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Theorem ply1mulgsum 32472
Description: The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.)
Hypotheses
Ref Expression
ply1mulgsum.p  |-  P  =  (Poly1 `  R )
ply1mulgsum.b  |-  B  =  ( Base `  P
)
ply1mulgsum.a  |-  A  =  (coe1 `  K )
ply1mulgsum.c  |-  C  =  (coe1 `  L )
ply1mulgsum.x  |-  X  =  (var1 `  R )
ply1mulgsum.pm  |-  .X.  =  ( .r `  P )
ply1mulgsum.sm  |-  .x.  =  ( .s `  P )
ply1mulgsum.rm  |-  .*  =  ( .r `  R )
ply1mulgsum.m  |-  M  =  (mulGrp `  P )
ply1mulgsum.e  |-  .^  =  (.g
`  M )
Assertion
Ref Expression
ply1mulgsum  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( K  .X.  L )  =  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) ) ) )
Distinct variable groups:    A, l    B, l    C, l    K, l    L, l    R, l    A, k    B, k    C, k   
k, K    k, L    R, k    .* , k, l    k, X    .^ , k    .x. , k    P, k    .* , l
Allowed substitution hints:    P( l)    .x. ( l)    .X. ( k, l)    .^ ( l)    M( k, l)    X( l)

Proof of Theorem ply1mulgsum
Dummy variables  n  i  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ply1mulgsum.p . . . . . . 7  |-  P  =  (Poly1 `  R )
2 ply1mulgsum.pm . . . . . . 7  |-  .X.  =  ( .r `  P )
3 ply1mulgsum.rm . . . . . . 7  |-  .*  =  ( .r `  R )
4 ply1mulgsum.b . . . . . . 7  |-  B  =  ( Base `  P
)
51, 2, 3, 4coe1mul 18181 . . . . . 6  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  (coe1 `  ( K  .X.  L ) )  =  ( m  e.  NN0  |->  ( R 
gsumg  ( i  e.  ( 0 ... m ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( m  -  i
) ) ) ) ) ) )
65adantr 465 . . . . 5  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  (coe1 `  ( K  .X.  L ) )  =  ( m  e.  NN0  |->  ( R  gsumg  ( i  e.  ( 0 ... m ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( m  -  i
) ) ) ) ) ) )
76fveq1d 5874 . . . 4  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  ( (coe1 `  ( K  .X.  L ) ) `
 n )  =  ( ( m  e. 
NN0  |->  ( R  gsumg  ( i  e.  ( 0 ... m )  |->  ( ( (coe1 `  K ) `  i )  .*  (
(coe1 `  L ) `  ( m  -  i
) ) ) ) ) ) `  n
) )
8 eqidd 2468 . . . . 5  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  ( m  e. 
NN0  |->  ( R  gsumg  ( i  e.  ( 0 ... m )  |->  ( ( (coe1 `  K ) `  i )  .*  (
(coe1 `  L ) `  ( m  -  i
) ) ) ) ) )  =  ( m  e.  NN0  |->  ( R 
gsumg  ( i  e.  ( 0 ... m ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( m  -  i
) ) ) ) ) ) )
9 oveq2 6303 . . . . . . . 8  |-  ( m  =  n  ->  (
0 ... m )  =  ( 0 ... n
) )
10 oveq1 6302 . . . . . . . . . 10  |-  ( m  =  n  ->  (
m  -  i )  =  ( n  -  i ) )
1110fveq2d 5876 . . . . . . . . 9  |-  ( m  =  n  ->  (
(coe1 `  L ) `  ( m  -  i
) )  =  ( (coe1 `  L ) `  ( n  -  i
) ) )
1211oveq2d 6311 . . . . . . . 8  |-  ( m  =  n  ->  (
( (coe1 `  K ) `  i )  .*  (
(coe1 `  L ) `  ( m  -  i
) ) )  =  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( n  -  i
) ) ) )
139, 12mpteq12dv 4531 . . . . . . 7  |-  ( m  =  n  ->  (
i  e.  ( 0 ... m )  |->  ( ( (coe1 `  K ) `  i )  .*  (
(coe1 `  L ) `  ( m  -  i
) ) ) )  =  ( i  e.  ( 0 ... n
)  |->  ( ( (coe1 `  K ) `  i
)  .*  ( (coe1 `  L ) `  (
n  -  i ) ) ) ) )
1413oveq2d 6311 . . . . . 6  |-  ( m  =  n  ->  ( R  gsumg  ( i  e.  ( 0 ... m ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( m  -  i
) ) ) ) )  =  ( R 
