Theorem idpm2idmp 22688

Description: The transformation of the identity polynomial matrix into polynomials over matrices results in the identity of the polynomials over matrices. (Contributed by AV, 18-Oct-2019.) (Revised 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
idpm2idmp	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐶)) = (1r‘𝑄))

Proof of Theorem idpm2idmp

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2mpval.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
2		pm2mpval.c	. . . . 5 ⊢ 𝐶 = (𝑁 Mat 𝑃)
3	1, 2	pmatring 22579	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
4		pm2mpval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5		eqid 2729	. . . . 5 ⊢ (1r‘𝐶) = (1r‘𝐶)
6	4, 5	ringidcl 20174	. . . 4 ⊢ (𝐶 ∈ Ring → (1r‘𝐶) ∈ 𝐵)
7	3, 6	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) ∈ 𝐵)
8		pm2mpval.m	. . . 4 ⊢ ∗ = ( ·𝑠 ‘𝑄)
9		pm2mpval.e	. . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑄))
10		pm2mpval.x	. . . 4 ⊢ 𝑋 = (var1‘𝐴)
11		pm2mpval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
12		pm2mpval.q	. . . 4 ⊢ 𝑄 = (Poly1‘𝐴)
13		pm2mpval.t	. . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
14	1, 2, 4, 8, 9, 10, 11, 12, 13	pm2mpfval 22683	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (1r‘𝐶) ∈ 𝐵) → (𝑇‘(1r‘𝐶)) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((1r‘𝐶) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
15	7, 14	mpd3an3 1464	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐶)) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((1r‘𝐶) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
16		eqid 2729	. . . . . . 7 ⊢ (0g‘𝐴) = (0g‘𝐴)
17		eqid 2729	. . . . . . 7 ⊢ (1r‘𝐴) = (1r‘𝐴)
18	1, 2, 5, 11, 16, 17	decpmatid 22657	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑘 ∈ ℕ0) → ((1r‘𝐶) decompPMat 𝑘) = if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)))
19	18	3expa 1118	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r‘𝐶) decompPMat 𝑘) = if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)))
20	19	oveq1d 7402	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (((1r‘𝐶) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) = (if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)))
21	20	mpteq2dva 5200	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑘 ∈ ℕ0 ↦ (((1r‘𝐶) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋))))
22	21	oveq2d 7403	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((1r‘𝐶) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)))))
23		ovif 7487	. . . . . 6 ⊢ (if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)) = if(𝑘 = 0, ((1r‘𝐴) ∗ (𝑘 ↑ 𝑋)), ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)))
24	11	matring 22330	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
25	12	ply1sca 22137	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
26	24, 25	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄))
27	26	adantr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → 𝐴 = (Scalar‘𝑄))
28	27	fveq2d 6862	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (1r‘𝐴) = (1r‘(Scalar‘𝑄)))
29	28	oveq1d 7402	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r‘𝐴) ∗ (𝑘 ↑ 𝑋)) = ((1r‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)))
30	12	ply1lmod 22136	. . . . . . . . . 10 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
31	24, 30	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
32		eqid 2729	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
33		eqid 2729	. . . . . . . . . . 11 ⊢ (Base‘𝑄) = (Base‘𝑄)
34	12, 10, 32, 9, 33	ply1moncl 22157	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑄))
35	24, 34	sylan 580	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑄))
36		eqid 2729	. . . . . . . . . 10 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
37		eqid 2729	. . . . . . . . . 10 ⊢ (1r‘(Scalar‘𝑄)) = (1r‘(Scalar‘𝑄))
38	33, 36, 8, 37	lmodvs1 20796	. . . . . . . . 9 ⊢ ((𝑄 ∈ LMod ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑄)) → ((1r‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)) = (𝑘 ↑ 𝑋))
