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Theorem chcoeffeqlem 19840
Description: Lemma for chcoeffeq 19841. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)
Hypotheses
Ref Expression
chcoeffeq.a  |-  A  =  ( N Mat  R )
chcoeffeq.b  |-  B  =  ( Base `  A
)
chcoeffeq.p  |-  P  =  (Poly1 `  R )
chcoeffeq.y  |-  Y  =  ( N Mat  P )
chcoeffeq.r  |-  .X.  =  ( .r `  Y )
chcoeffeq.s  |-  .-  =  ( -g `  Y )
chcoeffeq.0  |-  .0.  =  ( 0g `  Y )
chcoeffeq.t  |-  T  =  ( N matToPolyMat  R )
chcoeffeq.c  |-  C  =  ( N CharPlyMat  R )
chcoeffeq.k  |-  K  =  ( C `  M
)
chcoeffeq.g  |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `  0 ) ) ) ) ,  if ( n  =  (
s  +  1 ) ,  ( T `  ( b `  s
) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `  ( b `
 ( n  - 
1 ) ) ) 
.-  ( ( T `
 M )  .X.  ( T `  ( b `
 n ) ) ) ) ) ) ) )
chcoeffeq.w  |-  W  =  ( Base `  Y
)
chcoeffeq.1  |-  .1.  =  ( 1r `  A )
chcoeffeq.m  |-  .*  =  ( .s `  A )
chcoeffeq.u  |-  U  =  ( N cPolyMatToMat  R )
Assertion
Ref Expression
chcoeffeqlem  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( U `  ( G `  n ) ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  =  ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( ( (coe1 `  K ) `  n )  .*  .1.  ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  ->  A. n  e.  NN0  ( U `  ( G `
 n ) )  =  ( ( (coe1 `  K ) `  n
)  .*  .1.  )
) )
Distinct variable groups:    A, n    B, n    n, G    n, K    n, M    n, N    R, n    U, n    n, Y    .1. , n    .* , n    n, b   
n, s
Allowed substitution hints:    A( s, b)    B( s, b)    C( n, s, b)    P( n, s, b)    R( s, b)    T( n, s, b)    .X. ( n, s, b)    U( s, b)    .1. ( s, b)    G( s, b)    .* ( s,
b)    K( s, b)    M( s, b)    .- ( n, s, b)    N( s, b)    W( n, s, b)    Y( s, b)    .0. ( n, s, b)

Proof of Theorem chcoeffeqlem
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 eqid 2429 . . . . 5  |-  (Poly1 `  A
)  =  (Poly1 `  A
)
2 eqid 2429 . . . . 5  |-  (var1 `  A
)  =  (var1 `  A
)
3 eqid 2429 . . . . 5  |-  (.g `  (mulGrp `  (Poly1 `  A ) ) )  =  (.g `  (mulGrp `  (Poly1 `  A ) ) )
4 crngring 17726 . . . . . . . 8  |-  ( R  e.  CRing  ->  R  e.  Ring )
5 chcoeffeq.a . . . . . . . . 9  |-  A  =  ( N Mat  R )
65matring 19399 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
74, 6sylan2 476 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  Ring )
873adant3 1025 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  A  e.  Ring )
98adantr 466 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  A  e.  Ring )
10 chcoeffeq.b . . . . 5  |-  B  =  ( Base `  A
)
11 eqid 2429 . . . . 5  |-  ( .s
`  (Poly1 `  A ) )  =  ( .s `  (Poly1 `  A ) )
12 eqid 2429 . . . . 5  |-  ( 0g
`  A )  =  ( 0g `  A
)
13 chcoeffeq.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
14 chcoeffeq.y . . . . . . . 8  |-  Y  =  ( N Mat  P )
15 chcoeffeq.t . . . . . . . 8  |-  T  =  ( N matToPolyMat  R )
16 chcoeffeq.r . . . . . . . 8  |-  .X.  =  ( .r `  Y )
17 chcoeffeq.s . . . . . . . 8  |-  .-  =  ( -g `  Y )
18 chcoeffeq.0 . . . . . . . 8  |-  .0.  =  ( 0g `  Y )
19 chcoeffeq.g . . . . . . . 8  |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `  0 ) ) ) ) ,  if ( n  =  (