gsumg  ( i  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( n  -  i
) ) ) ) ) )
1514adantl 466 . . . . 5  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  n  e.  NN0 )  /\  m  =  n )  ->  ( R  gsumg  ( i  e.  ( 0 ... m ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( m  -  i
) ) ) ) )  =  ( R 
gsumg  ( i  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( n  -  i
) ) ) ) ) )
16 simpr 461 . . . . 5  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
17 ovex 6320 . . . . . 6  |-  ( R 
gsumg  ( i  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( n  -  i
) ) ) ) )  e.  _V
1817a1i 11 . . . . 5  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  ( R  gsumg  ( i  e.  ( 0 ... n )  |->  ( ( (coe1 `  K ) `  i )  .*  (
(coe1 `  L ) `  ( n  -  i
) ) ) ) )  e.  _V )
198, 15, 16, 18fvmptd 5962 . . . 4  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  ( ( m  e.  NN0  |->  ( R 
gsumg  ( i  e.  ( 0 ... m ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( m  -  i
) ) ) ) ) ) `  n
)  =  ( R 
gsumg  ( i  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( n  -  i
) ) ) ) ) )
20 ply1mulgsum.x . . . . . 6  |-  X  =  (var1 `  R )
21 ply1mulgsum.e . . . . . . 7  |-  .^  =  (.g
`  M )
22 ply1mulgsum.m . . . . . . . 8  |-  M  =  (mulGrp `  P )
2322fveq2i 5875 . . . . . . 7  |-  (.g `  M
)  =  (.g `  (mulGrp `  P ) )
2421, 23eqtri 2496 . . . . . 6  |-  .^  =  (.g
`  (mulGrp `  P )
)
25 simp1 996 . . . . . . 7  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  R  e.  Ring )
2625adantr 465 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  R  e.  Ring )
27 eqid 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
28 ply1mulgsum.sm . . . . . 6  |-  .x.  =  ( .s `  P )
29 eqid 2467 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
30 ringcmn 17101 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. CMnd
)
31303ad2ant1 1017 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  R  e. CMnd )
3231ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  n  e.  NN0 )  /\  k  e. 
NN0 )  ->  R  e. CMnd )
33 fzfid 12063 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  n  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
0 ... k )  e. 
Fin )
34 simpll1 1035 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  n  e.  NN0 )  /\  k  e. 
NN0 )  ->  R  e.  Ring )
3534adantr 465 . . . . . . . . . 10  |-  ( ( ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  /\  k  e.  NN0 )  /\  l  e.  ( 0 ... k
) )  ->  R  e.  Ring )
36 simp2 997 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  K  e.  B )
3736ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  n  e.  NN0 )  /\  k  e. 
NN0 )  ->  K  e.  B )
38 elfznn0 11782 . . . . . . . . . . 11  |-  ( l  e.  ( 0 ... k )  ->  l  e.  NN0 )
39 ply1mulgsum.a . . . . . . . . . . . 12  |-  A  =  (coe1 `  K )
4039, 4, 1, 27coe1fvalcl 18121 . . . . . . . . . . 11  |-  ( ( K  e.  B  /\  l  e.  NN0 )  -> 
( A `  l
)  e.  ( Base `  R ) )
4137, 38, 40syl2an 477 . . . . . . . . . 10  |-  ( ( ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  /\  k  e.  NN0 )  /\  l  e.  ( 0 ... k
) )  ->  ( A `  l )  e.  ( Base `  R
) )
42 simp3 998 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  L  e.  B )
4342ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  n  e.  NN0 )  /\  k  e. 
