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Theorem pm2mpfval 21332
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 21331 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
11103adant3 1124 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
12 oveq1 7152 . . . . . 6 (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘))
1312oveq1d 7160 . . . . 5 (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) (𝑘 𝑋)) = ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))
1413mpteq2dv 5153 . . . 4 (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋))))
1514oveq2d 7161 . . 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 1130 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑀𝐵)
18 ovexd 7180 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))) ∈ V)
1911, 16, 17, 18fvmptd 6767 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 1079   = wceq 1528  wcel 2105  Vcvv 3492  cmpt 5137  cfv 6348  (class class class)co 7145  Fincfn 8497  0cn0 11885  Basecbs 16471   ·𝑠 cvsca 16557   Σg cgsu 16702  .gcmg 18162  mulGrpcmgp 19168  var1cv1 20272  Poly1cpl1 20273   Mat cmat 20944   decompPMat cdecpmat 21298   pMatToMatPoly cpm2mp 21328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-pm2mp 21329
This theorem is referenced by:  pm2mpcl  21333  pm2mpf1  21335  pm2mpcoe1  21336  idpm2idmp  21337  mp2pm2mp  21347  pm2mpghm  21352  pm2mpmhmlem2  21355  monmat2matmon  21360
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