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Theorem ply1coeOLD 18125
Description: Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) Obsolete as of 28-Sep-2019. Use ply1coe 18124 instead. (New usage is discouraged.)
Hypotheses
Ref Expression
ply1coeOLD.p  |-  P  =  (Poly1 `  R )
ply1coeOLD.x  |-  X  =  (var1 `  R )
ply1coeOLD.b  |-  B  =  ( Base `  P
)
ply1coeOLD.n  |-  .x.  =  ( .s `  P )
ply1coeOLD.m  |-  M  =  (mulGrp `  P )
ply1coeOLD.e  |-  .^  =  (.g
`  M )
ply1coeOLD.a  |-  A  =  (coe1 `  K )
ply1coeOLD.r  |-  R  e. 
_V
Assertion
Ref Expression
ply1coeOLD  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  K  =  ( P  gsumg  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) ) )
Distinct variable groups:    A, k    B, k    k, K    k, X   
.^ , k    R, k    .x. , k
Allowed substitution hints:    P( k)    M( k)

Proof of Theorem ply1coeOLD
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
2 psr1baslem 18011 . . 3  |-  ( NN0 
^m  1o )  =  { d  e.  ( NN0  ^m  1o )  |  ( `' d
" NN )  e. 
Fin }
3 eqid 2467 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
4 eqid 2467 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 1onn 7288 . . . 4  |-  1o  e.  om
65a1i 11 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  1o  e.  om )
7 ply1coeOLD.p . . . 4  |-  P  =  (Poly1 `  R )
8 eqid 2467 . . . 4  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
9 ply1coeOLD.b . . . 4  |-  B  =  ( Base `  P
)
107, 8, 9ply1bas 18021 . . 3  |-  B  =  ( Base `  ( 1o mPoly  R ) )
11 ply1coeOLD.n . . . 4  |-  .x.  =  ( .s `  P )
127, 1, 11ply1vsca 18054 . . 3  |-  .x.  =  ( .s `  ( 1o mPoly  R ) )
13 crngrng 17005 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
1413adantr 465 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  R  e.  Ring )
15 simpr 461 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  K  e.  B )
161, 2, 3, 4, 6, 10, 12, 14, 15mplcoe1 17914 . 2  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  K  =  ( ( 1o mPoly  R )  gsumg  ( a  e.  ( NN0  ^m  1o ) 
|->  ( ( K `  a )  .x.  (
b  e.  ( NN0 
^m  1o )  |->  if ( b  =  a ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) ) ) ) )
17 ply1coeOLD.a . . . . . . 7  |-  A  =  (coe1 `  K )
1817fvcoe1 18033 . . . . . 6  |-  ( ( K  e.  B  /\  a  e.  ( NN0  ^m  1o ) )  -> 
( K `  a
)  =  ( A `
 ( a `  (/) ) ) )
1918adantll 713 . . . . 5  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( K `  a )  =  ( A `  ( a `
 (/) ) ) )
205a1i 11 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  1o  e.  om )
21 eqid 2467 . . . . . . 7  |-  (mulGrp `  ( 1o mPoly  R ) )  =  (mulGrp `  ( 1o mPoly  R ) )
22 eqid 2467 . . . . . . 7  |-  (.g `  (mulGrp `  ( 1o mPoly  R )
) )  =  (.g `  (mulGrp `  ( 1o mPoly  R ) ) )
23 eqid 2467 . . . . . . 7  |-  ( 1o mVar  R )  =  ( 1o mVar  R )
24 simpll 753 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  R  e.  CRing
)
25 simpr 461 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  a  e.  ( NN0  ^m  1o ) )
261, 2, 3, 4, 20, 21, 22, 23, 24, 25mplcoe2 17919 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( b  e.  ( NN0  ^m  1o )  |->  if ( b  =  a ,  ( 1r `  R ) ,  ( 0g `  R ) ) )  =  ( (mulGrp `  ( 1o mPoly  R ) ) 
gsumg  ( c  e.  1o  |->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R )
) ) ( ( 1o mVar  R ) `  c ) ) ) ) )
27 df1o2 7142 . . . . . . . . 9  |-  1o  =  { (/) }
28 mpteq1 4527 . . . . . . . . 9  |-  ( 1o  =  { (/) }  ->  ( c  e.  1o  |->  ( ( a `  c
) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R
) `  c )
) )  =  ( c  e.  { (/) } 
|->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R )
) ) ( ( 1o mVar  R ) `  c ) ) ) )
2927, 28ax-mp 5 . . . . . . . 8  |-  ( c  e.  1o  |->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R
) `  c )
) )  =  ( c  e.  { (/) } 
|->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R )
) ) ( ( 1o mVar  R ) `  c ) ) )
3029oveq2i 6294 . . . . . . 7  |-  ( (mulGrp `  ( 1o mPoly  R )
)  gsumg  ( c  e.  1o  |->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R )
) ) ( ( 1o mVar  R ) `  c ) ) ) )  =  ( (mulGrp `  ( 1o mPoly  R )
)  gsumg  ( c  e.  { (/)
}  |->  ( ( a `
 c ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R ) `
 c ) ) ) )
311mplcrng 17902 . . . . . . . . . . . . 13  |-  ( ( 1o  e.  om  /\  R  e.  CRing )  -> 
( 1o mPoly  R )  e.  CRing )
325, 31mpan 670 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  ( 1o mPoly  R )  e.  CRing )
3332adantr 465 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  CRing )
3421crngmgp 17003 . . . . . . . . . . 11  |-  ( ( 1o mPoly  R )  e. 
