Theorem idpm2idmp 22695

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 22586	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
4		pm2mpval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5		eqid 2730	. . . . 5 ⊢ (1r‘𝐶) = (1r‘𝐶)
6	4, 5	ringidcl 20181	. . . 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 22690	. . 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 2730	. . . . . . 7 ⊢ (0g‘𝐴) = (0g‘𝐴)
17		eqid 2730	. . . . . . 7 ⊢ (1r‘𝐴) = (1r‘𝐴)
18	1, 2, 5, 11, 16, 17	decpmatid 22664	. . . . . 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 7405	. . . 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 5203	. . 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 7406	. 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 7490	. . . . . 6 ⊢ (if(𝑘 = 0, (1r‘𝐴), (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)) = if(𝑘 = 0, ((1r‘𝐴) ∗ (𝑘 ↑ 𝑋)), ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)))
24	11	matring 22337	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
25	12	ply1sca 22144	. . . . . . . . . . . 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 6865	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (1r‘𝐴) = (1r‘(Scalar‘𝑄)))
29	28	oveq1d 7405	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r‘𝐴) ∗ (𝑘 ↑ 𝑋)) = ((1r‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)))
30	12	ply1lmod 22143	. . . . . . . . . 10 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
31	24, 30	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
32		eqid 2730	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
33		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝑄) = (Base‘𝑄)
34	12, 10, 32, 9, 33	ply1moncl 22164	. . . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
37		eqid 2730	. . . . . . . . . 10 ⊢ (1r‘(Scalar‘𝑄)) = (1r‘(Scalar‘𝑄))
38	33, 36, 8, 37	lmodvs1 20803	. . . . . . . . 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 2765	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((1r‘𝐴) ∗ (𝑘 ↑ 𝑋)) = (𝑘 ↑ 𝑋))
41	27	fveq2d 6865	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → (0g‘𝐴) = (0g‘(Scalar‘𝑄)))
42	41	oveq1d 7405	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)) = ((0g‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)))
43		eqid 2730	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
44		eqid 2730	. . . . . . . . . 10 ⊢ (0g‘𝑄) = (0g‘𝑄)
45	33, 36, 8, 43, 44	lmod0vs 20808	. . . . . . . . 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 2765	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑘 ∈ ℕ0) → ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑄))
48	40, 47	ifeq12d 4513	. . . . . 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 2777	. . . . 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 5203	. . . 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 7406	. . 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 22139	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
53		ringmnd 20159	. . . . 5 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ Mnd)
54	24, 52, 53	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Mnd)
55		nn0ex 12455	. . . . 5 ⊢ ℕ0 ∈ V
56	55	a1i 11	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ℕ0 ∈ V)
57		0nn0 12464	. . . . 5 ⊢ 0 ∈ ℕ0
58	57	a1i 11	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ ℕ0)
59		eqid 2730	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 ↑ 𝑋), (0g‘𝑄))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 ↑ 𝑋), (0g‘𝑄)))
60	35	ralrimiva 3126	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑘 ∈ ℕ0 (𝑘 ↑ 𝑋) ∈ (Base‘𝑄))
61	44, 54, 56, 58, 59, 60	gsummpt1n0 19902	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (𝑘 ↑ 𝑋), (0g‘𝑄)))) = ⦋0 / 𝑘⦌(𝑘 ↑ 𝑋))
62		c0ex 11175	. . . . 5 ⊢ 0 ∈ V
63		csbov1g 7437	. . . . 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 4400	. . . . . 6 ⊢ (0 ∈ V → ⦋0 / 𝑘⦌𝑘 = 0)
66	62, 65	mp1i 13	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ⦋0 / 𝑘⦌𝑘 = 0)
67	66	oveq1d 7405	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (⦋0 / 𝑘⦌𝑘 ↑ 𝑋) = (0 ↑ 𝑋))
68	12, 10, 32, 9	ply1idvr1 22188	. . . . 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 2769	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ⦋0 / 𝑘⦌(𝑘 ↑ 𝑋) = (1r‘𝑄))
71	51, 61, 70	3eqtrd 2769	. 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 2769	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐶)) = (1r‘𝑄))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⦋csb 3865  ifcif 4491   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390  Fincfn 8921  0cc0 11075  ℕ₀cn0 12449  Basecbs 17186  Scalarcsca 17230   ·_𝑠 cvsca 17231  0_gc0g 17409   Σ_g cgsu 17410  Mndcmnd 18668  ._gcmg 19006  mulGrpcmgp 20056  1_rcur 20097  Ringcrg 20149  LModclmod 20773  var₁cv1 22067  Poly₁cpl1 22068   Mat cmat 22301   decompPMat cdecpmat 22656   pMatToMatPoly cpm2mp 22686

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-ofr 7657  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-fzo 13623  df-seq 13974  df-hash 14303  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17411  df-gsum 17412  df-prds 17417  df-pws 17419  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-mulg 19007  df-subg 19062  df-ghm 19152  df-cntz 19256  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-subrng 20462  df-subrg 20486  df-lmod 20775  df-lss 20845  df-sra 21087  df-rgmod 21088  df-dsmm 21648  df-frlm 21663  df-psr 21825  df-mvr 21826  df-mpl 21827  df-opsr 21829  df-psr1 22071  df-vr1 22072  df-ply1 22073  df-coe1 22074  df-mamu 22285  df-mat 22302  df-decpmat 22657  df-pm2mp 22687

This theorem is referenced by:  pm2mpmhm  22714