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Theorem ply1coe 16368
Description: Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypotheses
Ref Expression
ply1coe.p  |-  P  =  (Poly1 `  R )
ply1coe.x  |-  X  =  (var1 `  R )
ply1coe.b  |-  B  =  ( Base `  P
)
ply1coe.n  |-  .x.  =  ( .s `  P )
ply1coe.m  |-  M  =  (mulGrp `  P )
ply1coe.e  |-  .^  =  (.g
`  M )
ply1coe.a  |-  A  =  (coe1 `  K )
ply1coe.r  |-  R  e. 
_V
Assertion
Ref Expression
ply1coe  |-  ( ( 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 ply1coe
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
2 psr1baslem 16264 . . 3  |-  ( NN0 
^m  1o )  =  { d  e.  ( NN0  ^m  1o )  |  ( `' d
" NN )  e. 
Fin }
3 eqid 2283 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
4 eqid 2283 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 1onn 6637 . . . 4  |-  1o  e.  om
65a1i 10 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  1o  e.  om )
7 ply1coe.p . . . 4  |-  P  =  (Poly1 `  R )
8 eqid 2283 . . . 4  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
9 ply1coe.b . . . 4  |-  B  =  ( Base `  P
)
107, 8, 9ply1bas 16274 . . 3  |-  B  =  ( Base `  ( 1o mPoly  R ) )
11 ply1coe.n . . . 4  |-  .x.  =  ( .s `  P )
127, 1, 11ply1vsca 16304 . . 3  |-  .x.  =  ( .s `  ( 1o mPoly  R ) )
13 crngrng 15351 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
1413adantr 451 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  R  e.  Ring )
15 simpr 447 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  K  e.  B )
161, 2, 3, 4, 6, 10, 12, 14, 15mplcoe1 16209 . 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 ply1coe.a . . . . . . 7  |-  A  =  (coe1 `  K )
1817fvcoe1 16288 . . . . . 6  |-  ( ( K  e.  B  /\  a  e.  ( NN0  ^m  1o ) )  -> 
( K `  a
)  =  ( A `
 ( a `  (/) ) ) )
1918adantll 694 . . . . 5  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( K `  a )  =  ( A `  ( a `
 (/) ) ) )
205a1i 10 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  1o  e.  om )
21 eqid 2283 . . . . . . 7  |-  (mulGrp `  ( 1o mPoly  R ) )  =  (mulGrp `  ( 1o mPoly  R ) )
22 eqid 2283 . . . . . . 7  |-  (.g `  (mulGrp `  ( 1o mPoly  R )
) )  =  (.g `  (mulGrp `  ( 1o mPoly  R ) ) )
23 eqid 2283 . . . . . . 7  |-  ( 1o mVar  R )  =  ( 1o mVar  R )
24 simpll 730 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  R  e.  CRing
)
25 simpr 447 . . . . . . 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 16211 . . . . . 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 6491 . . . . . . . . 9  |-  1o  =  { (/) }
28 mpteq1 4100 . . . . . . . . 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 8 . . . . . . . 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 5869 . . . . . . 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 16197 . . . . . . . . . . . . 13  |-  ( ( 1o  e.  om  /\  R  e.  CRing )  -> 
( 1o mPoly  R )  e.  CRing )
325, 31mpan 651 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  ( 1o mPoly  R )  e.  CRing )
3332adantr 451 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  CRing )
3421crngmgp 15349 . . . . . . . . . . 11  |-  ( ( 1o mPoly  R )  e. 
CRing  ->  (mulGrp `  ( 1o mPoly  R ) )  e. CMnd )
3533, 34syl 15 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (mulGrp `  ( 1o mPoly  R )
)  e. CMnd )
3635adantr 451 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  (mulGrp `  ( 1o mPoly  R ) )  e. CMnd
)
37 cmnmnd 15104 . . . . . . . . 9  |-  ( (mulGrp `  ( 1o mPoly  R )
)  e. CMnd  ->  (mulGrp `  ( 1o mPoly  R ) )  e.  Mnd )
3836, 37syl 15 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  (mulGrp `  ( 1o mPoly  R ) )  e. 