CRing  ->  (mulGrp `  ( 1o mPoly  R ) )  e. CMnd )
3533, 34syl 16 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (mulGrp `  ( 1o mPoly  R )
)  e. CMnd )
3635adantr 465 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  (mulGrp `  ( 1o mPoly  R ) )  e. CMnd
)
37 cmnmnd 16616 . . . . . . . . 9  |-  ( (mulGrp `  ( 1o mPoly  R )
)  e. CMnd  ->  (mulGrp `  ( 1o mPoly  R ) )  e.  Mnd )
3836, 37syl 16 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  (mulGrp `  ( 1o mPoly  R ) )  e. 
Mnd )
39 0ex 4577 . . . . . . . . 9  |-  (/)  e.  _V
4039a1i 11 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  (/)  e.  _V )
41 ply1coeOLD.e . . . . . . . . . . . 12  |-  .^  =  (.g
`  M )
4221, 10mgpbas 16946 . . . . . . . . . . . . 13  |-  B  =  ( Base `  (mulGrp `  ( 1o mPoly  R )
) )
4342a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  =  ( Base `  (mulGrp `  ( 1o mPoly  R )
) ) )
44 ply1coeOLD.m . . . . . . . . . . . . . 14  |-  M  =  (mulGrp `  P )
4544, 9mgpbas 16946 . . . . . . . . . . . . 13  |-  B  =  ( Base `  M
)
4645a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  =  ( Base `  M
) )
47 ssv 3524 . . . . . . . . . . . . 13  |-  B  C_  _V
4847a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  C_ 
_V )
49 ovex 6308 . . . . . . . . . . . . 13  |-  ( a ( +g  `  (mulGrp `  ( 1o mPoly  R )
) ) b )  e.  _V
5049a1i 11 . . . . . . . . . . . 12  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  ( a  e. 
_V  /\  b  e.  _V ) )  ->  (
a ( +g  `  (mulGrp `  ( 1o mPoly  R )
) ) b )  e.  _V )
51 eqid 2467 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  P )  =  ( .r `  P
)
527, 1, 51ply1mulr 18055 . . . . . . . . . . . . . . . 16  |-  ( .r
`  P )  =  ( .r `  ( 1o mPoly  R ) )
5321, 52mgpplusg 16944 . . . . . . . . . . . . . . 15  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R )
) )
5444, 51mgpplusg 16944 . . . . . . . . . . . . . . 15  |-  ( .r
`  P )  =  ( +g  `  M
)
5553, 54eqtr3i 2498 . . . . . . . . . . . . . 14  |-  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) )  =  ( +g  `  M
)
5655oveqi 6296 . . . . . . . . . . . . 13  |-  ( a ( +g  `  (mulGrp `  ( 1o mPoly  R )
) ) b )  =  ( a ( +g  `  M ) b )
5756a1i 11 . . . . . . . . . . . 12  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  ( a  e. 
