MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ply1coe Unicode version

Theorem ply1coe 16639
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 2404 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
2 psr1baslem 16538 . . 3  |-  ( NN0 
^m  1o )  =  { d  e.  ( NN0  ^m  1o )  |  ( `' d
" NN )  e. 
Fin }
3 eqid 2404 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
4 eqid 2404 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 1onn 6841 . . . 4  |-  1o  e.  om
65a1i 11 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  1o  e.  om )
7 ply1coe.p . . . 4  |-  P  =  (Poly1 `  R )
8 eqid 2404 . . . 4  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
9 ply1coe.b . . . 4  |-  B  =  ( Base `  P
)
107, 8, 9ply1bas 16548 . . 3  |-  B  =  ( Base `  ( 1o mPoly  R ) )
11 ply1coe.n . . . 4  |-  .x.  =  ( .s `  P )
127, 1, 11ply1vsca 16575 . . 3  |-  .x.  =  ( .s `  ( 1o mPoly  R ) )
13 crngrng 15629 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
1413adantr 452 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  R  e.  Ring )
15 simpr 448 . . 3  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  K  e.  B )
161, 2, 3, 4, 6, 10, 12, 14, 15mplcoe1 16483 . 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 16560 . . . . . 6  |-  ( ( K  e.  B  /\  a  e.  ( NN0  ^m  1o ) )  -> 
( K `  a
)  =  ( A `
 ( a `  (/) ) ) )
1918adantll 695 . . . . 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 2404 . . . . . . 7  |-  (mulGrp `  ( 1o mPoly  R ) )  =  (mulGrp `  ( 1o mPoly  R ) )
22 eqid 2404 . . . . . . 7  |-  (.g `  (mulGrp `  ( 1o mPoly  R )
) )  =  (.g `  (mulGrp `  ( 1o mPoly  R ) ) )
23 eqid 2404 . . . . . . 7  |-  ( 1o mVar  R )  =  ( 1o mVar  R )
24 simpll 731 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  R  e.  CRing
)
25 simpr 448 . . . . . . 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 16485 . . . . . 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 6695 . . . . . . . . 9  |-  1o  =  { (/) }
28 mpteq1 4249 . . . . . . . . 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 6051 . . . . . . 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 16471 . . . . . . . . . . . . 13  |-  ( ( 1o  e.  om  /\  R  e.  CRing )  -> 
( 1o mPoly  R )  e.  CRing )
325, 31mpan 652 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  ( 1o mPoly  R )  e.  CRing )
3332adantr 452 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  CRing )
3421crngmgp 15627 . . . . . . . . . . 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 452 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  (mulGrp `  ( 1o mPoly  R ) )  e. CMnd
)
37 cmnmnd 15382 . . . . . . . . 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 4299 . . . . . . . . 9  |-  (/)  e.  _V
4039a1i 11 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  (/)  e.  _V )
41 ply1coe.e . . . . . . . . . . . 12  |-  .^  =  (.g
`  M )
4221, 10mgpbas 15609 . . . . . . . . . . . . 13  |-  B  =  ( Base `  (mulGrp `  ( 1o mPoly  R )
) )
4342a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  =  ( Base `  (mulGrp `  ( 1o mPoly  R )
) ) )
44 ply1coe.m . . . . . . . . . . . . . 14  |-  M  =  (mulGrp `  P )
4544, 9mgpbas 15609 . . . . . . . . . . . . 13  |-  B  =  ( Base `  M
)
4645a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  =  ( Base `  M
) )
47 ssv 3328 . . . . . . . . . . . . 13  |-  B  C_  _V
4847a1i 11 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  B  C_ 
_V )
49 ovex 6065 . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  P )  =  ( .r `  P
)
527, 1, 51ply1mulr 16576 . . . . . . . . . . . . . . . 16  |-  ( .r
`  P )  =  ( .r `  ( 1o mPoly  R ) )
5321, 52mgpplusg 15607 . . . . . . . . . . . . . . 15  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R )
) )
5444, 51mgpplusg 15607 . . . . . . . . . . . . . . 15  |-  ( .r
`  P )  =  ( +g  `  M
)
5553, 54eqtr3i 2426 . . . . . . . . . . . . . 14  |-  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) )  =  ( +g  `  M
)
5655oveqi 6053 . . . . . . . . . . . . 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 14878 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (.g `  (mulGrp `  ( 1o mPoly  R
) ) )  = 
.^  )
5958oveqd 6057 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  =  ( ( a `  (/) )  .^  X ) )
6059adantr 452 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  =  ( ( a `  (/) )  .^  X ) )
617ply1crng 16551 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  P  e.  CRing
)
6261adantr 452 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  P  e.  CRing )
63 crngrng 15629 . . . . . . . . . . . 12  |-  ( P  e.  CRing  ->  P  e.  Ring )
6444rngmgp 15625 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  M  e. 