Mnd )
39 0ex 4150 . . . . . . . . 9  |-  (/)  e.  _V
4039a1i 10 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  (/)  e.  _V )
41 ply1coe.e . . . . . . . . . . . 12  |-  .^  =  (.g
`  M )
4221, 10mgpbas 15331 . . . . . . . . . . . . 13  |-  B  =  ( Base `  (mulGrp `  ( 1o mPoly  R )
) )
4342a1i 10 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  =  ( Base `  (mulGrp `  ( 1o mPoly  R )
) ) )
44 ply1coe.m . . . . . . . . . . . . . 14  |-  M  =  (mulGrp `  P )
4544, 9mgpbas 15331 . . . . . . . . . . . . 13  |-  B  =  ( Base `  M
)
4645a1i 10 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  =  ( Base `  M
) )
47 ssv 3198 . . . . . . . . . . . . 13  |-  B  C_  _V
4847a1i 10 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  C_ 
_V )
49 ovex 5883 . . . . . . . . . . . . 13  |-  ( a ( +g  `  (mulGrp `  ( 1o mPoly  R )
) ) b )  e.  _V
5049a1i 10 . . . . . . . . . . . 12  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  ( a  e. 
_V  /\  b  e.  _V ) )  ->  (
a ( +g  `  (mulGrp `  ( 1o mPoly  R )
) ) b )  e.  _V )
51 eqid 2283 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  P )  =  ( .r `  P
)
527, 1, 51ply1mulr 16305 . . . . . . . . . . . . . . . 16  |-  ( .r
`  P )  =  ( .r `  ( 1o mPoly  R ) )
5321, 52mgpplusg 15329 . . . . . . . . . . . . . . 15  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R )
) )
5444, 51mgpplusg 15329 . . . . . . . . . . . . . . 15  |-  ( .r
`  P )  =  ( +g  `  M
)
5553, 54eqtr3i 2305 . . . . . . . . . . . . . 14  |-  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) )  =  ( +g  `  M
)
5655oveqi 5871 . . . . . . . . . . . . 13  |-  ( a ( +g  `  (mulGrp `  ( 1o mPoly  R )
) ) b )  =  ( a ( +g  `  M ) b )
5756a1i 10 . . . . . . . . . . . 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 14600 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (.g `  (mulGrp `  ( 1o mPoly  R
) ) )  = 
.^  )
5958oveqd 5875 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  =  ( ( a `  (/) )  .^  X ) )
6059adantr 451 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  =  ( ( a `  (/) )  .^  X ) )
617ply1crng 16277 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  P  e.  CRing
)
6261adantr 451 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  P  e.  CRing )
63 crngrng 15351 . . . . . . . . . . . 12  |-  ( P  e.  CRing  ->  P  e.  Ring )
6444rngmgp 15347 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  M  e. 
Mnd )
6562, 63, 643syl 18 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  M  e.  Mnd )
6665adantr 451 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  M  e.  Mnd )
67 elmapi 6792 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  1o )  ->  a : 1o --> NN0 )
68 0lt1o 6503 . . . . . . . . . . . 12  |-  (/)  e.  1o
69 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( a : 1o --> NN0  /\  (/) 
e.  1o )  -> 
( a `  (/) )  e. 
NN0 )
7067, 68, 69sylancl 643 . . . . . . . . . . 11  |-  ( a  e.  ( NN0  ^m  1o )  ->  ( a `
 (/) )  e.  NN0 )
7170adantl 452 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( a `  (/) )  e.  NN0 )
72 ply1coe.x . . . . . . . . . . . . 13  |-  X  =  (var1 `  R )
7372, 7, 9vr1cl 16294 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  X  e.  B )
7414, 73syl 15 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  X  e.  B )
7574adantr 451 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  X  e.  B )
7645, 41mulgnn0cl 14583 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( a `  (/) )  e. 
NN0  /\  X  e.  B )  ->  (
( a `  (/) )  .^  X )  e.  B
)
7766, 71, 75, 76syl3anc 1182 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) )  .^  X )  e.  B
)
7860, 77eqeltrd 2357 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  e.  B )
79 fveq2 5525 . . . . . . . . . 10  |-  ( c  =  (/)  ->  ( a `
 c )  =  ( a `  (/) ) )
80 fveq2 5525 . . . . . . . . . . 11  |-  ( c  =  (/)  ->  ( ( 1o mVar  R ) `  c )  =  ( ( 1o mVar  R ) `
 (/) ) )
8172vr1val 16271 . . . . . . . . . . 11  |-  X  =  ( ( 1o mVar  R
) `  (/) )
8280, 81syl6eqr 2333 . . . . . . . . . 10  |-  ( c  =  (/)  ->  ( ( 1o mVar  R ) `  c )  =  X )
8379, 82oveq12d 5876 . . . . . . . . 9  |-  ( c  =  (/)  ->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R
) `  c )
)  =  ( ( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X ) )
8442, 83gsumsn 15220 . . . . . . . 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 1182 . . . . . . 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 2327 . . . . . 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 2319 . . . . 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 5876 . . . 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 4106 . . 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 5874 . 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 9971 . . . . . 6  |-  NN0  e.  _V
9291mptex 5746 . . . . 5  |-  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  e.  _V
9392a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  e.  _V )
94 fvex 5539 . . . . . 6  |-  (Poly1 `  R
)  e.  _V
957, 94eqeltri 2353 . . . . 5  |-  P  e. 