s  +  1 ) ,  ( T `  ( b `  s
) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `  ( b `
 ( n  - 
1 ) ) ) 
.-  ( ( T `
 M )  .X.  ( T `  ( b `
 n ) ) ) ) ) ) ) )
20 eqid 2429 . . . . . . . 8  |-  ( N ConstPolyMat  R )  =  ( N ConstPolyMat  R )
21 eqid 2429 . . . . . . . 8  |-  ( .s
`  Y )  =  ( .s `  Y
)
22 eqid 2429 . . . . . . . 8  |-  ( 1r
`  Y )  =  ( 1r `  Y
)
23 eqid 2429 . . . . . . . 8  |-  (var1 `  R
)  =  (var1 `  R
)
24 eqid 2429 . . . . . . . 8  |-  ( ( (var1 `  R ) ( .s `  Y ) ( 1r `  Y
) )  .-  ( T `  M )
)  =  ( ( (var1 `  R ) ( .s `  Y ) ( 1r `  Y
) )  .-  ( T `  M )
)
25 eqid 2429 . . . . . . . 8  |-  ( N maAdju  P )  =  ( N maAdju  P )
26 chcoeffeq.w . . . . . . . 8  |-  W  =  ( Base `  Y
)
27 chcoeffeq.u . . . . . . . 8  |-  U  =  ( N cPolyMatToMat  R )
285, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1, 2, 11, 3, 27cpmadumatpolylem1 19836 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  s  e.  NN )  /\  b  e.  ( B  ^m  (
0 ... s ) ) )  ->  ( U  o.  G )  e.  ( B  ^m  NN0 )
)
2928anasss 651 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( U  o.  G )  e.  ( B  ^m  NN0 )
)
305, 10, 13, 14, 16, 17, 18, 15, 19, 20chfacfisfcpmat 19810 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  G : NN0 --> ( N ConstPolyMat  R ) )
314, 30syl3anl2 1313 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  G : NN0 --> ( N ConstPolyMat  R ) )
3231adantr 466 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  ( U  o.  G )  e.  ( B  ^m  NN0 ) )  ->  G : NN0 --> ( N ConstPolyMat  R ) )
33 fvco3 5958 . . . . . . . . . 10  |-  ( ( G : NN0 --> ( N ConstPolyMat  R )  /\  l  e. 
NN0 )  ->  (
( U  o.  G
) `  l )  =  ( U `  ( G `  l ) ) )
3433eqcomd 2437 . . . . . . . . 9  |-  ( ( G : NN0 --> ( N ConstPolyMat  R )  /\  l  e. 
NN0 )  ->  ( U `  ( G `  l ) )  =  ( ( U  o.  G ) `  l
) )
3532, 34sylan 473 . . . . . . . 8  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  /\  ( U  o.  G )  e.  ( B  ^m  NN0 )
)  /\  l  e.  NN0 )  ->  ( U `  ( G `  l
) )  =  ( ( U  o.  G
) `  l )
)
36 elmapi 7501 . . . . . . . . . 10  |-  ( ( U  o.  G )  e.  ( B  ^m  NN0 )  ->  ( U  o.  G ) : NN0 --> B )
3736adantl 467 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  ( U  o.  G )  e.  ( B  ^m  NN0 ) )  ->  ( U  o.  G ) : NN0 --> B )
3837ffvelrnda 6037 . . . . . . . 8  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  /\  ( U  o.  G )  e.  ( B  ^m  NN0 )
)  /\  l  e.  NN0 )  ->  ( ( U  o.  G ) `  l )  e.  B
)
3935, 38eqeltrd 2517 . . . . . . 7  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  /\  ( U  o.  G )  e.  ( B  ^m  NN0 )
)  /\  l  e.  NN0 )  ->  ( U `  ( G `  l
) )  e.  B
)
4039ralrimiva 2846 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  ( U  o.  G )  e.  ( B  ^m  NN0 ) )  ->  A. l  e.  NN0  ( U `  ( G `  l ) )  e.  B )
4129, 40mpdan 672 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  A. l  e.  NN0  ( U `  ( G `
 l ) )  e.  B )
424anim2i 571 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
43423adant3 1025 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
4443adantr 466 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( N  e. 