Mnd )
6562, 63, 643syl 19 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  M  e.  Mnd )
6665adantr 452 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  M  e.  Mnd )
67 elmapi 6997 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  1o )  ->  a : 1o --> NN0 )
68 0lt1o 6707 . . . . . . . . . . . 12  |-  (/)  e.  1o
69 ffvelrn 5827 . . . . . . . . . . . 12  |-  ( ( a : 1o --> NN0  /\  (/) 
e.  1o )  -> 
( a `  (/) )  e. 
NN0 )
7067, 68, 69sylancl 644 . . . . . . . . . . 11  |-  ( a  e.  ( NN0  ^m  1o )  ->  ( a `
 (/) )  e.  NN0 )
7170adantl 453 . . . . . . . . . 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 16566 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  X  e.  B )
7414, 73syl 16 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  X  e.  B )
7574adantr 452 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  X  e.  B )
7645, 41mulgnn0cl 14861 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( a `  (/) )  e. 
NN0  /\  X  e.  B )  ->  (
( a `  (/) )  .^  X )  e.  B
)
7766, 71, 75, 76syl3anc 1184 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) )  .^  X )  e.  B
)
7860, 77eqeltrd 2478 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  ( NN0  ^m  1o ) )  ->  ( (
a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X )  e.  B )
79 fveq2 5687 . . . . . . . . . 10  |-  ( c  =  (/)  ->  ( a `
 c )  =  ( a `  (/) ) )
80 fveq2 5687 . . . . . . . . . . 11  |-  ( c  =  (/)  ->  ( ( 1o mVar  R ) `  c )  =  ( ( 1o mVar  R ) `
 (/) ) )
8172vr1val 16545 . . . . . . . . . . 11  |-  X  =  ( ( 1o mVar  R
) `  (/) )
8280, 81syl6eqr 2454 . . . . . . . . . 10  |-  ( c  =  (/)  ->  ( ( 1o mVar  R ) `  c )  =  X )
8379, 82oveq12d 6058 . . . . . . . . 9  |-  ( c  =  (/)  ->  ( ( a `  c ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R
) `  c )
)  =  ( ( a `  (/) ) (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) X ) )
8442, 83gsumsn 15498 . . . . . . . 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 1184 . . . . . . 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 2448 . . . . . 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 2440 . . . . 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 6058 . . . 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 4255 . . 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 6056 . 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 10183 . . . . . 6  |-  NN0  e.  _V
9291mptex 5925 . . . . 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 5701 . . . . . 6  |-  (Poly1 `  R
)  e.  _V
957, 94eqeltri 2474 . . . . 5  |-  P  e. 
_V
9695a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  P  e.  _V )
97 ovex 6065 . . . . 5  |-  ( 1o mPoly  R )  e.  _V
9897a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  _V )
999, 10eqtr3i 2426 . . . . 5  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
10099a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( Base `  P )  =  ( Base `  ( 1o mPoly  R ) ) )
101 eqid 2404 . . . . . 6  |-  ( +g  `  P )  =  ( +g  `  P )
1027, 1, 101ply1plusg 16574 . . . . 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 14731 . . 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 2404 . . . 4  |-  ( 0g
`  ( 1o mPoly  R
) )  =  ( 0g `  ( 1o mPoly  R ) )
1061mpllmod 16469 . . . . . 6  |-  ( ( 1o  e.  om  /\  R  e.  Ring )  -> 
( 1o mPoly  R )  e.  LMod )
1076, 14, 106syl2anc 643 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( 1o mPoly  R )  e.  LMod )
108 lmodcmn 15947 . . . . 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 452 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( 1o mPoly  R
)  e.  LMod )
112 eqid 2404 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
11317, 9, 7, 112coe1f 16564 . . . . . . . 8  |-  ( K  e.  B  ->  A : NN0 --> ( Base `  R
) )
114113adantl 453 . . . . . . 7  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A : NN0 --> ( Base `  R
) )
115114ffvelrnda 5829 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  (
Base `  R )
)
11665adantr 452 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  M  e.  Mnd )
117 simpr 448 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
11874adantr 452 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  X  e.  B
)
11945, 41mulgnn0cl 14861 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  k  e.  NN0  /\  X  e.  B )  ->  (
k  .^  X )  e.  B )
120116, 117, 118, 119syl3anc 1184 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( k  .^  X )  e.  B
)
121 ply1coe.r . . . . . . . 8  |-  R  e. 
_V
122 simpl 444 . . . . . . . . 9  |-  ( ( 1o  e.  om  /\  R  e.  _V )  ->  1o  e.  om )
123 simpr 448 . . . . . . . . 9  |-  ( ( 1o  e.  om  /\  R  e.  _V )  ->  R  e.  _V )
1241, 122, 123mplsca 16463 . . . . . . . 8  |-  ( ( 1o  e.  om  /\  R  e.  _V )  ->  R  =  (Scalar `  ( 1o mPoly  R ) ) )
1255, 121, 124mp2an 654 . . . . . . 7  |-  R  =  (Scalar `  ( 1o mPoly  R ) )
12610, 125, 12, 112lmodvscl 15922 . . . . . 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 1184 . . . . 5  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( ( A `
 k )  .x.  ( k  .^  X
) )  e.  B
)
128 eqid 2404 . . . . 5  |-  ( k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  =  ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) )
129127, 128fmptd 5852 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) ) : NN0 --> B )
13017, 9, 7, 3coe1sfi 16565 . . . . . 6  |-  ( K  e.  B  ->  ( `' A " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin )
131130adantl 453 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( `' A " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin )
132114feqmptd 5738 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  A  =  ( k  e. 
