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Theorem ply1coeOLD 18212
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 18211 instead. (New usage is discouraged.) (Proof modification 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 2443 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
2 psr1baslem 18098 . . 3  |-  ( NN0 
^m  1o )  =  { d  e.  ( NN0  ^m  1o )  |  ( `' d
" NN )  e. 
Fin }
3 eqid 2443 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
4 eqid 2443 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 1onn 7290 . . . 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 2443 . . . 4  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
9 ply1coeOLD.b . . . 4  |-  B  =  ( Base `  P
)
107, 8, 9ply1bas 18108 . . 3  |-  B  =  ( Base `  ( 1o mPoly  R ) )
11 ply1coeOLD.n . . . 4  |-  .x.  =  ( .s `  P )
127, 1, 11ply1vsca 18141 . . 3  |-  .x.  =  ( .s `  ( 1o mPoly  R ) )
13 crngring 17083 . . . 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 18001 . 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 18120 . . . . . 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 2443 . . . . . . 7  |-  (mulGrp `  ( 1o mPoly  R ) )  =  (mulGrp `  ( 1o mPoly  R ) )
22 eqid 2443 . . . . . . 7  |-  (.g `  (mulGrp `  ( 1o mPoly  R )
) )  =  (.g `  (mulGrp `  ( 1o mPoly  R ) ) )
23 eqid 2443 . . . . . . 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 18006 . . . . . 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 7144 . . . . . . . . 9  |-  1o  =  { (/) }
28 mpteq1 4517 . . . . . . . . 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 6292 . . . . . . 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 17989 . . . . . . . . . . . . 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 17080 . . . . . . . . . . 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 16687 . . . . . . . . 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 4567 . . . . . . . . 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 17021 . . . . . . . . . . . . 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 17021 . . . . . . . . . . . . 13  |-  B  =  ( Base `  M
)
4645a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  =  ( Base `  M
) )
47 ssv 3509 . . . . . . . . . . . . 13  |-  B  C_  _V
4847a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  C_ 
_V )
49 ovex 6309 . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  P )  =  ( .r `  P
)
527, 1, 51ply1mulr 18142 . . . . . . . . . . . . . . . 16  |-  ( .r
`  P )  =  ( .r `  ( 1o mPoly  R ) )
5321, 52mgpplusg 17019 . . . . . . . . . . . . . . 15  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R )
) )
5444, 51mgpplusg 17019 . . . . . . . . . . . . . . 15  |-  ( .r
`  P )  =  ( +g  `  M
)
5553, 54eqtr3i 2474 . . . . . . . . . . . . . 14  |-  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) )  =  ( +g  `  M
)
5655oveqi 6294 . . . . . . . . . . . . 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 16049 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (.g `  (mulGrp `  ( 1o mPoly  R
) ) )  = 
.^  )
5958oveqd 6298 . . . . . . . . . 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 18111 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  P  e.  CRing
)
6261adantr 465 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  P  e.  CRing )
63 crngring 17083 . . . . . . . . . . . 12  |-  ( P  e.  CRing  ->  P  e.  Ring )
6444ringmgp 17078 . . . . . . . . . . . 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 7442 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  1o )  ->  a : 1o --> NN0 )
68 0lt1o 7156 . . . . . . . . . . . 12  |-  (/)  e.  1o
69 ffvelrn 6014 . . . . . . . . . . . 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 18132 . . . . . . . . . . . 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 16032 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( a `  (/) )  e. 
NN0  /\  X  e.  B )  ->  (
( a `  (/) )  .^  X )  e.  B
)
7766, 71, 75, 76syl3anc 1229 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) )  .^  X )  e.  B
)
7860, 77eqeltrd 2531 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  e.  B )
79 fveq2 5856 . . . . . . . . . 10  |-  ( c  =  (/)  ->  ( a `
 c )  =  ( a `  (/) ) )
80 fveq2 5856 . . . . . . . . . . 11  |-  ( c  =  (/)  ->  ( ( 1o mVar  R ) `  c )  =  ( ( 1o mVar  R ) `
 (/) ) )
8172vr1val 18105 . . . . . . . . . . 11  |-  X  =  ( ( 1o mVar  R
) `  (/) )
8280, 81syl6eqr 2502 . . . . . . . . . 10  |-  ( c  =  (/)  ->  ( ( 1o mVar  R ) `  c )  =  X )
8379, 82oveq12d 6299 . . . . . . . . 9  |-  ( c  =  (/)  ->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R
) `  c )
)  =  ( ( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X ) )
8442, 83gsumsn 16855 . . . . . . . 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 1229 . . . . . . 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 2496 . . . . . 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 2488 . . . . 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 6299 . . . 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 4523 . . 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 6297 . 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 10807 . . . . . 6  |-  NN0  e.  _V
9291mptex 6128 . . . . 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 5866 . . . . . 6  |-  (Poly1 `  R
)  e.  _V
957, 94eqeltri 2527 . . . . 5  |-  P  e. 
