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Theorem pmattomply1 31253
Description: Transform a matrix over polynomials into a polynomial over matrices. (Contributed by AV, 4-Oct-2019.)
Hypotheses
Ref Expression
pmattomply1.p  |-  P  =  (Poly1 `  R )
pmattomply1.c  |-  C  =  ( N Mat  P )
pmattomply1.b  |-  B  =  ( Base `  C
)
pmattomply1.f  |-  F  =  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )
pmattomply1.m  |-  .*  =  ( .s `  Q )
pmattomply1.e  |-  .^  =  (.g
`  (mulGrp `  Q )
)
pmattomply1.x  |-  X  =  (var1 `  A )
pmattomply1.a  |-  A  =  ( N Mat  R )
pmattomply1.q  |-  Q  =  (Poly1 `  A )
pmattomply1.l  |-  L  =  ( Base `  Q
)
pmattomply1.t  |-  T  =  ( m  e.  B  |->  ( Q  gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) ) )
Assertion
Ref Expression
pmattomply1  |-  ( M  e.  B  ->  ( T `  M )  =  ( Q  gsumg  ( k  e.  NN0  |->  ( ( M F k )  .*  ( k  .^  X ) ) ) ) )
Distinct variable groups:    B, k, m    B, i, j, k   
i, M, j, k   
i, N, j, k    R, i, j, k    m, F    m, M    Q, m    m, X    .* , m    .^ , m
Allowed substitution hints:    A( i, j, k, m)    C( i,
j, k, m)    P( i, j, k, m)    Q( i, j, k)    R( m)    T( i, j, k, m)    .^ ( i, j, k)    F( i, j, k)    .* ( i,
j, k)    L( i,
j, k, m)    N( m)    X( i, j, k)

Proof of Theorem pmattomply1
StepHypRef Expression
1 pmattomply1.t . . 3  |-  T  =  ( m  e.  B  |->  ( Q  gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) ) )
21a1i 11 . 2  |-  ( M  e.  B  ->  T  =  ( m  e.  B  |->  ( Q  gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  ( k  .^  X ) ) ) ) ) )
3 oveq1 6194 . . . . . 6  |-  ( m  =  M  ->  (
m F k )  =  ( M F k ) )
43oveq1d 6202 . . . . 5  |-  ( m  =  M  ->  (
( m F k )  .*  ( k 
.^  X ) )  =  ( ( M F k )  .*  ( k  .^  X
) ) )
54mpteq2dv 4474 . . . 4  |-  ( m  =  M  ->  (
k  e.  NN0  |->  ( ( m F k )  .*  ( k  .^  X ) ) )  =  ( k  e. 
NN0  |->  ( ( M F k )  .*  ( k  .^  X
) ) ) )
65oveq2d 6203 . . 3  |-  ( m  =  M  ->  ( Q  gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) )  =  ( Q  gsumg  ( k  e.  NN0  |->  ( ( M F k )  .*  (
k  .^  X )
) ) ) )
76adantl 466 . 2  |-  ( ( M  e.  B  /\  m  =  M )  ->  ( Q  gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) )  =  ( Q  gsumg  ( k  e.  NN0  |->  ( ( M F k )  .*  (
k  .^  X )
) ) ) )
8 id 22 . 2  |-  ( M  e.  B  ->  M  e.  B )
9 ovex 6212 . . 3  |-  ( Q 
gsumg  ( k  e.  NN0  |->  ( ( M F k )  .*  (
k  .^  X )
) ) )  e. 
_V
109a1i 11 . 2  |-  ( M  e.  B  ->  ( Q  gsumg  ( k  e.  NN0  |->  ( ( M F k )  .*  (
k  .^  X )
) ) )  e. 
_V )
112, 7, 8, 10fvmptd 5875 1  |-  ( M  e.  B  ->  ( T `  M )  =  ( Q  gsumg  ( k  e.  NN0  |->  ( ( M F k )  .*  ( k  .^  X ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3065    |-> cmpt 4445   ` cfv 5513  (class class class)co 6187    |-> cmpt2 6189   NN0cn0 10677   Basecbs 14273   .scvsca 14341    gsumg cgsu 14478  .gcmg 15513  mulGrpcmgp 16693  var1cv1 17736  Poly1cpl1 17737  coe1cco1 17738   Mat cmat 18386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fv 5521  df-ov 6190
This theorem is referenced by:  pmattomply1rn  31254  pmattomply1coe1  31256  idpmattoidmply1  31257  mp2pm2mp  31263  pmattomply1f1  31264  pmattomply1ghm  31267  pmattomply1mhmlem2  31271  monmat2matmon  31275
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