Theorem pm2mpfval 22740

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 22739	. . 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 7365	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘))
13	12	oveq1d 7373	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) = ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))
14	13	mpteq2dv 5192	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))
15	14	oveq2d 7374	. . 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 7393	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ V)
19	11, 16, 17, 18	fvmptd 6948	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 1541   ∈ wcel 2113  Vcvv 3440   ↦ cmpt 5179  ‘cfv 6492  (class class class)co 7358  Fincfn 8883  ℕ₀cn0 12401  Basecbs 17136   ·_𝑠 cvsca 17181   Σ_g cgsu 17360  ._gcmg 18997  mulGrpcmgp 20075  var₁cv1 22116  Poly₁cpl1 22117   Mat cmat 22351   decompPMat cdecpmat 22706   pMatToMatPoly cpm2mp 22736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-pm2mp 22737

This theorem is referenced by:  pm2mpcl  22741  pm2mpf1  22743  pm2mpcoe1  22744  idpm2idmp  22745  mp2pm2mp  22755  pm2mpghm  22760  pm2mpmhmlem2  22763  monmat2matmon  22768