Fin  /\  R  e.  Ring ) )
455, 10, 20, 27cpm2mf 19707 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  U : ( N ConstPolyMat  R ) --> B )
4644, 45syl 17 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  U : ( N ConstPolyMat  R ) --> B )
47 fcompt 6074 . . . . . . 7  |-  ( ( U : ( N ConstPolyMat  R ) --> B  /\  G : NN0 --> ( N ConstPolyMat  R ) )  ->  ( U  o.  G )  =  ( l  e.  NN0  |->  ( U `
 ( G `  l ) ) ) )
4846, 31, 47syl2anc 665 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( U  o.  G )  =  ( l  e.  NN0  |->  ( U `
 ( G `  l ) ) ) )
495, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1, 2, 11, 3, 27cpmadumatpolylem2 19837 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  s  e.  NN )  /\  b  e.  ( B  ^m  (
0 ... s ) ) )  ->  ( U  o.  G ) finSupp  ( 0g
`  A ) )
5049anasss 651 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( U  o.  G ) finSupp  ( 0g `  A ) )
5148, 50eqbrtrrd 4448 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( l  e. 
NN0  |->  ( U `  ( G `  l ) ) ) finSupp  ( 0g
`  A ) )
52 simpll1 1044 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  l  e.  NN0 )  ->  N  e.  Fin )
5343ad2ant2 1027 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  R  e.  Ring )
5453ad2antrr 730 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  l  e.  NN0 )  ->  R  e.  Ring )
55 chcoeffeq.k . . . . . . . . . 10  |-  K  =  ( C `  M
)
56 chcoeffeq.c . . . . . . . . . . 11  |-  C  =  ( N CharPlyMat  R )
57 eqid 2429 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  P )
5856, 5, 10, 13, 57chpmatply1 19787 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( C `  M )  e.  ( Base `  P
) )
5955, 58syl5eqel 2521 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  K  e.  ( Base `  P
) )
60 eqid 2429 . . . . . . . . . 10  |-  (coe1 `  K
)  =  (coe1 `  K
)
61 eqid 2429 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
6260, 57, 13, 61coe1fvalcl 18740 . . . . . . . . 9  |-  ( ( K  e.  ( Base `  P )  /\  l  e.  NN0 )  ->  (
(coe1 `  K ) `  l )  e.  (
Base `  R )
)
6359, 62sylan 473 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  l  e.  NN0 )  ->  (
(coe1 `  K ) `  l )  e.  (
Base `  R )
)
6463adantlr 719 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  l  e.  NN0 )  ->  (
(coe1 `  K ) `  l )  e.  (
Base `  R )
)
65 chcoeffeq.1 . . . . . . . . . 10  |-  .1.  =  ( 1r `  A )
6610, 65ringidcl 17736 . . . . . . . . 9  |-  ( A  e.  Ring  ->  .1.  e.  B )
678, 66syl 17 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  .1.  e.  B )
6867ad2antrr 730 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  l  e.  NN0 )  ->  .1.  e.  B )
69 chcoeffeq.m . . . . . . . 8  |-  .*  =  ( .s `  A )
7061, 5, 10, 69matvscl 19387 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( ( (coe1 `  K
) `  l )  e.  ( Base `  R
)  /\  .1.  e.  B ) )  -> 
( ( (coe1 `  K
