, 11, 10, 32coe1fval3 17676 . . 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 2443 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
36 eqid 2443 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
37 simpl1 991 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  k  e.  NN0 )  ->  R  e.  Ring )
38 rngcmn 16687 . . . . . 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 11806 . . . . . 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 1027 . . . . . . 7  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  R  e.  Ring )
43 simpll2 1028 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  F  e.  B )
44 eqid 2443 . . . . . . . . . 10  |-  (coe1 `  F
)  =  (coe1 `  F
)
4544, 11, 10, 35coe1f2 17677 . . . . . . . . 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 11493 . . . . . . . . 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 5856 . . . . . . 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 1029 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  F  e.  B  /\  G  e.  B
)  /\  k  e.  NN0 )  /\  x  e.  ( 0 ... k
) )  ->  G  e.  B )
51 eqid 2443 . . . . . . . . . 10  |-  (coe1 `  G
)  =  (coe1 `  G
)
5251, 11, 10, 35coe1f2 17677 . . . . . . . . 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 11499 . . . . . . . . 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 5856 . . . . . . 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 16670 . . . . . . 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 1218 . . . . . 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 2443 . . . . . 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 5879 . . . . 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 2994 . . . . . . . . 9  |-  ( 0 ... k )  e. 
_V
6261mptex 5960 . . . . . . . 8  |-  ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )  e.  _V
63 funmpt 5466 . . . . . . . 8  |-  Fun  (
x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )
64 fvex 5713 . . . . . . . 8  |-  ( 0g
`  R )  e. 
_V
6562, 63, 643pm3.2i 1166 . . . . . . 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 6715 . . . . . . . . 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 5346 . . . . . . . . 9  |-  dom  (
x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) 
C_  ( 0 ... k )
6866, 67sstri 3377 . . . . . . . 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 7647 . . . . . . 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 2443 . . . . . . 7  |-  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o 
X.  { k } ) }  =  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) }
7473coe1mul2lem2 17734 . . . . . 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 16410 . . . 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 4307 . . . . . . . . . . 11  |-  ( d  =  c  ->  (
d  oR  <_ 
( 1o  X.  {
k } )  <->  c  oR  <_  ( 1o  X.  { k } ) ) )
7877elrab 3129 . . . . . . . . . 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 3126 . . . . . . . . . 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 17733 . . . . . . . . 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 2444 . . . . . . 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 2444 . . . . . . 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 5703 . . . . . . . 8  |-  ( x  =  ( c `  (/) )  ->  ( (coe1 `  F ) `  x
)  =  ( (coe1 `  F ) `  (
c `  (/) ) ) )
90 oveq2 6111 . . . . . . . . 9  |-  ( x  =  ( c `  (/) )  ->  ( k  -  x )  =  ( k  -  ( c `
 (/) ) ) )
9190fveq2d 5707 . . . . . . . 8  |-  ( x  =  ( c `  (/) )  ->  ( (coe1 `  G ) `  (
k  -  x ) )  =  ( (coe1 `  G ) `  (
k  -  ( c `
 (/) ) ) ) )
9289, 91oveq12d 6121 . . . . . . 7  |-  ( x  =  ( c `  (/) )  ->  ( (
(coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) )  =  ( ( (coe1 `  F
) `  ( c `  (/) ) )  .x.  ( (coe1 `  G ) `  ( k  -  (
c `  (/) ) ) ) ) )
9386, 87, 88, 92fmptco 5888 . . . . . 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 1028 . . . . . . . . 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 17675 . . . . . . . . 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 6944 . . . . . . . . . . . . . 14  |-  1o  =  { (/) }
98 0ex 4434 . . . . . . . . . . . . . 14  |-  (/)  e.  _V
9997, 2, 98mapsnconst 7270 . . . . . . . . . . . . 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 6119 . . . . . . . . . . 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 2987 . . . . . . . . . . . . 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 5713 . . . . . . . . . . . . 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 6357 . . . . . . . . . . 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 2475 . . . . . . . . . 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 5707 . . . . . . . . 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 1029 . . . . . . . . . 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 11499 . . . . . . . . . . 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 17674 . . . . . . . . . 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 2478 . . . . . . . 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 6121 . . . . . . 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 4390 . . . . . 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 2478 . . . . 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 6119 . . . 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 2475 . . 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 4390 . 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 2485 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 965    = wceq 1369    e. wcel 1756   {crab 2731   _Vcvv 2984    C_ wss 3340   (/)c0 3649   {csn 3889   class class class wbr 4304    e. cmpt 4362   Oncon0 4731    X. cxp 4850   dom cdm 4852    o. ccom 4856   Fun wfun 5424   -->wf 5426   -1-1-onto->wf1o 5429   ` cfv 5430  (class class class)co 6103    oFcof 6330    oRcofr 6331   supp csupp 6702   1oc1o 6925    ^m cmap 7226   Fincfn 7322   finSupp cfsupp 7632   0cc0 9294    <_ cle 9431    - cmin 9607   NN0cn0 10591   ...cfz 11449   Basecbs 14186   .rcmulr 14251   0gc0g 14390    gsumg cgsu 14391  CMndccmn 16289   Ringcrg 16657   mPwSer cmps 17430  PwSer1cps1 17643  coe1cco1 17646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-ofr 6333  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-fzo 11561  df-seq 11819  df-hash 12116  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-sca 14266  df-vsca 14267  df-tset 14269  df-ple 14270  df-0g 14392  df-gsum 14393  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-mhm 15476  df-submnd 15477  df-grp 15557  df-minusg 15558  df-mulg 15560  df-ghm 15757  df-cntz 15847  df-cmn 16291  df-abl 16292  df-mgp 16604  df-ur 16616  df-rng 16659  df-psr 17435  df-opsr 17439  df-psr1 17648  df-coe1 17651
This theorem is referenced by:  coe1mul  17736
  Copyright terms: Public domain W3C validator