_V
9695a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  P  e.  _V )
97 ovex 5883 . . . . 5  |-  ( 1o mPoly  R )  e.  _V
9897a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  _V )
999, 10eqtr3i 2305 . . . . 5  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
10099a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( Base `  P )  =  ( Base `  ( 1o mPoly  R ) ) )
101 eqid 2283 . . . . . 6  |-  ( +g  `  P )  =  ( +g  `  P )
1027, 1, 101ply1plusg 16303 . . . . 5  |-  ( +g  `  P )  =  ( +g  `  ( 1o mPoly  R ) )
103102a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( +g  `  P )  =  ( +g  `  ( 1o mPoly  R ) ) )
10493, 96, 98, 100, 103gsumpropd 14453 . . 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 2283 . . . 4  |-  ( 0g
`  ( 1o mPoly  R
) )  =  ( 0g `  ( 1o mPoly  R ) )
1061mpllmod 16195 . . . . . 6  |-  ( ( 1o  e.  om  /\  R  e.  Ring )  -> 
( 1o mPoly  R )  e.  LMod )
1076, 14, 106syl2anc 642 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  LMod )
108 lmodcmn 15673 . . . . 5  |-  ( ( 1o mPoly  R )  e. 
LMod  ->  ( 1o mPoly  R
)  e. CMnd )
109107, 108syl 15 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e. CMnd )
11091a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  NN0  e.  _V )
111107adantr 451 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( 1o mPoly  R
)  e.  LMod )
112 eqid 2283 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
11317, 9, 7, 112coe1f 16292 . . . . . . . 8  |-  ( K  e.  B  ->  A : NN0 --> ( Base `  R
) )
114113adantl 452 . . . . . . 7  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A : NN0 --> ( Base `  R
) )
115 ffvelrn 5663 . . . . . . 7  |-  ( ( A : NN0 --> ( Base `  R )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  ( Base `  R
) )
116114, 115sylan 457 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  (
Base `  R )
)
11765adantr 451 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  M  e.  Mnd )
118 simpr 447 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
11974adantr 451 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  X  e.  B
)
12045, 41mulgnn0cl 14583 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  k  e.  NN0  /\  X  e.  B )  ->  (
k  .^  X )  e.  B )
121117, 118, 119, 120syl3anc 1182 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( k  .^  X )  e.  B
)
122 ply1coe.r . . . . . . . 8  |-  R  e. 
_V
123 simpl 443 . . . . . . . . 9  |-  ( ( 1o  e.  om  /\  R  e.  _V )  ->  1o  e.  om )
124 simpr 447 . . . . . . . . 9  |-  ( ( 1o  e.  om  /\  R  e.  _V )  ->  R  e.  _V )
1251, 123, 124mplsca 16189 . . . . . . . 8  |-  ( ( 1o  e.  om  /\  R  e.  _V )  ->  R  =  (Scalar `  ( 1o mPoly  R ) ) )
1265, 122, 125mp2an 653 . . . . . . 7  |-  R  =  (Scalar `  ( 1o mPoly  R ) )
12710, 126, 12, 112lmodvscl 15644 . . . . . 6  |-  ( ( ( 1o mPoly  R )  e.  LMod  /\  ( A `  k )  e.  (
Base `  R )  /\  ( k  .^  X
)  e.  B )  ->  ( ( A `
 k )  .x.  ( k  .^  X
) )  e.  B
)
128111, 116, 121, 127syl3anc 1182 . . . . 5  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( ( A `
 k )  .x.  ( k  .^  X
) )  e.  B
)
129 eqid 2283 . . . . 5  |-  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  =  ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) )
130128, 129fmptd 5684 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) : NN0 --> B )
13117, 9, 7, 3coe1sfi 16293 . . . . . 6  |-  ( K  e.  B  ->  ( `' A " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin )
132131adantl 452 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( `' A " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin )
133114feqmptd 5575 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A  =  ( k  e. 
