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Theorem mply1topmatval 19413
Description: A polynomial over matrices transformed into a polynomial matrix.  I is the inverse function of the transformation  T of polynomial matrices into polynomials over matrices:  ( T `  ( I `
 O ) )  =  O ) (see mp2pm2mp 19420). (Contributed by AV, 6-Oct-2019.)
Hypotheses
Ref Expression
mply1topmat.a  |-  A  =  ( N Mat  R )
mply1topmat.q  |-  Q  =  (Poly1 `  A )
mply1topmat.l  |-  L  =  ( Base `  Q
)
mply1topmat.p  |-  P  =  (Poly1 `  R )
mply1topmat.m  |-  .x.  =  ( .s `  P )
mply1topmat.e  |-  E  =  (.g `  (mulGrp `  P
) )
mply1topmat.y  |-  Y  =  (var1 `  R )
mply1topmat.i  |-  I  =  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
Assertion
Ref Expression
mply1topmatval  |-  ( ( N  e.  V  /\  O  e.  L )  ->  ( I `  O
)  =  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
Distinct variable groups:    i, N, j, p    E, p    L, p    P, p    V, p    Y, p    i, O, j, k, p    .x. , k, p
Allowed substitution hints:    A( i, j, k, p)    P( i,
j, k)    Q( i,
j, k, p)    R( i, j, k, p)    .x. ( i,
j)    E( i, j, k)    I( i, j, k, p)    L( i, j, k)    N( k)    V( i, j, k)    Y( i, j, k)

Proof of Theorem mply1topmatval
StepHypRef Expression
1 mply1topmat.i . . 3  |-  I  =  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
21a1i 11 . 2  |-  ( ( N  e.  V  /\  O  e.  L )  ->  I  =  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) ) )
3 fveq2 5791 . . . . . . . . 9  |-  ( p  =  O  ->  (coe1 `  p )  =  (coe1 `  O ) )
43fveq1d 5793 . . . . . . . 8  |-  ( p  =  O  ->  (
(coe1 `  p ) `  k )  =  ( (coe1 `  O ) `  k ) )
54oveqd 6235 . . . . . . 7  |-  ( p  =  O  ->  (
i ( (coe1 `  p
) `  k )
j )  =  ( i ( (coe1 `  O
) `  k )
j ) )
65oveq1d 6233 . . . . . 6  |-  ( p  =  O  ->  (
( i ( (coe1 `  p ) `  k
) j )  .x.  ( k E Y ) )  =  ( ( i ( (coe1 `  O ) `  k
) j )  .x.  ( k E Y ) ) )
76mpteq2dv 4471 . . . . 5  |-  ( p  =  O  ->  (
k  e.  NN0  |->  ( ( i ( (coe1 `  p
) `  k )
j )  .x.  (
k E Y ) ) )  =  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O
) `  k )
j )  .x.  (
k E Y ) ) ) )
87oveq2d 6234 . . . 4  |-  ( p  =  O  ->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) )  =  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )
98mpt2eq3dv 6284 . . 3  |-  ( p  =  O  ->  (
i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )  =  ( i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
109adantl 464 . 2  |-  ( ( ( N  e.  V  /\  O  e.  L
)  /\  p  =  O )  ->  (
i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )  =  ( i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
11 simpr 459 . 2  |-  ( ( N  e.  V  /\  O  e.  L )  ->  O  e.  L )
12 simpl 455 . . 3  |-  ( ( N  e.  V  /\  O  e.  L )  ->  N  e.  V )
13 mpt2exga 6797 . . 3  |-  ( ( N  e.  V  /\  N  e.  V )  ->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )  e.  _V )
1412, 13syldan 468 . 2  |-  ( ( N  e.  V  /\  O  e.  L )  ->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )  e.  _V )
152, 10, 11, 14fvmptd 5879 1  |-  ( ( N  e.  V  /\  O  e.  L )  ->  ( I `  O
)  =  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   _Vcvv 3051    |-> cmpt 4442   ` cfv 5513  (class class class)co 6218    |-> cmpt2 6220   NN0cn0 10734   Basecbs 14657   .scvsca 14729    gsumg cgsu 14871  .gcmg 16196  mulGrpcmgp 17277  var1cv1 18351  Poly1cpl1 18352  coe1cco1 18353   Mat cmat 19017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722
This theorem is referenced by:  mply1topmatcl  19414
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