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Theorem ply1mulgsum 30992
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 17833 . . . . . 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 5793 . . . 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 2452 . . . . 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 6200 . . . . . . . 8  |-  ( m  =  n  ->  (
0 ... m )  =  ( 0 ... n
) )
10 oveq1 6199 . . . . . . . . . 10  |-  ( m  =  n  ->  (
m  -  i )  =  ( n  -  i ) )
1110fveq2d 5795 . . . . . . . . 9  |-  ( m  =  n  ->  (
(coe1 `  L ) `  ( m  -  i
) )  =  ( (coe1 `  L ) `  ( n  -  i
) ) )
1211oveq2d 6208 . . . . . . . 8  |-  ( m  =  n  ->  (
( (coe1 `  K ) `  i )  .*  (
(coe1 `  L ) `  ( m  -  i
) ) )  =  ( ( (coe1 `  K
) `  i )  .*  ( (coe1 `  L ) `  ( n  -  i
) ) ) )
139, 12mpteq12dv 4470 . . . . . . 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 6208 . . . . . 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 6217 . . . . . 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 5880 . . . 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 5794 . . . . . . 7  |-  (.g `  M
)  =  (.g `  (mulGrp `  P ) )
2421, 23eqtri 2480 . . . . . 6  |-  .^  =  (.g
`  (mulGrp `  P )
)
25 simp1 988 . . . . . . 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 2451 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
28 ply1mulgsum.sm . . . . . 6  |-  .x.  =  ( .s `  P )
29 eqid 2451 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
30 rngcmn 16783 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. CMnd
)
31303ad2ant1 1009 . . . . . . . . 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 11898 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  n  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
0 ... k )  e. 
Fin )
34 simpll1 1027 . . . . . . . . . . 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 989 . . . . . . . . . . . 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 11584 . . . . . . . . . . 11  |-  ( l  e.  ( 0 ... k )  ->  l  e.  NN0 )
39 ply1mulgsum.a . . . . . . . . . . . 12  |-  A  =  (coe1 `  K )
4039, 4, 1, 27coe1fvalcl 30974 . . . . . . . . . . 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 990 . . . . . . . . . . . 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 11590 . . . . . . . . . . 11  |-  ( l  e.  ( 0 ... k )  ->  (
k  -  l )  e.  NN0 )
45 ply1mulgsum.c . . . . . . . . . . . 12  |-  C  =  (coe1 `  L )
4645, 4, 1, 27coe1fvalcl 30974 . . . . . . . . . . 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, 3rngcl 16766 . . . . . . . . . 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 1219 . . . . . . . . 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 2822 . . . . . . . 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 16565 . . . . . . 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 2822 . . . . . 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 30990 . . . . . . 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 30987 . . . . 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 3073 . . . . . 6  |-  n  e. 
_V
57 csbov2g 6228 . . . . . . 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 6200 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
0 ... k )  =  ( 0 ... n
) )
60 oveq1 6199 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
k  -  l )  =  ( n  -  l ) )
6160fveq2d 5795 . . . . . . . . . . . 12  |-  ( k  =  n  ->  ( C `  ( k  -  l ) )  =  ( C `  ( n  -  l
) ) )
6261oveq2d 6208 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
( A `  l
)  .*  ( C `
 ( k  -  l ) ) )  =  ( ( A `
 l )  .*  ( C `  (
n  -  l ) ) ) )
6359, 62mpteq12dv 4470 . . . . . . . . . 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 3414 . . . . . . . 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 6208 . . . . . . 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 2492 . . . . . 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 5791 . . . . . . . . . 10  |-  ( l  =  i  ->  ( A `  l )  =  ( A `  i ) )
7039fveq1i 5792 . . . . . . . . . 10  |-  ( A `
 i )  =  ( (coe1 `  K ) `  i )
7169, 70syl6eq 2508 . . . . . . . . 9  |-  ( l  =  i  ->  ( A `  l )  =  ( (coe1 `  K
) `  i )
)
72 oveq2 6200 . . . . . . . . . . 11  |-  ( l  =  i  ->  (
n  -  l )  =  ( n  -  i ) )
7372fveq2d 5795 . . . . . . . . . 10  |-  ( l  =  i  ->  ( C `  ( n  -  l ) )  =  ( C `  ( n  -  i
) ) )
7445fveq1i 5792 . . . . . . . . . 10  |-  ( C `
 ( n  -  i ) )  =  ( (coe1 `  L ) `  ( n  -  i
) )
7573, 74syl6eq 2508 . . . . . . . . 9  |-  ( l  =  i  ->  ( C `  ( n  -  l ) )  =  ( (coe1 `  L
) `  ( n  -  i ) ) )
7671, 75oveq12d 6210 . . . . . . . 8  |-  ( l  =  i  ->  (
( A `  l
)  .*  ( C `
 ( n  -  l ) ) )  =  ( ( (coe1 `  K ) `  i
)  .*  ( (coe1 `  L ) `  (
n  -  i ) ) ) )
7776cbvmptv 4483 . . . . . . 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 6208 . . . . 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 2497 . . . 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 2496 . . 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 2822 . 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 ) )
831ply1rng 17812 . . . 4  |-  ( R  e.  Ring  ->  P  e. 