) `  l )  .*  .1.  )  e.  B
)
7152, 54, 64, 68, 70syl22anc 1265 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  l  e.  NN0 )  ->  (
( (coe1 `  K ) `  l )  .*  .1.  )  e.  B )
7271ralrimiva 2846 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  A. l  e.  NN0  ( ( (coe1 `  K
) `  l )  .*  .1.  )  e.  B
)
73 nn0ex 10875 . . . . . . 7  |-  NN0  e.  _V
7473a1i 11 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  NN0  e.  _V )
755matlmod 19385 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
764, 75sylan2 476 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  LMod )
77763adant3 1025 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  A  e.  LMod )
7877adantr 466 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  A  e.  LMod )
79 eqidd 2430 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  (Scalar `  A )  =  (Scalar `  A )
)
80 fvex 5891 . . . . . . 7  |-  ( (coe1 `  K ) `  l
)  e.  _V
8180a1i 11 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  l  e.  NN0 )  ->  (
(coe1 `  K ) `  l )  e.  _V )
82 eqid 2429 . . . . . 6  |-  ( 0g
`  (Scalar `  A )
)  =  ( 0g
`  (Scalar `  A )
)
835matsca2 19376 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  =  (Scalar `  A
) )
84833adant3 1025 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  R  =  (Scalar `  A )
)
8584, 53eqeltrrd 2518 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (Scalar `  A )  e.  Ring )
8684eqcomd 2437 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (Scalar `  A )  =  R )
8786fveq2d 5885 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (Poly1 `  (Scalar `  A ) )  =  (Poly1 `  R ) )
8887, 13syl6eqr 2488 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (Poly1 `  (Scalar `  A ) )  =  P )
8988fveq2d 5885 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( Base `  (Poly1 `  (Scalar `  A
) ) )  =  ( Base `  P
) )
9059, 89eleqtrrd 2520 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  K  e.  ( Base `  (Poly1 `  (Scalar `  A ) ) ) )
91 eqid 2429 . . . . . . . . 9  |-  (Poly1 `  (Scalar `  A ) )  =  (Poly1 `  (Scalar `  A
) )
92 eqid 2429 . . . . . . . . 9  |-  ( Base `  (Poly1 `  (Scalar `  A
) ) )  =  ( Base `  (Poly1 `  (Scalar `  A ) ) )
9391, 92, 82mptcoe1fsupp 18743 . . . . . . . 8  |-  ( ( (Scalar `  A )  e.  Ring  /\  K  e.  ( Base `  (Poly1 `  (Scalar `  A ) ) ) )  ->  ( l  e.  NN0  |->  ( (coe1 `  K
) `  l )
) finSupp  ( 0g `  (Scalar `  A ) ) )
9485, 90, 93syl2anc 665 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
l  e.  NN0  |->  ( (coe1 `  K ) `  l
) ) finSupp  ( 0g `  (Scalar `  A )
) )
9594adantr 466 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( l  e. 
NN0  |->  ( (coe1 `  K
) `  l )
) finSupp  ( 0g `  (Scalar `  A ) ) )
9674, 78, 79, 10, 81, 68, 12, 82, 69, 95mptscmfsupp0 18088 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( l  e. 
