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Theorem pm2mpfval 20420
 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 20419 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
11103adant3 1074 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
12 oveq1 6556 . . . . . 6 (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘))
1312oveq1d 6564 . . . . 5 (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) (𝑘 𝑋)) = ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))
1413mpteq2dv 4673 . . . 4 (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋))))
1514oveq2d 6565 . . 3 (𝑚 = 𝑀 → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))))
1615adantl 481 . 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 1056 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑀𝐵)
18 ovex 6577 . . 3 (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))) ∈ V
1918a1i 11 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))) ∈ V)
2011, 16, 17, 19fvmptd 6197 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 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℕ0cn0 11169  Basecbs 15695   ·𝑠 cvsca 15772   Σg cgsu 15924  .gcmg 17363  mulGrpcmgp 18312  var1cv1 19367  Poly1cpl1 19368   Mat cmat 20032   decompPMat cdecpmat 20386   pMatToMatPoly cpm2mp 20416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm2mp 20417 This theorem is referenced by:  pm2mpcl  20421  pm2mpf1  20423  pm2mpcoe1  20424  idpm2idmp  20425  mp2pm2mp  20435  pm2mpghm  20440  pm2mpmhmlem2  20443  monmat2matmon  20448
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