NN0  |->  ( A `  k ) ) )
134133cnveqd 4857 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  `' A  =  `' (
k  e.  NN0  |->  ( A `
 k ) ) )
135134imaeq1d 5011 . . . . . . 7  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( `' A " ( _V 
\  { ( 0g
`  R ) } ) )  =  ( `' ( k  e. 
NN0  |->  ( A `  k ) ) "
( _V  \  {
( 0g `  R
) } ) ) )
136 eqimss2 3231 . . . . . . 7  |-  ( ( `' A " ( _V 
\  { ( 0g
`  R ) } ) )  =  ( `' ( k  e. 
NN0  |->  ( A `  k ) ) "
( _V  \  {
( 0g `  R
) } ) )  ->  ( `' ( k  e.  NN0  |->  ( A `
 k ) )
" ( _V  \  { ( 0g `  R ) } ) )  C_  ( `' A " ( _V  \  { ( 0g `  R ) } ) ) )
137135, 136syl 15 . . . . . 6  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( `' ( k  e. 
NN0  |->  ( A `  k ) ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  ( `' A " ( _V  \  {
( 0g `  R
) } ) ) )
13810, 126, 12, 3, 105lmod0vs 15663 . . . . . . 7  |-  ( ( ( 1o mPoly  R )  e.  LMod  /\  a  e.  B )  ->  (
( 0g `  R
)  .x.  a )  =  ( 0g `  ( 1o mPoly  R ) ) )
139107, 138sylan 457 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  B
)  ->  ( ( 0g `  R )  .x.  a )  =  ( 0g `  ( 1o mPoly  R ) ) )
140 fvex 5539 . . . . . . 7  |-  ( A `
 k )  e. 
_V
141140a1i 10 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  _V )
142137, 139, 141, 121suppssov1 6075 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( `' ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) ) "
( _V  \  {
( 0g `  ( 1o mPoly  R ) ) } ) )  C_  ( `' A " ( _V 
\  { ( 0g
`  R ) } ) ) )
143 ssfi 7083 . . . . 5  |-  ( ( ( `' A "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin  /\  ( `' ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) ) "
( _V  \  {
( 0g `  ( 1o mPoly  R ) ) } ) )  C_  ( `' A " ( _V 
\  { ( 0g
`  R ) } ) ) )  -> 
( `' ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )
" ( _V  \  { ( 0g `  ( 1o mPoly  R ) ) } ) )  e. 
Fin )
144132, 142, 143syl2anc 642 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( `' ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) ) "
( _V  \  {
( 0g `  ( 1o mPoly  R ) ) } ) )  e.  Fin )
145 eqid 2283 . . . . . 6  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) )
14627, 91, 39, 145mapsnf1o2 6815 . . . . 5  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) : ( NN0  ^m  1o )
-1-1-onto-> NN0
147146a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) : ( NN0  ^m  1o ) -1-1-onto-> NN0 )
14810, 105, 109, 110, 130, 144, 147gsumf1o 15199 . . 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 `  (/) ) ) ) ) )
149 eqidd 2284 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) )  =  ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) )
150 eqidd 2284 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  =  ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) ) )
151 fveq2 5525 . . . . . 6  |-  ( k  =  ( a `  (/) )  ->  ( A `  k )  =  ( A `  ( a `
 (/) ) ) )
152 oveq1 5865 . . . . . 6  |-  ( k  =  ( a `  (/) )  ->  ( k  .^  X )  =  ( ( a `  (/) )  .^  X ) )
153151, 152oveq12d 5876 . . . . 5  |-  ( k  =  ( a `  (/) )  ->  ( ( A `  k )  .x.  ( k  .^  X
) )  =  ( ( A `  (
a `  (/) ) ) 
.x.  ( ( a `
 (/) )  .^  X
) ) )
15471, 149, 150, 153fmptco 5691 . . . 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 )
) ) )
155154oveq2d 5874 . . 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
) ) ) ) )
156104, 148, 1553eqtrrd 2320 . 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 ) ) ) ) )
15716, 90, 1563eqtrd 2319 1  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  K  =  ( P  gsumg  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   ifcif 3565   {csn 3640    e. cmpt 4077   omcom 4656   `'ccnv 4688   "cima 4692    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1oc1o 6472    ^m cmap 6772   Fincfn 6863   NN0cn0 9965   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361  .gcmg 14366  CMndccmn 15089  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338   1rcur 15339   LModclmod 15627   mVar cmvr 16088   mPoly cmpl 16089  PwSer1cps1 16250  var1cv1 16251  Poly1cpl1 16252  coe1cco1 16255
This theorem is referenced by:  plypf1  19594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-psr 16098  df-mvr 16099  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-coe1 16262
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