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

Step	Hyp	Ref	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 ) ) ) ) ) )
2	1	a1i 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 ) )
4	3	fveq1d 5793	. . . . . . . 8 |- ( p = O -> ( (coe1 ` p ) ` k ) = ( (coe1 ` O ) ` k ) )
5	4	oveqd 6235	. . . . . . 7 |- ( p = O -> ( i ( (coe1 ` p ) ` k ) j ) = ( i ( (coe1 ` O ) ` k ) j ) )
6	5	oveq1d 6233	. . . . . 6 |- ( p = O -> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) = ( ( i ( (coe1 ` O ) ` k ) j ) .x. ( k E Y ) ) )
7	6	mpteq2dv 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 ) ) ) )
8	7	oveq2d 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 ) ) ) ) )
9	8	mpt2eq3dv 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 ) ) ) ) ) )
10	9	adantl 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 )
14	12, 13	syldan 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 )
15	2, 10, 11, 14	fvmptd 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 ) ) ) ) ) )

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   gsum_g 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