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

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 5800	. . . . . . . . 9 |- ( p = O -> (coe1 ` p ) = (coe1 ` O ) )
4	3	fveq1d 5802	. . . . . . . 8 |- ( p = O -> ( (coe1 ` p ) ` k ) = ( (coe1 ` O ) ` k ) )
5	4	oveqd 6218	. . . . . . 7 |- ( p = O -> ( i ( (coe1 ` p ) ` k ) j ) = ( i ( (coe1 ` O ) ` k ) j ) )
6	5	oveq1d 6216	. . . . . 6 |- ( p = O -> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) = ( ( i ( (coe1 ` O ) ` k ) j ) .x. ( k E Y ) ) )
7	6	mpteq2dv 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 ) ) ) )
8	7	oveq2d 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 ) ) ) ) )
9	8	mpt2eq3dv 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 ) ) ) ) ) )
10	9	adantl 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 ) ) ) ) )
14	13	mpt2exg 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 )
15	12, 14	syldan 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 )
16	2, 10, 11, 15	fvmptd 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 ) ) ) ) ) )

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   gsum_g 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