NN0  |->  ( ( (coe1 `  K ) `  l
)  .*  .1.  )
) finSupp  ( 0g `  A
) )
97 fveq2 5881 . . . . . . . . . 10  |-  ( n  =  l  ->  ( G `  n )  =  ( G `  l ) )
9897fveq2d 5885 . . . . . . . . 9  |-  ( n  =  l  ->  ( U `  ( G `  n ) )  =  ( U `  ( G `  l )
) )
99 oveq1 6312 . . . . . . . . 9  |-  ( n  =  l  ->  (
n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) )  =  ( l (.g `  (mulGrp `  (Poly1 `  A
) ) ) (var1 `  A ) ) )
10098, 99oveq12d 6323 . . . . . . . 8  |-  ( n  =  l  ->  (
( U `  ( G `  n )
) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) )  =  ( ( U `  ( G `
 l ) ) ( .s `  (Poly1 `  A ) ) ( l (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) )
101100cbvmptv 4518 . . . . . . 7  |-  ( n  e.  NN0  |->  ( ( U `  ( G `
 n ) ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) )  =  ( l  e.  NN0  |->  ( ( U `  ( G `
 l ) ) ( .s `  (Poly1 `  A ) ) ( l (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) )
102101oveq2i 6316 . . . . . 6  |-  ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( U `  ( G `  n ) ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  =  ( (Poly1 `  A )  gsumg  ( l  e.  NN0  |->  ( ( U `  ( G `
 l ) ) ( .s `  (Poly1 `  A ) ) ( l (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )
103102a1i 11 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( (Poly1 `  A
)  gsumg  ( n  e.  NN0  |->  ( ( U `  ( G `  n ) ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  =  ( (Poly1 `  A )  gsumg  ( l  e.  NN0  |->  ( ( U `  ( G `
 l ) ) ( .s `  (Poly1 `  A ) ) ( l (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) ) )
104 fveq2 5881 . . . . . . . . . 10  |-  ( n  =  l  ->  (
(coe1 `  K ) `  n )  =  ( (coe1 `  K ) `  l ) )
105104oveq1d 6320 . . . . . . . . 9  |-  ( n  =  l  ->  (
( (coe1 `  K ) `  n )  .*  .1.  )  =  ( (
(coe1 `  K ) `  l )  .*  .1.  ) )
106105, 99oveq12d 6323 . . . . . . . 8  |-  ( n  =  l  ->  (
( ( (coe1 `  K
) `  n )  .*  .1.  ) ( .s
`  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) )  =  ( ( ( (coe1 `  K ) `  l )  .*  .1.  ) ( .s `  (Poly1 `  A ) ) ( l (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) )
107106cbvmptv 4518 . . . . . . 7  |-  ( n  e.  NN0  |->  ( ( ( (coe1 `  K ) `  n )  .*  .1.  ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) )  =  ( l  e.  NN0  |->  ( ( ( (coe1 `  K ) `  l )  .*  .1.  ) ( .s `  (Poly1 `  A ) ) ( l (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) )
108107oveq2i 6316 . . . . . 6  |-  ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( ( (coe1 `  K ) `  n
)  .*  .1.  )
( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  =  ( (Poly1 `  A )  gsumg  ( l  e.  NN0  |->  ( ( ( (coe1 `  K ) `  l )  .*  .1.  ) ( .s `  (Poly1 `  A ) ) ( l (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )
109108a1i 11 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( (Poly1 `  A
)  gsumg  ( n  e.  NN0  |->  ( ( ( (coe1 `  K ) `  n
)  .*  .1.  )
( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  =  ( (Poly1 `  A )  gsumg  ( l  e.  NN0  |->  ( ( ( (coe1 `  K ) `  l )  .*  .1.  ) ( .s `  (Poly1 `  A ) ) ( l (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) ) )
1101, 2, 3, 9, 10, 11, 12, 41, 51, 72, 96, 103, 109gsumply1eq 18834 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( U `  ( G `  n ) ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  =  ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( ( (coe1 `  K ) `  n )  .*  .1.  ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  <->  A. l  e.  NN0  ( U `  ( G `  l ) )  =  ( ( (coe1 `  K ) `  l )  .*  .1.  ) ) )
111110biimpa 486 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  (
(Poly1 `
