Theorem mply1topmatval 20609

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 20616). (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 6191	. . . . . . . . 9 |- ( p = O -> (coe1 ` p ) = (coe1 ` O ) )
4	3	fveq1d 6193	. . . . . . . 8 |- ( p = O -> ( (coe1 ` p ) ` k ) = ( (coe1 ` O ) ` k ) )
5	4	oveqd 6667	. . . . . . 7 |- ( p = O -> ( i ( (coe1 ` p ) ` k ) j ) = ( i ( (coe1 ` O ) ` k ) j ) )
6	5	oveq1d 6665	. . . . . 6 |- ( p = O -> ( ( i ( (coe1 ` p ) ` k ) j ) .x. ( k E Y ) ) = ( ( i ( (coe1 ` O ) ` k ) j ) .x. ( k E Y ) ) )
7	6	mpteq2dv 4745	. . . . 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 6666	. . . 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 6721	. . 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 482	. 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 477	. 2 |- ( ( N e. V /\ O e. L ) -> O e. L )
12		simpl 473	. . 3 |- ( ( N e. V /\ O e. L ) -> N e. V )
13		mpt2exga 7246	. . 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 487	. 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 6288	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 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   NN0cn0 11292   Basecbs 15857   .scvsca 15945   gsum_g cgsu 16101  ._gcmg 17540  mulGrpcmgp 18489  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  mply1topmatcl  20610