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Theorem ply1coe 16384
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 2296 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
2 psr1baslem 16280 . . 3  |-  ( NN0 
^m  1o )  =  { d  e.  ( NN0  ^m  1o )  |  ( `' d
" NN )  e. 
Fin }
3 eqid 2296 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
4 eqid 2296 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 1onn 6653 . . . 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 2296 . . . 4  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
9 ply1coe.b . . . 4  |-  B  =  ( Base `  P
)
107, 8, 9ply1bas 16290 . . 3  |-  B  =  ( Base `  ( 1o mPoly  R ) )
11 ply1coe.n . . . 4  |-  .x.  =  ( .s `  P )
127, 1, 11ply1vsca 16320 . . 3  |-  .x.  =  ( .s `  ( 1o mPoly  R ) )
13 crngrng 15367 . . . 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 16225 . 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 16304 . . . . . 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 2296 . . . . . . 7  |-  (mulGrp `  ( 1o mPoly  R ) )  =  (mulGrp `  ( 1o mPoly  R ) )
22 eqid 2296 . . . . . . 7  |-  (.g `  (mulGrp `  ( 1o mPoly  R )
) )  =  (.g `  (mulGrp `  ( 1o mPoly  R ) ) )
23 eqid 2296 . . . . . . 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 16227 . . . . . 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 6507 . . . . . . . . 9  |-  1o  =  { (/) }
28 mpteq1 4116 . . . . . . . . 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 5885 . . . . . . 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 16213 . . . . . . . . . . . . 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 15365 . . . . . . . . . . 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 15120 . . . . . . . . 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 4166 . . . . . . . . 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 15347 . . . . . . . . . . . . 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 15347 . . . . . . . . . . . . 13  |-  B  =  ( Base `  M
)
4645a1i 10 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  =  ( Base `  M
) )
47 ssv 3211 . . . . . . . . . . . . 13  |-  B  C_  _V
4847a1i 10 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  C_ 
_V )
49 ovex 5899 . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  P )  =  ( .r `  P
)
527, 1, 51ply1mulr 16321 . . . . . . . . . . . . . . . 16  |-  ( .r
`  P )  =  ( .r `  ( 1o mPoly  R ) )
5321, 52mgpplusg 15345 . . . . . . . . . . . . . . 15  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R )
) )
5444, 51mgpplusg 15345 . . . . . . . . . . . . . . 15  |-  ( .r
`  P )  =  ( +g  `  M
)
5553, 54eqtr3i 2318 . . . . . . . . . . . . . 14  |-  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) )  =  ( +g  `  M
)
5655oveqi 5887 . . . . . . . . . . . . 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 14616 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (.g `  (mulGrp `  ( 1o mPoly  R
) ) )  = 
.^  )
5958oveqd 5891 . . . . . . . . . 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 16293 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  P  e.  CRing
)
6261adantr 451 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  P  e.  CRing )
63 crngrng 15367 . . . . . . . . . . . 12  |-  ( P  e.  CRing  ->  P  e.  Ring )
6444rngmgp 15363 . . . . . . . . . . . 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 6808 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  1o )  ->  a : 1o --> NN0 )
68 0lt1o 6519 . . . . . . . . . . . 12  |-  (/)  e.  1o
69 ffvelrn 5679 . . . . . . . . . . . 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 16310 . . . . . . . . . . . 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 14599 . . . . . . . . . 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 2370 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  e.  B )
79 fveq2 5541 . . . . . . . . . 10  |-  ( c  =  (/)  ->  ( a `
 c )  =  ( a `  (/) ) )
80 fveq2 5541 . . . . . . . . . . 11  |-  ( c  =  (/)  ->  ( ( 1o mVar  R ) `  c )  =  ( ( 1o mVar  R ) `
 (/) ) )
8172vr1val 16287 . . . . . . . . . . 11  |-  X  =  ( ( 1o mVar  R
) `  (/) )
8280, 81syl6eqr 2346 . . . . . . . . . 10  |-  ( c  =  (/)  ->  ( ( 1o mVar  R ) `  c )  =  X )
8379, 82oveq12d 5892 . . . . . . . . 9  |-  ( c  =  (/)  ->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R
) `  c )
)  =  ( ( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X ) )
8442, 83gsumsn 15236 . . . . . . . 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 2340 . . . . . 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 2332 . . . . 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 5892 . . . 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 4122 . . 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 5890 . 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 9987 . . . . . 6  |-  NN0  e.  _V
9291mptex 5762 . . . . 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 5555 . . . . . 6  |-  (Poly1 `  R
)  e.  _V
957, 94eqeltri 2366 . . . . 5  |-  P  e. 
