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Theorem pm2mpfval 21404
 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 21403 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
11103adant3 1129 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
12 oveq1 7146 . . . . . 6 (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘))
1312oveq1d 7154 . . . . 5 (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) (𝑘 𝑋)) = ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))
1413mpteq2dv 5129 . . . 4 (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋))))
1514oveq2d 7155 . . 3 (𝑚 = 𝑀 → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))))
1615adantl 485 . 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 1135 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑀𝐵)
18 ovexd 7174 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))) ∈ V)
1911, 16, 17, 18fvmptd 6756 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 1084   = wceq 1538   ∈ wcel 2112  Vcvv 3444   ↦ cmpt 5113  ‘cfv 6328  (class class class)co 7139  Fincfn 8496  ℕ0cn0 11889  Basecbs 16478   ·𝑠 cvsca 16564   Σg cgsu 16709  .gcmg 18219  mulGrpcmgp 19235  var1cv1 20808  Poly1cpl1 20809   Mat cmat 21015   decompPMat cdecpmat 21370   pMatToMatPoly cpm2mp 21400 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-pm2mp 21401 This theorem is referenced by:  pm2mpcl  21405  pm2mpf1  21407  pm2mpcoe1  21408  idpm2idmp  21409  mp2pm2mp  21419  pm2mpghm  21424  pm2mpmhmlem2  21427  monmat2matmon  21432
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