_V  /\  b  e.  _V ) )  ->  (
a ( +g  `  (mulGrp `  ( 1o mPoly  R )
) ) b )  =  ( a ( +g  `  M ) b ) )
5822, 41, 43, 46, 48, 50, 57mulgpropd 15982 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (.g `  (mulGrp `  ( 1o mPoly  R
) ) )  = 
.^  )
5958oveqd 6300 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  =  ( ( a `  (/) )  .^  X ) )
6059adantr 465 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  =  ( ( a `  (/) )  .^  X ) )
617ply1crng 18024 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  P  e.  CRing
)
6261adantr 465 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  P  e.  CRing )
63 crngrng 17005 . . . . . . . . . . . 12  |-  ( P  e.  CRing  ->  P  e.  Ring )
6444rngmgp 17001 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  M  e. 
Mnd )
6562, 63, 643syl 20 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  M  e.  Mnd )
6665adantr 465 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  M  e.  Mnd )
67 elmapi 7440 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  1o )  ->  a : 1o --> NN0 )
68 0lt1o 7154 . . . . . . . . . . . 12  |-  (/)  e.  1o
69 ffvelrn 6018 . . . . . . . . . . . 12  |-  ( ( a : 1o --> NN0  /\  (/) 
e.  1o )  -> 
( a `  (/) )  e. 
NN0 )
7067, 68, 69sylancl 662 . . . . . . . . . . 11  |-  ( a  e.  ( NN0  ^m  1o )  ->  ( a `
 (/) )  e.  NN0 )
7170adantl 466 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( a `  (/) )  e.  NN0 )
72 ply1coeOLD.x . . . . . . . . . . . . 13  |-  X  =  (var1 `  R )
7372, 7, 9vr1cl 18045 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  X  e.  B )
7414, 73syl 16 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  X  e.  B )
7574adantr 465 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  X  e.  B )
7645, 41mulgnn0cl 15965 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( a `  (/) )  e. 
NN0  /\  X  e.  B )  ->  (
( a `  (/) )  .^  X )  e.  B
)
7766, 71, 75, 76syl3anc 1228 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) )  .^  X )  e.  B
)
7860, 77eqeltrd 2555 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  e.  B )
79 fveq2 5865 . . . . . . . . . 10  |-  ( c  =  (/)  ->  ( a `
 c )  =  ( a `  (/) ) )
80 fveq2 5865 . . . . . . . . . . 11  |-  ( c  =  (/)  ->  ( ( 1o mVar  R ) `  c )  =  ( ( 1o mVar  R ) `
 (/) ) )
8172vr1val 18018 . . . . . . . . . . 11  |-  X  =  ( ( 1o mVar  R
) `  (/) )
8280, 81syl6eqr 2526 . . . . . . . . . 10  |-  ( c  =  (/)  ->  ( ( 1o mVar  R ) `  c )  =  X )
8379, 82oveq12d 6301 . . . . . . . . 9  |-  ( c  =  (/)  ->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R
) `  c )
)  =  ( ( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X ) )
8442, 83gsumsn 16781 . . . . . . . 8  |-  ( ( (mulGrp `  ( 1o mPoly  R ) )  e.  Mnd  /\  (/)  e.  _V  /\  (
( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  e.  B )  ->  ( (mulGrp `  ( 1o mPoly  R ) ) 
gsumg  ( c  e.  { (/)
}  |->  ( ( a `
 c ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R ) `
 c ) ) ) )  =  ( ( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X ) )
8538, 40, 78, 84syl3anc 1228 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (mulGrp `  ( 1o mPoly  R )
)  gsumg  ( c  e.  { (/)
}  |->  ( ( a `
 c ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R ) `
 c ) ) ) )  =  ( ( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X ) )
8630, 85syl5eq 2520 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (mulGrp `  ( 1o mPoly  R )
)  gsumg  ( c  e.  1o  |->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R )
) ) ( ( 1o mVar  R ) `  c ) ) ) )  =  ( ( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X ) )
8726, 86, 603eqtrd 2512 . . . . 5  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( b  e.  ( NN0  ^m  1o )  |->  if ( b  =  a ,  ( 1r `  R ) ,  ( 0g `  R ) ) )  =  ( ( a `
 (/) )  .^  X
) )
8819, 87oveq12d 6301 . . . 4  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( ( K `  a )  .x.  ( b  e.  ( NN0  ^m  1o ) 
|->  if ( b  =  a ,  ( 1r
`  R ) ,  ( 0g `  R
) ) ) )  =  ( ( A `
 ( a `  (/) ) )  .x.  (
( a `  (/) )  .^  X ) ) )
8988mpteq2dva 4533 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
a  e.  ( NN0 
^m  1o )  |->  ( ( K `  a
)  .x.  ( b  e.  ( NN0  ^m  1o )  |->  if ( b  =  a ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) ) )  =  ( a  e.  ( NN0 
^m  1o )  |->  ( ( A `  (
a `  (/) ) ) 
.x.  ( ( a `
 (/) )  .^  X
) ) ) )
9089oveq2d 6299 . 2  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( 1o mPoly  R )  gsumg  ( a  e.  ( NN0 
^m  1o )  |->  ( ( K `  a
)  .x.  ( b  e.  ( NN0  ^m  1o )  |->  if ( b  =  a ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) ) ) )  =  ( ( 1o mPoly  R
)  gsumg  ( a  e.  ( NN0  ^m  1o ) 
|->  ( ( A `  ( a `  (/) ) ) 
.x.  ( ( a `
 (/) )  .^  X
) ) ) ) )
91 nn0ex 10800 . . . . . 6  |-  NN0  e.  _V
9291mptex 6130 . . . . 5  |-  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  e.  _V
9392a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  e.  _V )
94 fvex 5875 . . . . . 6  |-  (Poly1 `  R
)  e.  _V
957, 94eqeltri 2551 . . . . 5  |-  P  e. 
