Theorem pm2mpfval 21665

Description: A polynomial matrix transformed into a polynomial over matrices. (Contributed by AV, 4-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
pm2mpfval	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))

Distinct variable groups:   𝑘,𝑁   𝑅,𝑘   𝑘,𝑀

Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐶(𝑘)   𝑃(𝑘)   𝑄(𝑘)   𝑇(𝑘)   ↑ (𝑘)   ∗ (𝑘)   𝑉(𝑘)   𝑋(𝑘)

Proof of Theorem pm2mpfval

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2mpval.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
2		pm2mpval.c	. . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃)
3		pm2mpval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4		pm2mpval.m	. . . 4 ⊢ ∗ = ( ·𝑠 ‘𝑄)
5		pm2mpval.e	. . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑄))
6		pm2mpval.x	. . . 4 ⊢ 𝑋 = (var1‘𝐴)
7		pm2mpval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
8		pm2mpval.q	. . . 4 ⊢ 𝑄 = (Poly1‘𝐴)
9		pm2mpval.t	. . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	pm2mpval 21664	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))
11	10	3adant3 1134	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))
12		oveq1 7209	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘))
13	12	oveq1d 7217	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) = ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))
14	13	mpteq2dv 5140	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))
15	14	oveq2d 7218	. . 3 ⊢ (𝑚 = 𝑀 → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
16	15	adantl 485	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
17		simp3 1140	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
18		ovexd 7237	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ V)
19	11, 16, 17, 18	fvmptd 6814	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))

Syntax hints:   → wi 4   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  Vcvv 3401   ↦ cmpt 5124  ‘cfv 6369  (class class class)co 7202  Fincfn 8615  ℕ₀cn0 12073  Basecbs 16684   ·_𝑠 cvsca 16771   Σ_g cgsu 16917  ._gcmg 18460  mulGrpcmgp 19476  var₁cv1 21069  Poly₁cpl1 21070   Mat cmat 21276   decompPMat cdecpmat 21631   pMatToMatPoly cpm2mp 21661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pr 5311

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-ov 7205  df-oprab 7206  df-mpo 7207  df-pm2mp 21662

This theorem is referenced by:  pm2mpcl  21666  pm2mpf1  21668  pm2mpcoe1  21669  idpm2idmp  21670  mp2pm2mp  21680  pm2mpghm  21685  pm2mpmhmlem2  21688  monmat2matmon  21693