_V
9695a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  P  e.  _V )
97 ovex 5899 . . . . 5  |-  ( 1o mPoly  R )  e.  _V
9897a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  _V )
999, 10eqtr3i 2318 . . . . 5  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
10099a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( Base `  P )  =  ( Base `  ( 1o mPoly  R ) ) )
101 eqid 2296 . . . . . 6  |-  ( +g  `  P )  =  ( +g  `  P )
1027, 1, 101ply1plusg 16319 . . . . 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 14469 . . 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 2296 . . . 4  |-  ( 0g
`  ( 1o mPoly  R
) )  =  ( 0g `  ( 1o mPoly  R ) )
1061mpllmod 16211 . . . . . 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 15689 . . . . 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 2296 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
11317, 9, 7, 112coe1f 16308 . . . . . . . 8  |-  ( K  e.  B  ->  A : NN0 --> ( Base `  R
) )
114113adantl 452 . . . . . . 7  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A : NN0 --> ( Base `  R
) )
115 ffvelrn 5679 . . . . . . 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 14599 . . . . . . 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 16205 . . . . . . . 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 15660 . . . . . 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 2296 . . . . 5  |-  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  =  ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) )
130128, 129fmptd 5700 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) : NN0 --> B )
13117, 9, 7, 3coe1sfi 16309 . . . . . 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 5591 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A  =  ( k  e. 
NN0  |->  ( A `  k ) ) )
134133cnveqd 4873 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  `' A  =  `' (
k  e.  NN0  |->  ( A `
 k ) ) )
135134imaeq1d 5027 . . . . . . 7  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( `' A " ( _V 
\  { ( 0g
`  R ) } ) )  =  ( `' ( k  e. 
NN0  |->  ( A `  k ) ) "
( _V  \  {
( 0g `  R
) } ) ) )
136 eqimss2 3244 . . . . . . 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 15679 . . . . . . 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 5555 . . . . . . 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 6091 . . . . 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 7099 . . . . 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 2296 . . . . . 6  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) )
14627, 91, 39, 145mapsnf1o2 6831 . . . . 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 15215 . . 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 2297 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) )  =  ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) )
150 eqidd 2297 . . . . 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 5541 . . . . . 6  |-  ( k  =  ( a `  (/) )  ->  ( A `  k )  =  ( A `  ( a `
 (/) ) ) )
152 oveq1 5881 . . . . . 6  |-  ( k  =  ( a `  (/) )  ->  ( k  .^  X )  =  ( ( a `  (/) )  .^  X ) )
153151, 152oveq12d 5892 . . . . 5  |-  ( k  =  ( a `  (/) )  ->  ( ( A `  k )  .x.  ( k  .^  X
) )  =  ( ( A `  (
a `  (/) ) ) 
.x.  ( ( a `
 (/) )  .^  X
) ) )
15471, 149, 150, 153fmptco 5707 . . . 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 5890 . . 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 2333 . 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 2332 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 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   ifcif 3578   {csn 3653    e. cmpt 4093   omcom 4672   `'ccnv 4704   "cima 4708    o. ccom 4709   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1oc1o 6488    ^m cmap 6788   Fincfn 6879   NN0cn0 9981   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377  .gcmg 14382  CMndccmn 15105  mulGrpcmgp 15341   Ringcrg 15353   CRingccrg 15354   1rcur 15355   LModclmod 15643   mVar cmvr 16104   mPoly cmpl 16105  PwSer1cps1 16266  var1cv1 16267  Poly1cpl1 16268  coe1cco1 16271
This theorem is referenced by:  plypf1  19610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-subrg 15559  df-lmod 15645  df-lss 15706  df-psr 16114  df-mvr 16115  df-mpl 16116  df-opsr 16122  df-psr1 16273  df-vr1 16274  df-ply1 16275  df-coe1 16278
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