_V
9695a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  P  e.  _V )
97 ovex 6308 . . . . 5  |-  ( 1o mPoly  R )  e.  _V
9897a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  _V )
999, 10eqtr3i 2498 . . . . 5  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
10099a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( Base `  P )  =  ( Base `  ( 1o mPoly  R ) ) )
101 eqid 2467 . . . . . 6  |-  ( +g  `  P )  =  ( +g  `  P )
1027, 1, 101ply1plusg 18053 . . . . 5  |-  ( +g  `  P )  =  ( +g  `  ( 1o mPoly  R ) )
103102a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( +g  `  P )  =  ( +g  `  ( 1o mPoly  R ) ) )
10493, 96, 98, 100, 103gsumpropd 15823 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( P  gsumg  ( k  e.  NN0  |->  ( ( A `  k )  .x.  (
k  .^  X )
) ) )  =  ( ( 1o mPoly  R
)  gsumg  ( k  e.  NN0  |->  ( ( A `  k )  .x.  (
k  .^  X )
) ) ) )
105 eqid 2467 . . . 4  |-  ( 0g
`  ( 1o mPoly  R
) )  =  ( 0g `  ( 1o mPoly  R ) )
1061mpllmod 17900 . . . . . 6  |-  ( ( 1o  e.  om  /\  R  e.  Ring )  -> 
( 1o mPoly  R )  e.  LMod )
1075, 14, 106sylancr 663 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  LMod )
108 lmodcmn 17353 . . . . 5  |-  ( ( 1o mPoly  R )  e. 
LMod  ->  ( 1o mPoly  R
)  e. CMnd )
109107, 108syl 16 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e. CMnd )
11091a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  NN0  e.  _V )
111107adantr 465 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( 1o mPoly  R
)  e.  LMod )
112 eqid 2467 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
11317, 9, 7, 112coe1f 18037 . . . . . . . 8  |-  ( K  e.  B  ->  A : NN0 --> ( Base `  R
) )
114113adantl 466 . . . . . . 7  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A : NN0 --> ( Base `  R
) )
115114ffvelrnda 6020 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  (
Base `  R )
)
11665adantr 465 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  M  e.  Mnd )
117 simpr 461 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
11874adantr 465 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  X  e.  B
)
11945, 41mulgnn0cl 15965 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  k  e.  NN0  /\  X  e.  B )  ->  (
k  .^  X )  e.  B )
120116, 117, 118, 119syl3anc 1228 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( k  .^  X )  e.  B
)
121 ply1coeOLD.r . . . . . . . 8  |-  R  e. 