NN0 )  ->  L  e.  B )
44 fznn0sub 11728 . . . . . . . . . . 11  |-  ( l  e.  ( 0 ... k )  ->  (
k  -  l )  e.  NN0 )
45 ply1mulgsum.c . . . . . . . . . . . 12  |-  C  =  (coe1 `  L )
4645, 4, 1, 27coe1fvalcl 18121 . . . . . . . . . . 11  |-  ( ( L  e.  B  /\  ( k  -  l
)  e.  NN0 )  ->  ( C `  (
k  -  l ) )  e.  ( Base `  R ) )
4743, 44, 46syl2an 477 . . . . . . . . . 10  |-  ( ( ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  /\  k  e.  NN0 )  /\  l  e.  ( 0 ... k
) )  ->  ( C `  ( k  -  l ) )  e.  ( Base `  R
) )
4827, 3ringcl 17084 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  ( A `  l )  e.  ( Base `  R
)  /\  ( C `  ( k  -  l
) )  e.  (
Base `  R )
)  ->  ( ( A `  l )  .*  ( C `  (
k  -  l ) ) )  e.  (
Base `  R )
)
4935, 41, 47, 48syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  /\  k  e.  NN0 )  /\  l  e.  ( 0 ... k
) )  ->  (
( A `  l
)  .*  ( C `
 ( k  -  l ) ) )  e.  ( Base `  R
) )
5049ralrimiva 2881 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  n  e.  NN0 )  /\  k  e. 
NN0 )  ->  A. l  e.  ( 0 ... k
) ( ( A `
 l )  .*  ( C `  (
k  -  l ) ) )  e.  (
Base `  R )
)
5127, 32, 33, 50gsummptcl 16867 . . . . . . 7  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  n  e.  NN0 )  /\  k  e. 
NN0 )  ->  ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  e.  ( Base `  R
) )
5251ralrimiva 2881 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  A. k  e.  NN0  ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  e.  ( Base `  R
) )
531, 4, 39, 45, 20, 2, 28, 3, 22, 21ply1mulgsumlem3 32470 . . . . . . 7  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  (
k  e.  NN0  |->  ( R 
gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) ) ) finSupp 
( 0g `  R
) )
5453adantr 465 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  ( k  e. 
NN0  |->  ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) ) ) finSupp  ( 0g
`  R ) )
551, 4, 20, 24, 26, 27, 28, 29, 52, 54, 16gsummoncoe1 18216 . . . . 5  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  ( (coe1 `  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) ) ) ) `  n )  =  [_ n  /  k ]_ ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) ) )
56 vex 3121 . . . . . 6  |-  n  e. 
_V
57 csbov2g 6331 . . . . . . 7  |-  ( n  e.  _V  ->  [_ n  /  k ]_ ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  =  ( R  gsumg  [_ n  /  k ]_ ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) ) )
58 id 22 . . . . . . . . 9  |-  ( n  e.  _V  ->  n  e.  _V )
59 oveq2 6303 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
0 ... k )  =  ( 0 ... n
) )
60 oveq1 6302 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
k  -  l )  =  ( n  -  l ) )
6160fveq2d 5876 . . . . . . . . . . . 12  |-  ( k  =  n  ->  ( C `  ( k  -  l ) )  =  ( C `  ( n  -  l
) ) )
6261oveq2d 6311 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
( A `  l
)  .*  ( C `
 ( k  -  l ) ) )  =  ( ( A `
 l )  .*  ( C `  (
n  -  l ) ) ) )
6359, 62mpteq12dv 4531 . . . . . . . . . 10  |-  ( k  =  n  ->  (
l  e.  ( 0 ... k )  |->  ( ( A `  l
)  .*  ( C `
 ( k  -  l ) ) ) )  =  ( l  e.  ( 0 ... n )  |->  ( ( A `  l )  .*  ( C `  ( n  -  l
) ) ) ) )
6463adantl 466 . . . . . . . . 9  |-  ( ( n  e.  _V  /\  k  =  n )  ->  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) )  =  ( l  e.  ( 0 ... n )  |->  ( ( A `  l
