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Theorem coe1mul2 18437
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 5780 . . . . 5  |-  ( k  e.  NN0  ->  ( 1o 
X.  { k } ) : 1o --> NN0 )
2 nn0ex 10822 . . . . . 6  |-  NN0  e.  _V
3 1on 7155 . . . . . . 7  |-  1o  e.  On
43elexi 3119 . . . . . 6  |-  1o  e.  _V
52, 4elmap 7466 . . . . 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 2458 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (
k  e.  NN0  |->  ( 1o 
X.  { k } ) )  =  ( k  e.  NN0  |->  ( 1o 
X.  { k } ) ) )
9 eqid 2457 . . . 4  |-  ( 1o mPwSer  R )  =  ( 1o mPwSer  R )
10 coe1mul2.s . . . . 5  |-  S  =  (PwSer1 `  R )
11 coe1mul2.b . . . . 5  |-  B  =  ( Base `  S
)
1210, 11, 9psr1bas2 18356 . . . 4  |-  B  =  ( Base `  ( 1o mPwSer  R ) )
13 coe1mul2.u . . . 4  |-  .x.  =  ( .r `  R )
14 coe1mul2.t . . . . 5  |-  .xb  =  ( .r `  S )
1510, 9, 14psr1mulr 18392 . . . 4  |-  .xb  =  ( .r `  ( 1o mPwSer  R ) )
16 psr1baslem 18351 . . . 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 18165 . . 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 4460 . . . . . 6  |-  ( b  =  ( 1o  X.  { k } )  ->  ( d  oR  <_  b  <->  d  oR  <_  ( 1o  X.  { k } ) ) )
2120rabbidv 3101 . . . . 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 6303 . . . . . . 7  |-  ( b  =  ( 1o  X.  { k } )  ->  ( b  oF  -  c )  =  ( ( 1o 
X.  { k } )  oF  -  c ) )
2322fveq2d 5876 . . . . . 6  |-  ( b  =  ( 1o  X.  { k } )  ->  ( G `  ( b  oF  -  c ) )  =  ( G `  ( ( 1o  X.  { k } )  oF  -  c
) ) )
2423oveq2d 6312 . . . . 5  |-  ( b  =  ( 1o  X.  { k } )  ->  ( ( F `
 c )  .x.  ( G `  ( b  oF  -  c
) ) )  =  ( ( F `  c )  .x.  ( G `  ( ( 1o  X.  { k } )  oF  -  c ) ) ) )
2521, 24mpteq12dv 4535 . . . 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 6312 . . 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 6065 . 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 ) ) ) ) ) ) )
2810psr1ring 18415 . . . 4  |-  ( R  e.  Ring  ->  S  e. 
Ring )
2911, 14ringcl 17339 . . . 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 2457 . . . 4  |-  (coe1 `  ( F  .xb  G ) )  =  (coe1 `  ( F  .xb  G ) )
32 eqid 2457 . . . 4  |-  ( k  e.  NN0  |->  ( 1o 
X.  { k } ) )  =  ( k  e.  NN0  |->  ( 1o 
X.  { k } ) )
3331, 11, 10, 32coe1fval3 18374 . . 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 2457 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
36 eqid 2457 . . . . 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 ringcmn 17356 . . . . . 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 12085 . . . . . 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 2457 . . . . . . . . . 10  |-  (coe1 `  F
)  =  (coe1 `  F
)
4544, 11, 10, 35coe1f2 18375 . . . . . . . . 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 11797 . . . . . . . . 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 6033 . . . . . . 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 2457 . . . . . . . . . 10  |-  (coe1 `  G
)  =  (coe1 `  G
)
5251, 11, 10, 35coe1f2 18375 . . . . . . . . 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 11742 . . . . . . . . 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 6033 . . . . . . 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, 13ringcl 17339 . . . . . . 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 2457 . . . . . 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 6056 . . . . 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 3119 . . . . . . . . 9  |-  ( 0 ... k )  e. 