NN0  |->  ( A `  k ) ) )
133132cnveqd 5007 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  `' A  =  `' (
k  e.  NN0  |->  ( A `
 k ) ) )
134133imaeq1d 5161 . . . . . . 7  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( `' A " ( _V 
\  { ( 0g
`  R ) } ) )  =  ( `' ( k  e. 
NN0  |->  ( A `  k ) ) "
( _V  \  {
( 0g `  R
) } ) ) )
135 eqimss2 3361 . . . . . . 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 ) } ) ) )
136134, 135syl 16 . . . . . 6  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( `' ( k  e. 
NN0  |->  ( A `  k ) ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  ( `' A " ( _V  \  {
( 0g `  R
) } ) ) )
13710, 125, 12, 3, 105lmod0vs 15938 . . . . . . 7  |-  ( ( ( 1o mPoly  R )  e.  LMod  /\  a  e.  B )  ->  (
( 0g `  R
)  .x.  a )  =  ( 0g `  ( 1o mPoly  R ) ) )
138107, 137sylan 458 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  a  e.  B
)  ->  ( ( 0g `  R )  .x.  a )  =  ( 0g `  ( 1o mPoly  R ) ) )
139 fvex 5701 . . . . . . 7  |-  ( A `
 k )  e. 
_V
140139a1i 11 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  K  e.  B )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  _V )
141136, 138, 140, 120suppssov1 6261 . . . . 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 ) } ) ) )
142 ssfi 7288 . . . . 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 )
143131, 141, 142syl2anc 643 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  ( `' ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) ) "
( _V  \  {
( 0g `  ( 1o mPoly  R ) ) } ) )  e.  Fin )
144 eqid 2404 . . . . . 6  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) )
14527, 91, 39, 144mapsnf1o2 7020 . . . . 5  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) : ( NN0  ^m  1o )
-1-1-onto-> NN0
146145a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) : ( NN0  ^m  1o ) -1-1-onto-> NN0 )
14710, 105, 109, 110, 129, 143, 146gsumf1o 15477 . . 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 `  (/) ) ) ) ) )
148 eqidd 2405 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) )  =  ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) )
149 eqidd 2405 . . . . 5  |-  ( ( R  e.  CRing  /\  K  e.  B )  ->  (
k  e.  NN0  |->  ( ( A `  k ) 
.x.  ( k  .^  X ) ) )  =  ( k  e. 
NN0  |->  ( ( A `
 k )  .x.  ( k  .^  X
) ) ) )
150 fveq2 5687 . . . . . 6  |-  ( k  =  ( a `  (/) )  ->  ( A `  k )  =  ( A `  ( a `
 (/) ) ) )
151 oveq1 6047 . . . . . 6  |-  ( k  =  ( a `  (/) )  ->  ( k  .^  X )  =  ( ( a `  (/) )  .^  X ) )
152150, 151oveq12d 6058 . . . . 5  |-  ( k  =  ( a `  (/) )  ->  ( ( A `  k )  .x.  ( k  .^  X
) )  =  ( ( A `  (
a `  (/) ) ) 
.x.  ( ( a `
 (/) )  .^  X
) ) )
15371, 148, 149, 152fmptco 5860 . . . 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 )
) ) )
154153oveq2d 6056 . . 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
) ) ) ) )
155104, 147, 1543eqtrrd 2441 . 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 ) ) ) ) )
15616, 90, 1553eqtrd 2440 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 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    C_ wss 3280   (/)c0 3588   ifcif 3699   {csn 3774    e. cmpt 4226   omcom 4804   `'ccnv 4836   "cima 4840    o. ccom 4841   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   1oc1o 6676    ^m cmap 6977   Fincfn 7068   NN0cn0 10177   Basecbs 13424   +g cplusg 13484   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488   0gc0g 13678    gsumg cgsu 13679   Mndcmnd 14639  .gcmg 14644  CMndccmn 15367  mulGrpcmgp 15603   Ringcrg 15615   CRingccrg 15616   1rcur 15617   LModclmod 15905   mVar cmvr 16362   mPoly cmpl 16363  PwSer1cps1 16524  var1cv1 16525  Poly1cpl1 16526  coe1cco1 16529
This theorem is referenced by:  plypf1  20084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-subrg 15821  df-lmod 15907  df-lss 15964  df-psr 16372  df-mvr 16373  df-mpl 16374  df-opsr 16380  df-psr1 16531  df-vr1 16532  df-ply1 16533  df-coe1 16536
  Copyright terms: Public domain W3C validator