)  .*  ( C `
 ( n  -  l ) ) ) ) )
6558, 64csbied 3467 . . . . . . . 8  |-  ( n  e.  _V  ->  [_ n  /  k ]_ (
l  e.  ( 0 ... k )  |->  ( ( A `  l
)  .*  ( C `
 ( k  -  l ) ) ) )  =  ( l  e.  ( 0 ... n )  |->  ( ( A `  l )  .*  ( C `  ( n  -  l
) ) ) ) )
6665oveq2d 6311 . . . . . . 7  |-  ( n  e.  _V  ->  ( R  gsumg  [_ n  /  k ]_ ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  =  ( R  gsumg  ( l  e.  ( 0 ... n ) 
|->  ( ( A `  l )  .*  ( C `  ( n  -  l ) ) ) ) ) )
6757, 66eqtrd 2508 . . . . . 6  |-  ( n  e.  _V  ->  [_ n  /  k ]_ ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  =  ( R  gsumg  ( l  e.  ( 0 ... n ) 
|->  ( ( A `  l )  .*  ( C `  ( n  -  l ) ) ) ) ) )
6856, 67mp1i 12 . . . . 5  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  [_ n  /  k ]_ ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  =  ( R  gsumg  ( l  e.  ( 0 ... n ) 
|->  ( ( A `  l )  .*  ( C `  ( n  -  l ) ) ) ) ) )
69 fveq2 5872 . . . . . . . . . 10  |-  ( l  =  i  ->  ( A `  l )  =  ( A `  i ) )
7039fveq1i 5873 . . . . . . . . . 10  |-  ( A `
 i )  =  ( (coe1 `  K ) `  i )
7169, 70syl6eq 2524 . . . . . . . . 9  |-  ( l  =  i  ->  ( A `  l )  =  ( (coe1 `  K
) `  i )
)
72 oveq2 6303 . . . . . . . . . . 11  |-  ( l  =  i  ->  (
n  -  l )  =  ( n  -  i ) )
7372fveq2d 5876 . . . . . . . . . 10  |-  ( l  =  i  ->  ( C `  ( n  -  l ) )  =  ( C `  ( n  -  i
) ) )
7445fveq1i 5873 . . . . . . . . . 10  |-  ( C `
 ( n  -  i ) )  =  ( (coe1 `  L ) `  ( n  -  i
) )
7573, 74syl6eq 2524 . . . . . . . . 9  |-  ( l  =  i  ->  ( C `  ( n  -  l ) )  =  ( (coe1 `  L
) `  ( n  -  i ) ) )
7671, 75oveq12d 6313 . . . . . . . 8  |-  ( l  =  i  ->  (
( A `  l
)  .*  ( C `
 ( n  -  l ) ) )  =  ( ( (coe1 `  K ) `  i
)  .*  ( (coe1 `  L ) `  (
n  -  i ) ) ) )
7776cbvmptv 4544 . . . . . . 7  |-  ( l  e.  ( 0 ... n )  |->  ( ( A `  l )  .*  ( C `  ( n  -  l
) ) ) )  =  ( i  e.  ( 0 ... n
)  |->  ( ( (coe1 `  K ) `  i
)  .*  ( (coe1 `  L ) `  (
n  -  i ) ) ) )
7877a1i 11 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  ( l  e.  ( 0 ... n
)  |->  ( ( A `
 l )  .*  ( C `  (
n  -  l ) ) ) )  =  ( i  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( n  -  i
) ) ) ) )
7978oveq2d 6311 . . . . 5  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  ( R  gsumg  ( l  e.  ( 0 ... n )  |->  ( ( A `  l )  .*  ( C `  ( n  -  l
) ) ) ) )  =  ( R 
gsumg  ( i  e.  ( 0 ... n ) 
|->  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( n  -  i
) ) ) ) ) )
8055, 68, 793eqtrrd 2513 . . . 4  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  ( R  gsumg  ( i  e.  ( 0 ... n )  |->  ( ( (coe1 `  K ) `  i )  .*  (
(coe1 `  L ) `  ( n  -  i
) ) ) ) )  =  ( (coe1 `  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) ) ) ) `  n ) )
817, 19, 803eqtrd 2512 . . 3  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  n  e.  NN0 )  ->  ( (coe1 `  ( K  .X.  L ) ) `
 n )  =  ( (coe1 `  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  .x.  ( k  .^  X
) ) ) ) ) `  n ) )
8281ralrimiva 2881 . 2  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  A. n  e.  NN0  ( (coe1 `  ( K  .X.  L ) ) `
 n )  =  ( (coe1 `  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  .x.  ( k  .^  X
) ) ) ) ) `  n ) )
831ply1ring 18159 . . . 4  |-  ( R  e.  Ring  ->  P  e. 