_V
122 simpl 457 . . . . . . . . 9  |-  ( ( 1o  e.  om  /\  R  e.  _V )  ->  1o  e.  om )
123 simpr 461 . . . . . . . . 9  |-  ( ( 1o  e.  om  /\  R  e.  _V )  ->  R  e.  _V )
1241, 122, 123mplsca 17894 . . . . . . . 8  |-  ( ( 1o  e.  om  /\  R  e.  _V )  ->  R  =  (Scalar `  ( 1o mPoly  R ) ) )
1255, 121, 124mp2an 672 . . . . . . 7  |-  R  =  (Scalar `  ( 1o mPoly  R ) )
12610, 125, 12, 112lmodvscl 17324 . . . . . 6  |-  ( ( ( 1o mPoly  R )  e.  LMod  /\  ( A `  k )  e.  (
Base `  R )  /\  ( k  .^  X
)  e.  B )  ->  ( ( A `
 k )  .x.  ( k  .^  X
) )  e.  B
)
127111, 115, 120, 126syl3anc 1228 . . . . 5  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( ( A `
 k )  .x.  ( k  .^  X
) )  e.  B
)
128 eqid 2467 . . . . 5  |-  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  =  ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) )
129127, 128fmptd 6044 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) : NN0 --> B )
130 funmpt 5623 . . . . . . 7  |-  Fun  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )
131 fvex 5875 . . . . . . 7  |-  ( 0g
`  ( 1o mPoly  R
) )  e.  _V
13292, 130, 1313pm3.2i 1174 . . . . . 6  |-  ( ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  e.  _V  /\  Fun  ( k  e.  NN0  |->  ( ( A `  k )  .x.  (
k  .^  X )
) )  /\  ( 0g `  ( 1o mPoly  R
) )  e.  _V )
133132a1i 11 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( k  e.  NN0  |->  ( ( A `  k )  .x.  (
k  .^  X )
) )  e.  _V  /\ 
Fun  ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) )  /\  ( 0g `  ( 1o mPoly  R ) )  e. 
_V ) )
13417, 9, 7, 3coe1sfi 18039 . . . . . . 7  |-  ( K  e.  B  ->  A finSupp  ( 0g `  R ) )
135134adantl 466 . . . . . 6  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A finSupp  ( 0g `  R ) )
136135fsuppimpd 7835 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( A supp  ( 0g `  R
) )  e.  Fin )
137114feqmptd 5919 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A  =  ( k  e. 
NN0  |->  ( A `  k ) ) )
138137eqcomd 2475 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( A `
 k ) )  =  A )
139138oveq1d 6298 . . . . . . 7  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( k  e.  NN0  |->  ( A `  k ) ) supp  ( 0g `  R ) )  =  ( A supp  ( 0g
`  R ) ) )
140 ssid 3523 . . . . . . 7  |-  ( A supp  ( 0g `  R
) )  C_  ( A supp  ( 0g `  R
) )
141139, 140syl6eqss 3554 . . . . . 6  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( k  e.  NN0  |->  ( A `  k ) ) supp  ( 0g `  R ) )  C_  ( A supp  ( 0g `  R ) ) )
14210, 125, 12, 3, 105lmod0vs 17340 . . . . . . 7  |-  ( ( ( 1o mPoly  R )  e.  LMod  /\  a  e.  B )  ->  (
( 0g `  R
)  .x.  a )  =  ( 0g `  ( 1o mPoly  R ) ) )
143107, 142sylan 471 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  B
)  ->  ( ( 0g `  R )  .x.  a )  =  ( 0g `  ( 1o mPoly  R ) ) )
144 fvex 5875 . . . . . . 7  |-  ( A `
 k )  e. 
_V
145144a1i 11 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  _V )
146 fvex 5875 . . . . . . 7  |-  ( 0g
`  R )  e. 
_V
147146a1i 11 . . . . . 6  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 0g `  R )  e. 
_V )
148141, 143, 145, 120, 147suppssov1 6932 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( k  e.  NN0  |->  ( ( A `  k )  .x.  (
k  .^  X )
) ) supp  ( 0g
`  ( 1o mPoly  R
) ) )  C_  ( A supp  ( 0g `  R ) ) )
149 suppssfifsupp 7843 . . . . 5  |-  ( ( ( ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) )  e. 
