Theorem pm2mpfval 22683

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 22682	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))
11	10	3adant3 1132	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))
12		oveq1 7394	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘))
13	12	oveq1d 7402	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) = ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))
14	13	mpteq2dv 5201	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))
15	14	oveq2d 7403	. . 3 ⊢ (𝑚 = 𝑀 → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
16	15	adantl 481	. 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 1138	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
18		ovexd 7422	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ V)
19	11, 16, 17, 18	fvmptd 6975	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  Fincfn 8918  ℕ₀cn0 12442  Basecbs 17179   ·_𝑠 cvsca 17224   Σ_g cgsu 17403  ._gcmg 18999  mulGrpcmgp 20049  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-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3045  df-rex 3054  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-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  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-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-pm2mp 22680

This theorem is referenced by:  pm2mpcl  22684  pm2mpf1  22686  pm2mpcoe1  22687  idpm2idmp  22688  mp2pm2mp  22698  pm2mpghm  22703  pm2mpmhmlem2  22706  monmat2matmon  22711