Ring )
844, 2ringcl 17084 . . . 4  |-  ( ( P  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( K  .X.  L )  e.  B )
8583, 84syl3an1 1261 . . 3  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( K  .X.  L )  e.  B )
86 eqid 2467 . . . 4  |-  ( 0g
`  P )  =  ( 0g `  P
)
87 ringcmn 17101 . . . . . 6  |-  ( P  e.  Ring  ->  P  e. CMnd
)
8883, 87syl 16 . . . . 5  |-  ( R  e.  Ring  ->  P  e. CMnd
)
89883ad2ant1 1017 . . . 4  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  P  e. CMnd )
90 nn0ex 10813 . . . . 5  |-  NN0  e.  _V
9190a1i 11 . . . 4  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  NN0  e.  _V )
921ply1lmod 18163 . . . . . . . 8  |-  ( R  e.  Ring  ->  P  e. 
LMod )
93923ad2ant1 1017 . . . . . . 7  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  P  e.  LMod )
9493adantr 465 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  P  e.  LMod )
9531adantr 465 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  R  e. CMnd )
96 fzfid 12063 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  ( 0 ... k )  e.  Fin )
97 simpll1 1035 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  k  e.  NN0 )  /\  l  e.  ( 0 ... k
) )  ->  R  e.  Ring )
9836adantr 465 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  K  e.  B
)
9998, 38, 40syl2an 477 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  k  e.  NN0 )  /\  l  e.  ( 0 ... k
) )  ->  ( A `  l )  e.  ( Base `  R
) )
10042adantr 465 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  L  e.  B
)
101100, 44, 46syl2an 477 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  k  e.  NN0 )  /\  l  e.  ( 0 ... k
) )  ->  ( C `  ( k  -  l ) )  e.  ( Base `  R
) )
10297, 99, 101, 48syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  k  e.  NN0 )  /\  l  e.  ( 0 ... k
) )  ->  (
( A `  l
)  .*  ( C `
 ( k  -  l ) ) )  e.  ( Base `  R
) )
103102ralrimiva 2881 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  A. l  e.  ( 0 ... k ) ( ( A `  l )  .*  ( C `  ( k  -  l ) ) )  e.  ( Base `  R ) )
10427, 95, 96, 103gsummptcl 16867 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  e.  ( Base `  R ) )
10525adantr 465 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  R  e.  Ring )
1061ply1sca 18164 . . . . . . . . 9  |-  ( R  e.  Ring  ->  R  =  (Scalar `  P )
)
107105, 106syl 16 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  R  =  (Scalar `  P ) )
108107fveq2d 5876 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  ( Base `  R
)  =  ( Base `  (Scalar `  P )
) )
109104, 108eleqtrd 2557 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  e.  ( Base `  (Scalar `  P )
) )
11022ringmgp 17076 . . . . . . . . . 10  |-  ( P  e.  Ring  ->  M  e. 
Mnd )
11183, 110syl 16 . . . . . . . . 9  |-  ( R  e.  Ring  ->  M  e. 