39	31, 35, 38	syl2an2r 685	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)) = (𝑘 ↑ 𝑋))
40	29, 39	eqtrd 2764	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r‘𝐴) ∗ (𝑘 ↑ 𝑋)) = (𝑘 ↑ 𝑋))
41	27	fveq2d 6862	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (0g‘𝐴) = (0g‘(Scalar‘𝑄)))
42	41	oveq1d 7402	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)) = ((0g‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)))
43		eqid 2729	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
44		eqid 2729	. . . . . . . . . 10 ⊢ (0g‘𝑄) = (0g‘𝑄)
45	33, 36, 8, 43, 44	lmod0vs 20801	. . . . . . . . 9 ⊢ ((𝑄 ∈ LMod ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑄)) → ((0g‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑄))
46	31, 35, 45	syl2an2r 685	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑄))
47	42, 46	eqtrd 2764	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑄))
48	40, 47	ifeq12d 4510	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 0, ((1r‘𝐴) ∗ (𝑘 ↑ 𝑋)), ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋))) = if(𝑘 = 0, (𝑘 ↑ 𝑋), (0g‘𝑄)))
49	23, 48	eqtrid 2776	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)) = if(𝑘 = 0, (𝑘 ↑ 𝑋), (0g‘𝑄)))
50	49	mpteq2dva 5200	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 ↑ 𝑋), (0g‘𝑄))))
51	50	oveq2d 7403	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 ↑ 𝑋), (0g‘𝑄)))))
52	12	ply1ring 22132	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
53		ringmnd 20152	. . . . 5 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ Mnd)
54	24, 52, 53	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Mnd)
55		nn0ex 12448	. . . . 5 ⊢ ℕ0 ∈ V
56	55	a1i 11	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ℕ0 ∈ V)
57		0nn0 12457	. . . . 5 ⊢ 0 ∈ ℕ0
58	57	a1i 11	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ ℕ0)
59		eqid 2729	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 ↑ 𝑋), (0g‘𝑄))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 ↑ 𝑋), (0g‘𝑄)))
60	35	ralrimiva 3125	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑘 ∈ ℕ0 (𝑘 ↑ 𝑋) ∈ (Base‘𝑄))
61	44, 54, 56, 58, 59, 60	gsummpt1n0 19895	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 ↑ 𝑋), (0g‘𝑄)))) = ⦋0 / 𝑘⦌(𝑘 ↑ 𝑋))
62		c0ex 11168	. . . . 5 ⊢ 0 ∈ V
63		csbov1g 7434	. . . . 5 ⊢ (0 ∈ V → ⦋0 / 𝑘⦌(𝑘 ↑ 𝑋) = (⦋0 / 𝑘⦌𝑘 ↑ 𝑋))
64	62, 63	mp1i 13	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ⦋0 / 𝑘⦌(𝑘 ↑ 𝑋) = (⦋0 / 𝑘⦌𝑘 ↑ 𝑋))
65		csbvarg 4397	. . . . . 6 ⊢ (0 ∈ V → ⦋0 / 𝑘⦌𝑘 = 0)
66	62, 65	mp1i 13	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ⦋0 / 𝑘⦌𝑘 = 0)
67	66	oveq1d 7402	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (⦋0 / 𝑘⦌𝑘 ↑ 𝑋) = (0 ↑ 𝑋))
68	12, 10, 32, 9	ply1idvr1 22181	. . . . 5 ⊢ (𝐴 ∈ Ring → (0 ↑ 𝑋) = (1r‘𝑄))
69	24, 68	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0 ↑ 𝑋) = (1r‘𝑄))
70	64, 67, 69	3eqtrd 2768	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ⦋0 / 𝑘⦌(𝑘 ↑ 𝑋) = (1r‘𝑄))
71	51, 61, 70	3eqtrd 2768	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)))) = (1r‘𝑄))
72	15, 22, 71	3eqtrd 2768	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐶)) = (1r‘𝑄))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⦋csb 3862  ifcif 4488   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  Fincfn 8918  0cc0 11068  ℕ₀cn0 12442  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402   Σ_g cgsu 17403  Mndcmnd 18661  ._gcmg 18999  mulGrpcmgp 20049  1_rcur 20090  Ringcrg 20142  LModclmod 20766  var₁cv1 22060  Poly₁cpl1 22061   Mat cmat 22294   decompPMat cdecpmat 22649   pMatToMatPoly cpm2mp 22679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-ghm 19145  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-subrng 20455  df-subrg 20479  df-lmod 20768  df-lss 20838  df-sra 21080  df-rgmod 21081  df-dsmm 21641  df-frlm 21656  df-psr 21818  df-mvr 21819  df-mpl 21820  df-opsr 21822  df-psr1 22064  df-vr1 22065  df-ply1 22066  df-coe1 22067  df-mamu 22278  df-mat 22295  df-decpmat 22650  df-pm2mp 22680

This theorem is referenced by:  pm2mpmhm  22707