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Theorem pm2mpfval 22145
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 = (Poly1𝑅)
pm2mpval.c 𝐶 = (𝑁 Mat 𝑃)
pm2mpval.b 𝐵 = (Base‘𝐶)
pm2mpval.m = ( ·𝑠𝑄)
pm2mpval.e = (.g‘(mulGrp‘𝑄))
pm2mpval.x 𝑋 = (var1𝐴)
pm2mpval.a 𝐴 = (𝑁 Mat 𝑅)
pm2mpval.q 𝑄 = (Poly1𝐴)
pm2mpval.t 𝑇 = (𝑁 pMatToMatPoly 𝑅)
Assertion
Ref Expression
pm2mpfval ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (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.
StepHypRef Expression
1 pm2mpval.p . . . 4 𝑃 = (Poly1𝑅)
2 pm2mpval.c . . . 4 𝐶 = (𝑁 Mat 𝑃)
3 pm2mpval.b . . . 4 𝐵 = (Base‘𝐶)
4 pm2mpval.m . . . 4 = ( ·𝑠𝑄)
5 pm2mpval.e . . . 4 = (.g‘(mulGrp‘𝑄))
6 pm2mpval.x . . . 4 𝑋 = (var1𝐴)
7 pm2mpval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
8 pm2mpval.q . . . 4 𝑄 = (Poly1𝐴)
9 pm2mpval.t . . . 4 𝑇 = (𝑁 pMatToMatPoly 𝑅)
101, 2, 3, 4, 5, 6, 7, 8, 9pm2mpval 22144 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
11103adant3 1132 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
12 oveq1 7364 . . . . . 6 (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘))
1312oveq1d 7372 . . . . 5 (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) (𝑘 𝑋)) = ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))
1413mpteq2dv 5207 . . . 4 (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋))))
1514oveq2d 7373 . . 3 (𝑚 = 𝑀 → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))))
1615adantl 482 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ 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 ∧ 𝑅𝑉𝑀𝐵) → 𝑀𝐵)
18 ovexd 7392 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))) ∈ V)
1911, 16, 17, 18fvmptd 6955 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cmpt 5188  cfv 6496  (class class class)co 7357  Fincfn 8883  0cn0 12413  Basecbs 17083   ·𝑠 cvsca 17137   Σg cgsu 17322  .gcmg 18872  mulGrpcmgp 19896  var1cv1 21547  Poly1cpl1 21548   Mat cmat 21754   decompPMat cdecpmat 22111   pMatToMatPoly cpm2mp 22141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-pm2mp 22142
This theorem is referenced by:  pm2mpcl  22146  pm2mpf1  22148  pm2mpcoe1  22149  idpm2idmp  22150  mp2pm2mp  22160  pm2mpghm  22165  pm2mpmhmlem2  22168  monmat2matmon  22173
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