Theorem pm2mpval 20600

Description: Value of the transformation of a polynomial matrix into a polynomial over matrices. (Contributed by AV, 5-Dec-2019.)

Hypotheses

Ref	Expression
pm2mpval.p	|- P = (Poly1 ` R )
pm2mpval.c	|- C = ( N Mat P )
pm2mpval.b	|- B = ( Base ` C )
pm2mpval.m	|- .* = ( .s ` Q )
pm2mpval.e	|- .^ = (.g ` (mulGrp ` Q ) )
pm2mpval.x	|- X = (var1 ` A )
pm2mpval.a	|- A = ( N Mat R )
pm2mpval.q	|- Q = (Poly1 ` A )
pm2mpval.t	|- T = ( N pMatToMatPoly R )

Assertion

Ref	Expression
pm2mpval	|- ( ( N e. Fin /\ R e. V ) -> T = ( m e. B |-> ( Q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) .* ( k .^ X ) ) ) ) ) )

Distinct variable groups:   B, m   k, N, m   R, k, m   m, V

Allowed substitution hints:   A( k, m)   B( k)   C( k, m)   P( k, m)   Q( k, m)   T( k, m)   .^ ( k, m)   .* ( k, m)   V( k)   X( k, m)

Proof of Theorem pm2mpval

Dummy variables n r a q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2mpval.t	. 2 |- T = ( N pMatToMatPoly R )
2		df-pm2mp 20598	. . . 4 |- pMatToMatPoly = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat (Poly1 ` r ) ) ) |-> [_ ( n Mat r ) / a ]_ [_ (Poly1 ` a ) / q ]_ ( q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` q ) ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) ) ) ) ) )
3	2	a1i 11	. . 3 |- ( ( N e. Fin /\ R e. V ) -> pMatToMatPoly = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat (Poly1 ` r ) ) ) |-> [_ ( n Mat r ) / a ]_ [_ (Poly1 ` a ) / q ]_ ( q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` q ) ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) ) ) ) ) ) )
4		simpl 473	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> n = N )
5		fveq2 6191	. . . . . . . . 9 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
6	5	adantl 482	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> (Poly1 ` r ) = (Poly1 ` R ) )
7	4, 6	oveq12d 6668	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( n Mat (Poly1 ` r ) ) = ( N Mat (Poly1 ` R ) ) )
8	7	fveq2d 6195	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat (Poly1 ` r ) ) ) = ( Base ` ( N Mat (Poly1 ` R ) ) ) )
9		pm2mpval.b	. . . . . . 7 |- B = ( Base ` C )
10		pm2mpval.c	. . . . . . . . 9 |- C = ( N Mat P )
11		pm2mpval.p	. . . . . . . . . 10 |- P = (Poly1 ` R )
12	11	oveq2i 6661	. . . . . . . . 9 |- ( N Mat P ) = ( N Mat (Poly1 ` R ) )
13	10, 12	eqtri 2644	. . . . . . . 8 |- C = ( N Mat (Poly1 ` R ) )
14	13	fveq2i 6194	. . . . . . 7 |- ( Base ` C ) = ( Base ` ( N Mat (Poly1 ` R ) ) )
15	9, 14	eqtri 2644	. . . . . 6 |- B = ( Base ` ( N Mat (Poly1 ` R ) ) )
16	8, 15	syl6eqr 2674	. . . . 5 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat (Poly1 ` r ) ) ) = B )
17	16	adantl 482	. . . 4 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> ( Base ` ( n Mat (Poly1 ` r ) ) ) = B )
18		ovex 6678	. . . . . 6 |- ( n Mat r ) e. _V
19		fvexd 6203	. . . . . . 7 |- ( a = ( n Mat r ) -> (Poly1 ` a ) e. _V )
20		simpr 477	. . . . . . . . 9 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> q = (Poly1 ` a ) )
21		fveq2 6191	. . . . . . . . . 10 |- ( a = ( n Mat r ) -> (Poly1 ` a ) = (Poly1 ` ( n Mat r ) ) )
22	21	adantr 481	. . . . . . . . 9 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> (Poly1 ` a ) = (Poly1 ` ( n Mat r ) ) )
23	20, 22	eqtrd 2656	. . . . . . . 8 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> q = (Poly1 ` ( n Mat r ) ) )
24	23	fveq2d 6195	. . . . . . . . . 10 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> ( .s ` q ) = ( .s ` (Poly1 ` ( n Mat r ) ) ) )
25		eqidd 2623	. . . . . . . . . 10 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> ( m decompPMat k ) = ( m decompPMat k ) )
26	23	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> (mulGrp ` q ) = (mulGrp ` (Poly1 ` ( n Mat r ) ) ) )
27	26	fveq2d 6195	. . . . . . . . . . 11 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> (.g ` (mulGrp ` q ) ) = (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) )
28		eqidd 2623	. . . . . . . . . . 11 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> k = k )