_V  /\  Fun  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  /\  ( 0g `  ( 1o mPoly  R ) )  e.  _V )  /\  ( ( A supp  ( 0g `  R ) )  e.  Fin  /\  (
( k  e.  NN0  |->  ( ( A `  k )  .x.  (
k  .^  X )
) ) supp  ( 0g
`  ( 1o mPoly  R
) ) )  C_  ( A supp  ( 0g `  R ) ) ) )  ->  ( k  e.  NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) ) finSupp  ( 0g `  ( 1o mPoly  R
) ) )
150133, 136, 148, 149syl12anc 1226 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) finSupp 
( 0g `  ( 1o mPoly  R ) ) )
151 eqid 2467 . . . . . 6  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) )
15227, 91, 39, 151mapsnf1o2 7466 . . . . 5  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) : ( NN0  ^m  1o )
-1-1-onto-> NN0
153152a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) : ( NN0  ^m  1o ) -1-1-onto-> NN0 )
15410, 105, 109, 110, 129, 150, 153gsumf1o 16724 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( 1o mPoly  R )  gsumg  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) )  =  ( ( 1o mPoly  R )  gsumg  ( ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  o.  ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) ) ) )
155 eqidd 2468 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) )  =  ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) )
156 eqidd 2468 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  =  ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) ) )
157 fveq2 5865 . . . . . 6  |-  ( k  =  ( a `  (/) )  ->  ( A `  k )  =  ( A `  ( a `
 (/) ) ) )
158 oveq1 6290 . . . . . 6  |-  ( k  =  ( a `  (/) )  ->  ( k  .^  X )  =  ( ( a `  (/) )  .^  X ) )
159157, 158oveq12d 6301 . . . . 5  |-  ( k  =  ( a `  (/) )  ->  ( ( A `  k )  .x.  ( k  .^  X
) )  =  ( ( A `  (
a `  (/) ) ) 
.x.  ( ( a `
 (/) )  .^  X
) ) )
16071, 155, 156, 159fmptco 6053 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( k  e.  NN0  |->  ( ( A `  k )  .x.  (
k  .^  X )
) )  o.  (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) )  =  ( a  e.  ( NN0  ^m  1o )  |->  ( ( A `  ( a `
 (/) ) )  .x.  ( ( a `  (/) )  .^  X )
) ) )
161160oveq2d 6299 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( 1o mPoly  R )  gsumg  ( ( k  e.  NN0  |->  ( ( A `  k )  .x.  (
k  .^  X )
) )  o.  (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) ) )  =  ( ( 1o mPoly  R )  gsumg  ( a  e.  ( NN0 
^m  1o )  |->  ( ( A `  (
a `  (/) ) ) 
.x.  ( ( a `
 (/) )  .^  X
) ) ) ) )
162104, 154, 1613eqtrrd 2513 . 2  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( 1o mPoly  R )  gsumg  ( a  e.  ( NN0 
^m  1o )  |->  ( ( A `  (
a `  (/) ) ) 
.x.  ( ( a `
 (/) )  .^  X
) ) ) )  =  ( P  gsumg  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) ) )
16316, 90, 1623eqtrd 2512 1  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  K  =  ( P  gsumg  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ifcif 3939   {csn 4027   class class class wbr 4447    |-> cmpt 4505    o. ccom 5003   Fun wfun 5581   -->wf 5583   -1-1-onto->wf1o 5586   ` cfv 5587  (class class class)co 6283   omcom 6679   supp csupp 6901   1oc1o 7123    ^m cmap 7420   Fincfn 7516   finSupp cfsupp 7828   NN0cn0 10794   Basecbs 14489   +g cplusg 14554   .rcmulr 14555  Scalarcsca 14557   .scvsca 14558   0gc0g 14694    gsumg cgsu 14695   Mndcmnd 15725  .gcmg 15730  CMndccmn 16601  mulGrpcmgp 16940   1rcur 16952   Ringcrg 16995   CRingccrg 16996   LModclmod 17307   mVar cmvr 17788   mPoly cmpl 17789  PwSer1cps1 18001  var1cv1 18002  Poly1cpl1 18003  coe1cco1 18004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-ofr 6524  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-oi 7934  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-fzo 11792  df-seq 12075  df-hash 12373  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-sca 14570  df-vsca 14571  df-tset 14573  df-ple 14574  df-0g 14696  df-gsum 14697  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-mhm 15783  df-submnd 15784  df-grp 15864  df-minusg 15865  df-sbg 15866  df-mulg 15867  df-subg 16000  df-ghm 16067  df-cntz 16157  df-cmn 16603  df-abl 16604  df-mgp 16941  df-ur 16953  df-srg 16957  df-rng 16997  df-cring 16998  df-subrg 17222  df-lmod 17309  df-lss 17374  df-psr 17792  df-mvr 17793  df-mpl 17794  df-opsr 17796  df-psr1 18006  df-vr1 18007  df-ply1 18008  df-coe1 18009
This theorem is referenced by: (None)
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