Mnd )
1121113ad2ant1 1017 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  M  e.  Mnd )
113112adantr 465 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  M  e.  Mnd )
114 simpr 461 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
11520, 1, 4vr1cl 18128 . . . . . . . . 9  |-  ( R  e.  Ring  ->  X  e.  B )
1161153ad2ant1 1017 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  X  e.  B )
117116adantr 465 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  X  e.  B
)
11822, 4mgpbas 17019 . . . . . . . 8  |-  B  =  ( Base `  M
)
119118, 21mulgnn0cl 16030 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  k  e.  NN0  /\  X  e.  B )  ->  (
k  .^  X )  e.  B )
120113, 114, 117, 119syl3anc 1228 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  ( k  .^  X )  e.  B
)
121 eqid 2467 . . . . . . 7  |-  (Scalar `  P )  =  (Scalar `  P )
122 eqid 2467 . . . . . . 7  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
1234, 121, 28, 122lmodvscl 17400 . . . . . 6  |-  ( ( P  e.  LMod  /\  ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  e.  ( Base `  (Scalar `  P ) )  /\  ( k  .^  X
)  e.  B )  ->  ( ( R 
gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  .x.  ( k  .^  X
) )  e.  B
)
12494, 109, 120, 123syl3anc 1228 . . . . 5  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  ( ( R 
gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  .x.  ( k  .^  X
) )  e.  B
)
125 eqid 2467 . . . . 5  |-  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  .x.  ( k  .^  X
) ) )  =  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) )
126124, 125fmptd 6056 . . . 4  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  (
k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  .x.  ( k  .^  X
) ) ) : NN0 --> B )
1271, 4, 39, 45, 20, 2, 28, 3, 22, 21ply1mulgsumlem4 32471 . . . 4  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  (
k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  .x.  ( k  .^  X
) ) ) finSupp  ( 0g `  P ) )
1284, 86, 89, 91, 126, 127gsumcl 16796 . . 3  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) ) )  e.  B
)
129 eqid 2467 . . . 4  |-  (coe1 `  ( K  .X.  L ) )  =  (coe1 `  ( K  .X.  L ) )
130 eqid 2467 . . . 4  |-  (coe1 `  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) ) ) )  =  (coe1 `  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  .x.  ( k  .^  X
) ) ) ) )
1311, 4, 129, 130ply1coe1eq 18210 . . 3  |-  ( ( R  e.  Ring  /\  ( K  .X.  L )  e.  B  /\  ( P 
gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) ) )  e.  B
)  ->  ( A. n  e.  NN0  ( (coe1 `  ( K  .X.  L
) ) `  n
)  =  ( (coe1 `  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) ) ) ) `  n )  <->  ( K  .X.  L )  =  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) ) ) ) )
13225, 85, 128, 131syl3anc 1228 . 2  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( A. n  e.  NN0  ( (coe1 `  ( K  .X.  L ) ) `  n )  =  ( (coe1 `  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k ) 
|->  ( ( A `  l )  .*  ( C `  ( k  -  l ) ) ) ) )  .x.  ( k  .^  X
) ) ) ) ) `  n )  <-> 
( K  .X.  L
)  =  ( P 
gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) ) ) ) )
13382, 132mpbid 210 1  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( K  .X.  L )  =  ( P  gsumg  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
) ) ) ) )  .x.  ( k 
.^  X ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   [_csb 3440   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   finSupp cfsupp 7841   0cc0 9504    - cmin 9817   NN0cn0 10807   ...cfz 11684   Basecbs 14507   .rcmulr 14573  Scalarcsca 14575   .scvsca 14576   0gc0g 14712    gsumg cgsu 14713   Mndcmnd 15793  .gcmg 15928  CMndccmn 16671  mulGrpcmgp 17013   Ringcrg 17070   LModclmod 17383  var1cv1 18085  Poly1cpl1 18086  coe1cco1 18087
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-ofr 6536  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-tset 14591  df-ple 14592  df-0g 14714  df-gsum 14715  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-ghm 16137  df-cntz 16227  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-srg 17030  df-ring 17072  df-subrg 17298  df-lmod 17385  df-lss 17450  df-psr 17875  df-mvr 17876  df-mpl 17877  df-opsr 17879  df-psr1 18089  df-vr1 18090  df-ply1 18091  df-coe1 18092
This theorem is referenced by: (None)
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