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Theorem mply1topmatval 31292
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 31299). (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 5800 . . . . . . . . 9  |-  ( p  =  O  ->  (coe1 `  p )  =  (coe1 `  O ) )
43fveq1d 5802 . . . . . . . 8  |-  ( p  =  O  ->  (
(coe1 `  p ) `  k )  =  ( (coe1 `  O ) `  k ) )
54oveqd 6218 . . . . . . 7  |-  ( p  =  O  ->  (
i ( (coe1 `  p
) `  k )
j )  =  ( i ( (coe1 `  O
) `  k )
j ) )
65oveq1d 6216 . . . . . 6  |-  ( p  =  O  ->  (
( i ( (coe1 `  p ) `  k
) j )  .x.  ( k E Y ) )  =  ( ( i ( (coe1 `  O ) `  k
) j )  .x.  ( k E Y ) ) )
76mpteq2dv 4488 . . . . 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 6217 . . . 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 6262 . . 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 466 . 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 461 . 2  |-  ( ( N  e.  V  /\  O  e.  L )  ->  O  e.  L )
12 simpl 457 . . 3  |-  ( ( N  e.  V  /\  O  e.  L )  ->  N  e.  V )
13 eqid 2454 . . . 4  |-  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  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 ) ) ) ) )
1413mpt2exg 6759 . . 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 )
1512, 14syldan 470 . 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 )
162, 10, 11, 15fvmptd 5889 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    |-> cmpt 4459   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   NN0cn0 10691   Basecbs 14293   .scvsca 14362    gsumg cgsu 14499  .gcmg 15534  mulGrpcmgp 16714  var1cv1 17757  Poly1cpl1 17758  coe1cco1 17759   Mat cmat 18406
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689
This theorem is referenced by:  mply1topmatcl  31293
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