_V
6261mptex 6144 . . . . . . . 8  |-  ( x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )  e.  _V
63 funmpt 5630 . . . . . . . 8  |-  Fun  (
x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) )
64 fvex 5882 . . . . . . . 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 6930 . . . . . . . . 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 5509 . . . . . . . . 9  |-  dom  (
x  e.  ( 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) ) ) 
C_  ( 0 ... k )
6866, 67sstri 3508 . . . . . . . 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 7862 . . . . . . 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 2457 . . . . . . 7  |-  { d  e.  ( NN0  ^m  1o )  |  d  oR  <_  ( 1o 
X.  { k } ) }  =  {
d  e.  ( NN0 
^m  1o )  |  d  oR  <_ 
( 1o  X.  {
k } ) }
7473coe1mul2lem2 18436 . . . . . 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 17051 . . . 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 4459 . . . . . . . . . . 11  |-  ( d  =  c  ->  (
d  oR  <_ 
( 1o  X.  {
k } )  <->  c  oR  <_  ( 1o  X.  { k } ) ) )
7877elrab 3257 . . . . . . . . . 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 755 . . . . . . . . 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 3254 . . . . . . . . . 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 18435 . . . . . . . . 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 2458 . . . . . . 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 2458 . . . . . . 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 5872 . . . . . . . 8  |-  ( x  =  ( c `  (/) )  ->  ( (coe1 `  F ) `  x
)  =  ( (coe1 `  F ) `  (
c `  (/) ) ) )
90 oveq2 6304 . . . . . . . . 9  |-  ( x  =  ( c `  (/) )  ->  ( k  -  x )  =  ( k  -  ( c `
 (/) ) ) )
9190fveq2d 5876 . . . . . . . 8  |-  ( x  =  ( c `  (/) )  ->  ( (coe1 `  G ) `  (
k  -  x ) )  =  ( (coe1 `  G ) `  (
k  -  ( c `
 (/) ) ) ) )
9289, 91oveq12d 6314 . . . . . . 7  |-  ( x  =  ( c `  (/) )  ->  ( (
(coe1 `  F ) `  x )  .x.  (
(coe1 `  G ) `  ( k  -  x
) ) )  =  ( ( (coe1 `  F
) `  ( c `  (/) ) )  .x.  ( (coe1 `  G ) `  ( k  -  (
c `  (/) ) ) ) ) )
9386, 87, 88, 92fmptco 6065 . . . . . 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 18373 . . . . . . . . 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 7160 . . . . . . . . . . . . . 14  |-  1o  =  { (/) }
98 0ex 4587 . . . . . . . . . . . . . 14  |-  (/)  e.  _V
9997, 2, 98mapsnconst 7483 . . . . . . . . . . . . 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 6312 . . . . . . . . . . 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 3112 . . . . . . . . . . . . 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 5882 . . . . . . . . . . . . 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 6564 . . . . . . . . . . 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 2498 . . . . . . . . . 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 5876 . . . . . . . . 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 11742 . . . . . . . . . . 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 18372 . . . . . . . . . 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 2501 . . . . . . . 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 6314 . . . . . . 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 4543 . . . . . 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 2501 . . . . 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 6312 . . . 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 2498 . . 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 4543 . 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 2508 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    C_ wss 3471   (/)c0 3793   {csn 4032   class class class wbr 4456    |-> cmpt 4515   Oncon0 4887    X. cxp 5006   dom cdm 5008    o. ccom 5012   Fun wfun 5588   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    oFcof 6537    oRcofr 6538   supp csupp 6917   1oc1o 7141    ^m cmap 7438   Fincfn 7535   finSupp cfsupp 7847   0cc0 9509    <_ cle 9646    - cmin 9824   NN0cn0 10816   ...cfz 11697   Basecbs 14644   .rcmulr 14713   0gc0g 14857    gsumg cgsu 14858  CMndccmn 16925   Ringcrg 17325   mPwSer cmps 18127  PwSer1cps1 18341  coe1cco1 18344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-tset 14731  df-ple 14732  df-0g 14859  df-gsum 14860  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-mulg 16187  df-ghm 16392  df-cntz 16482  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-psr 18132  df-opsr 18136  df-psr1 18346  df-coe1 18349
This theorem is referenced by:  coe1mul  18438
  Copyright terms: Public domain W3C validator