 A )  gsumg  ( n  e.  NN0  |->  ( ( U `  ( G `
 n ) ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  =  ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( ( (coe1 `  K ) `  n )  .*  .1.  ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) ) )  ->  A. l  e.  NN0  ( U `  ( G `
 l ) )  =  ( ( (coe1 `  K ) `  l
)  .*  .1.  )
)
11298, 105eqeq12d 2451 . . . 4  |-  ( n  =  l  ->  (
( U `  ( G `  n )
)  =  ( ( (coe1 `  K ) `  n )  .*  .1.  ) 
<->  ( U `  ( G `  l )
)  =  ( ( (coe1 `  K ) `  l )  .*  .1.  ) ) )
113112cbvralv 3062 . . 3  |-  ( A. n  e.  NN0  ( U `
 ( G `  n ) )  =  ( ( (coe1 `  K
) `  n )  .*  .1.  )  <->  A. l  e.  NN0  ( U `  ( G `  l ) )  =  ( ( (coe1 `  K ) `  l )  .*  .1.  ) )
114111, 113sylibr 215 . 2  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( s  e.  NN  /\  b  e.  ( B  ^m  (
0 ... s ) ) ) )  /\  (
(Poly1 `
 A )  gsumg  ( n  e.  NN0  |->  ( ( U `  ( G `
 n ) ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  =  ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( ( (coe1 `  K ) `  n )  .*  .1.  ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) ) )  ->  A. n  e.  NN0  ( U `  ( G `
 n ) )  =  ( ( (coe1 `  K ) `  n
)  .*  .1.  )
)
115114ex 435 1  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
) ) ) )  ->  ( ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( U `  ( G `  n ) ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  =  ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( ( (coe1 `  K ) `  n )  .*  .1.  ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  ->  A. n  e.  NN0  ( U `  ( G `
 n ) )  =  ( ( (coe1 `  K ) `  n
)  .*  .1.  )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087   ifcif 3915   class class class wbr 4426    |-> cmpt 4484    o. ccom 4858   -->wf 5597   ` cfv 5601  (class class class)co 6305    ^m cmap 7480   Fincfn 7577   finSupp cfsupp 7889   0cc0 9538   1c1 9539    + caddc 9541    < clt 9674    - cmin 9859   NNcn 10609   NN0cn0 10869   ...cfz 11782   Basecbs 15084   .rcmulr 15153  Scalarcsca 15155   .scvsca 15156   0gc0g 15297    gsumg cgsu 15298   -gcsg 16622  .gcmg 16623  mulGrpcmgp 17658   1rcur 17670   Ringcrg 17715   CRingccrg 17716   LModclmod 18026  var1cv1 18704  Poly1cpl1 18705  coe1cco1 18706   Mat cmat 19363   maAdju cmadu 19588   ConstPolyMat ccpmat 19658   matToPolyMat cmat2pmat 19659   cPolyMatToMat ccpmat2mat 19660   CharPlyMat cchpmat 19781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-xor 1401  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-ofr 6546  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-word 12651  df-lsw 12652  df-concat 12653  df-s1 12654  df-substr 12655  df-splice 12656  df-reverse 12657  df-s2 12929  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-0g 15299  df-gsum 15300  df-prds 15305  df-pws 15307  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mulg 16627  df-subg 16765  df-ghm 16832  df-gim 16874  df-cntz 16922  df-oppg 16948  df-symg 16970  df-pmtr 17034  df-psgn 17083  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-srg 17675  df-ring 17717  df-cring 17718  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-rnghom 17878  df-drng 17912  df-subrg 17941  df-lmod 18028  df-lss 18091  df-sra 18330  df-rgmod 18331  df-ascl 18473  df-psr 18515  df-mvr 18516  df-mpl 18517  df-opsr 18519  df-psr1 18708  df-vr1 18709  df-ply1 18710  df-coe1 18711  df-cnfld 18906  df-zring 18974  df-zrh 19006  df-dsmm 19226  df-frlm 19241  df-mamu 19340  df-mat 19364  df-mdet 19541  df-cpmat 19661  df-mat2pmat 19662  df-cpmat2mat 19663  df-chpmat 19782
This theorem is referenced by:  chcoeffeq  19841
  Copyright terms: Public domain W3C validator