29		fveq2 6191	. . . . . . . . . . . 12 |- ( a = ( n Mat r ) -> (var1 ` a ) = (var1 ` ( n Mat r ) ) )
30	29	adantr 481	. . . . . . . . . . 11 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> (var1 ` a ) = (var1 ` ( n Mat r ) ) )
31	27, 28, 30	oveq123d 6671	. . . . . . . . . 10 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) = ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) )
32	24, 25, 31	oveq123d 6671	. . . . . . . . 9 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> ( ( m decompPMat k ) ( .s ` q ) ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) ) = ( ( m decompPMat k ) ( .s ` (Poly1 ` ( n Mat r ) ) ) ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) ) )
33	32	mpteq2dv 4745	. . . . . . . 8 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` q ) ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) ) ) = ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` (Poly1 ` ( n Mat r ) ) ) ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) ) ) )
34	23, 33	oveq12d 6668	. . . . . . 7 |- ( ( a = ( n Mat r ) /\ q = (Poly1 ` a ) ) -> ( q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` q ) ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) ) ) ) = ( (Poly1 ` ( n Mat r ) ) gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` (Poly1 ` ( n Mat r ) ) ) ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) ) ) ) )
35	19, 34	csbied 3560	. . . . . 6 |- ( a = ( n Mat r ) -> [_ (Poly1 ` a ) / q ]_ ( q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` q ) ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) ) ) ) = ( (Poly1 ` ( n Mat r ) ) gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` (Poly1 ` ( n Mat r ) ) ) ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) ) ) ) )
36	18, 35	csbie 3559	. . . . 5 |- [_ ( n Mat r ) / a ]_ [_ (Poly1 ` a ) / q ]_ ( q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` q ) ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) ) ) ) = ( (Poly1 ` ( n Mat r ) ) gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` (Poly1 ` ( n Mat r ) ) ) ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) ) ) )
37		oveq12 6659	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
38	37	fveq2d 6195	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> (Poly1 ` ( n Mat r ) ) = (Poly1 ` ( N Mat R ) ) )
39		pm2mpval.q	. . . . . . . . 9 |- Q = (Poly1 ` A )
40		pm2mpval.a	. . . . . . . . . 10 |- A = ( N Mat R )
41	40	fveq2i 6194	. . . . . . . . 9 |- (Poly1 ` A ) = (Poly1 ` ( N Mat R ) )
42	39, 41	eqtri 2644	. . . . . . . 8 |- Q = (Poly1 ` ( N Mat R ) )
43	38, 42	syl6eqr 2674	. . . . . . 7 |- ( ( n = N /\ r = R ) -> (Poly1 ` ( n Mat r ) ) = Q )
44	38	fveq2d 6195	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> ( .s ` (Poly1 ` ( n Mat r ) ) ) = ( .s ` (Poly1 ` ( N Mat R ) ) ) )
45		pm2mpval.m	. . . . . . . . . . 11 |- .* = ( .s ` Q )
46	42	fveq2i 6194	. . . . . . . . . . 11 |- ( .s ` Q ) = ( .s ` (Poly1 ` ( N Mat R ) ) )
47	45, 46	eqtri 2644	. . . . . . . . . 10 |- .* = ( .s ` (Poly1 ` ( N Mat R ) ) )
48	44, 47	syl6eqr 2674	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( .s ` (Poly1 ` ( n Mat r ) ) ) = .* )
49		eqidd 2623	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( m decompPMat k ) = ( m decompPMat k ) )
50	38	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( n = N /\ r = R ) -> (mulGrp ` (Poly1 ` ( n Mat r ) ) ) = (mulGrp ` (Poly1 ` ( N Mat R ) ) ) )
51	50	fveq2d 6195	. . . . . . . . . . 11 |- ( ( n = N /\ r = R ) -> (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) = (.g ` (mulGrp ` (Poly1 ` ( N Mat R ) ) ) ) )
52		pm2mpval.e	. . . . . . . . . . . 12 |- .^ = (.g ` (mulGrp ` Q ) )
53	42	fveq2i 6194	. . . . . . . . . . . . 13 |- (mulGrp ` Q ) = (mulGrp ` (Poly1 ` ( N Mat R ) ) )
54	53	fveq2i 6194	. . . . . . . . . . . 12 |- (.g ` (mulGrp ` Q ) ) = (.g ` (mulGrp ` (Poly1 ` ( N Mat R ) ) ) )
55	52, 54	eqtri 2644	. . . . . . . . . . 11 |- .^ = (.g ` (mulGrp ` (Poly1 ` ( N Mat R ) ) ) )