_V
9695a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  P  e.  _V )
97 ovex 6309 . . . . 5  |-  ( 1o mPoly  R )  e.  _V
9897a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  _V )
999, 10eqtr3i 2474 . . . . 5  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
10099a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( Base `  P )  =  ( Base `  ( 1o mPoly  R ) ) )
101 eqid 2443 . . . . . 6  |-  ( +g  `  P )  =  ( +g  `  P )
1027, 1, 101ply1plusg 18140 . . . . 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 15773 . . 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 2443 . . . 4  |-  ( 0g
`  ( 1o mPoly  R
) )  =  ( 0g `  ( 1o mPoly  R ) )
1061mpllmod 17987 . . . . . 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 17432 . . . . 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 2443 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
11317, 9, 7, 112coe1f 18124 . . . . . . . 8  |-  ( K  e.  B  ->  A : NN0 --> ( Base `  R
) )
114113adantl 466 . . . . . . 7  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A : NN0 --> ( Base `  R
) )
115114ffvelrnda 6016 . . . . . 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 16032 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  k  e.  NN0  /\  X  e.  B )  ->  (
k  .^  X )  e.  B )
120116, 117, 118, 119syl3anc 1229 . . . . . 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 17981 . . . . . . . 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 17403 . . . . . 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 1229 . . . . 5  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( ( A `
 k )  .x.  ( k  .^  X
) )  e.  B
)
128 eqid 2443 . . . . 5  |-  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  =  ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) )
129127, 128fmptd 6040 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) : NN0 --> B )
130 funmpt 5614 . . . . . . 7  |-  Fun  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )
131 fvex 5866 . . . . . . 7  |-  ( 0g
`  ( 1o mPoly  R
) )  e.  _V
13292, 130, 1313pm3.2i 1175 . . . . . 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 18126 . . . . . . 7  |-  ( K  e.  B  ->  A finSupp  ( 0g `  R ) )
135134adantl 466 . . . . . 6  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A finSupp  ( 0g `  R ) )
136135fsuppimpd 7838 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( A supp  ( 0g `  R
) )  e.  Fin )
137114feqmptd 5911 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A  =  ( k  e. 
NN0  |->  ( A `  k ) ) )
138137eqcomd 2451 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( A `
 k ) )  =  A )
139138oveq1d 6296 . . . . . . 7  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( k  e.  NN0  |->  ( A `  k ) ) supp  ( 0g `  R ) )  =  ( A supp  ( 0g
`  R ) ) )
140 ssid 3508 . . . . . . 7  |-  ( A supp  ( 0g `  R
) )  C_  ( A supp  ( 0g `  R
) )
141139, 140syl6eqss 3539 . . . . . 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 17419 . . . . . . 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 5866 . . . . . . 7  |-  ( A `
 k )  e. 
_V
145144a1i 11 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  _V )
146 fvex 5866 . . . . . . 7  |-  ( 0g
`  R )  e. 
_V
147146a1i 11 . . . . . 6  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 0g `  R )  e. 
_V )
148141, 143, 145, 120, 147suppssov1 6934 . . . . 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 7846 . . . . 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 1227 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) finSupp 
( 0g `  ( 1o mPoly  R ) ) )
151 eqid 2443 . . . . . 6  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) )
15227, 91, 39, 151mapsnf1o2 7468 . . . . 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 16798 . . 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 2444 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) )  =  ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) )
156 eqidd 2444 . . . . 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 5856 . . . . . 6  |-  ( k  =  ( a `  (/) )  ->  ( A `  k )  =  ( A `  ( a `
 (/) ) ) )
158 oveq1 6288 . . . . . 6  |-  ( k  =  ( a `  (/) )  ->  ( k  .^  X )  =  ( ( a `  (/) )  .^  X ) )
159157, 158oveq12d 6299 . . . . 5  |-  ( k  =  ( a `  (/) )  ->  ( ( A `  k )  .x.  ( k  .^  X
) )  =  ( ( A `  (
a `  (/) ) ) 
.x.  ( ( a `
 (/) )  .^  X
) ) )
16071, 155, 156, 159fmptco 6049 . . . 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 6297 . . 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 2489 . 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 2488 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 974    = wceq 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461   (/)c0 3770   ifcif 3926   {csn 4014   class class class wbr 4437    |-> cmpt 4495    o. ccom 4993   Fun wfun 5572   -->wf 5574   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   omcom 6685   supp csupp 6903   1oc1o 7125    ^m cmap 7422   Fincfn 7518   finSupp cfsupp 7831   NN0cn0 10801   Basecbs 14509   +g cplusg 14574   .rcmulr 14575  Scalarcsca 14577   .scvsca 14578   0gc0g 14714    gsumg cgsu 14715   Mndcmnd 15793  .gcmg 15930  CMndccmn 16672  mulGrpcmgp 17015   1rcur 17027   Ringcrg 17072   CRingccrg 17073   LModclmod 17386   mVar cmvr 17875   mPoly cmpl 17876  PwSer1cps1 18088  var1cv1 18089  Poly1cpl1 18090  coe1cco1 18091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-fzo 11804  df-seq 12087  df-hash 12385  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-tset 14593  df-ple 14594  df-0g 14716  df-gsum 14717  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-mulg 15934  df-subg 16072  df-ghm 16139  df-cntz 16229  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-srg 17032  df-ring 17074  df-cring 17075  df-subrg 17301  df-lmod 17388  df-lss 17453  df-psr 17879  df-mvr 17880  df-mpl 17881  df-opsr 17883  df-psr1 18093  df-vr1 18094  df-ply1 18095  df-coe1 18096
This theorem is referenced by: (None)
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