Ring )
844, 2rngcl 16766 . . . 4  |-  ( ( P  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( K  .X.  L )  e.  B )
8583, 84syl3an1 1252 . . 3  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( K  .X.  L )  e.  B )
86 eqid 2451 . . . 4  |-  ( 0g
`  P )  =  ( 0g `  P
)
87 rngcmn 16783 . . . . . 6  |-  ( P  e.  Ring  ->  P  e. CMnd
)
8883, 87syl 16 . . . . 5  |-  ( R  e.  Ring  ->  P  e. CMnd
)
89883ad2ant1 1009 . . . 4  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  P  e. CMnd )
90 nn0ex 10688 . . . . 5  |-  NN0  e.  _V
9190a1i 11 . . . 4  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  NN0  e.  _V )
921ply1lmod 17816 . . . . . . . 8  |-  ( R  e.  Ring  ->  P  e. 
LMod )
93923ad2ant1 1009 . . . . . . 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 11898 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  ( 0 ... k )  e.  Fin )
97 simpll1 1027 . . . . . . . . . 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 1219 . . . . . . . . 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 2822 . . . . . . . 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 16565 . . . . . . 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 17817 . . . . . . . . 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 5795 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  ( Base `  R
)  =  ( Base `  (Scalar `  P )
) )
109104, 108eleqtrd 2541 . . . . . 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 )
) )
11022rngmgp 16759 . . . . . . . . . 10  |-  ( P  e.  Ring  ->  M  e. 
Mnd )
11183, 110syl 16 . . . . . . . . 9  |-  ( R  e.  Ring  ->  M  e. 
Mnd )
1121113ad2ant1 1009 . . . . . . . 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 17780 . . . . . . . . 9  |-  ( R  e.  Ring  ->  X  e.  B )
1161153ad2ant1 1009 . . . . . . . 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 16704 . . . . . . . 8  |-  B  =  ( Base `  M
)
119118, 21mulgnn0cl 15747 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  k  e.  NN0  /\  X  e.  B )  ->  (
k  .^  X )  e.  B )
120113, 114, 117, 119syl3anc 1219 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  k  e.  NN0 )  ->  ( k  .^  X )  e.  B
)
121 eqid 2451 . . . . . . 7  |-  (Scalar `  P )  =  (Scalar `  P )
122 eqid 2451 . . . . . . 7  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
1234, 121, 28, 122lmodvscl 17073 . . . . . 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 1219 . . . . 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 2451 . . . . 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 5968 . . . 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 30991 . . . 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 16503 . . 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 2451 . . . 4  |-  (coe1 `  ( K  .X.  L ) )  =  (coe1 `  ( K  .X.  L ) )
130 eqid 2451 . . . 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 30980 . . 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 1219 . 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3070   [_csb 3388   class class class wbr 4392    |-> cmpt 4450   ` cfv 5518  (class class class)co 6192   finSupp cfsupp 7723   0cc0 9385    - cmin 9698   NN0cn0 10682   ...cfz 11540   Basecbs 14278   .rcmulr 14343  Scalarcsca 14345   .scvsca 14346   0gc0g 14482    gsumg cgsu 14483   Mndcmnd 15513  .gcmg 15518  CMndccmn 16383  mulGrpcmgp 16698   Ringcrg 16753   LModclmod 17056  var1cv1 17741  Poly1cpl1 17742  coe1cco1 17743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-ofr 6423  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-1o 7022  df-2o 7023  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-ixp 7366  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fsupp 7724  df-oi 7827  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-seq 11910  df-hash 12207  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-sca 14358  df-vsca 14359  df-tset 14361  df-ple 14362  df-0g 14484  df-gsum 14485  df-mre 14628  df-mrc 14629  df-acs 14631  df-mnd 15519  df-mhm 15568  df-submnd 15569  df-grp 15649  df-minusg 15650  df-sbg 15651  df-mulg 15652  df-subg 15782  df-ghm 15849  df-cntz 15939  df-cmn 16385  df-abl 16386  df-mgp 16699  df-ur 16711  df-srg 16715  df-rng 16755  df-subrg 16971  df-lmod 17058  df-lss 17122  df-psr 17531  df-mvr 17532  df-mpl 17533  df-opsr 17535  df-psr1 17745  df-vr1 17746  df-ply1 17747  df-coe1 17748
This theorem is referenced by: (None)
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