56	51, 55	syl6eqr 2674	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) = .^ )
57		eqidd 2623	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> k = k )
58	37	fveq2d 6195	. . . . . . . . . . 11 |- ( ( n = N /\ r = R ) -> (var1 ` ( n Mat r ) ) = (var1 ` ( N Mat R ) ) )
59		pm2mpval.x	. . . . . . . . . . . 12 |- X = (var1 ` A )
60	40	fveq2i 6194	. . . . . . . . . . . 12 |- (var1 ` A ) = (var1 ` ( N Mat R ) )
61	59, 60	eqtri 2644	. . . . . . . . . . 11 |- X = (var1 ` ( N Mat R ) )
62	58, 61	syl6eqr 2674	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> (var1 ` ( n Mat r ) ) = X )
63	56, 57, 62	oveq123d 6671	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) = ( k .^ X ) )
64	48, 49, 63	oveq123d 6671	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( ( m decompPMat k ) ( .s ` (Poly1 ` ( n Mat r ) ) ) ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) ) = ( ( m decompPMat k ) .* ( k .^ X ) ) )
65	64	mpteq2dv 4745	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` (Poly1 ` ( n Mat r ) ) ) ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) ) ) = ( k e. NN0 |-> ( ( m decompPMat k ) .* ( k .^ X ) ) ) )
66	43, 65	oveq12d 6668	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( (Poly1 ` ( n Mat r ) ) gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` (Poly1 ` ( n Mat r ) ) ) ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) ) ) ) = ( Q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) .* ( k .^ X ) ) ) ) )
67	66	adantl 482	. . . . 5 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> ( (Poly1 ` ( n Mat r ) ) gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` (Poly1 ` ( n Mat r ) ) ) ( k (.g ` (mulGrp ` (Poly1 ` ( n Mat r ) ) ) ) (var1 ` ( n Mat r ) ) ) ) ) ) = ( Q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) .* ( k .^ X ) ) ) ) )
68	36, 67	syl5eq 2668	. . . 4 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> [_ ( n Mat r ) / a ]_ [_ (Poly1 ` a ) / q ]_ ( q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` q ) ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) ) ) ) = ( Q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) .* ( k .^ X ) ) ) ) )
69	17, 68	mpteq12dv 4733	. . 3 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> ( m e. ( Base ` ( n Mat (Poly1 ` r ) ) ) |-> [_ ( n Mat r ) / a ]_ [_ (Poly1 ` a ) / q ]_ ( q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) ( .s ` q ) ( k (.g ` (mulGrp ` q ) ) (var1 ` a ) ) ) ) ) ) = ( m e. B |-> ( Q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) .* ( k .^ X ) ) ) ) ) )
70		simpl 473	. . 3 |- ( ( N e. Fin /\ R e. V ) -> N e. Fin )
71		elex 3212	. . . 4 |- ( R e. V -> R e. _V )
72	71	adantl 482	. . 3 |- ( ( N e. Fin /\ R e. V ) -> R e. _V )
73		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
74	9, 73	eqeltri 2697	. . . . 5 |- B e. _V
75	74	mptex 6486	. . . 4 |- ( m e. B |-> ( Q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) .* ( k .^ X ) ) ) ) ) e. _V
76	75	a1i 11	. . 3 |- ( ( N e. Fin /\ R e. V ) -> ( m e. B |-> ( Q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) .* ( k .^ X ) ) ) ) ) e. _V )
77	3, 69, 70, 72, 76	ovmpt2d 6788	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( N pMatToMatPoly R ) = ( m e. B |-> ( Q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) .* ( k .^ X ) ) ) ) ) )
78	1, 77	syl5eq 2668	1 |- ( ( N e. Fin /\ R e. V ) -> T = ( m e. B |-> ( Q gsumg ( k e. NN0 |-> ( ( m decompPMat k ) .* ( k .^ X ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   NN0cn0 11292   Basecbs 15857   .scvsca 15945   gsum_g cgsu 16101  ._gcmg 17540  mulGrpcmgp 18489  var₁cv1 19546  Poly₁cpl1 19547   Mat cmat 20213   decompPMat cdecpmat 20567   pMatToMatPoly cpm2mp 20597

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-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-pr 4906

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-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-pm2mp 20598

This theorem is referenced by:  pm2mpfval  20601  pm2mpf  20603