MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cpmatpmat Structured version   Visualization version   GIF version

Theorem cpmatpmat 21246
Description: A constant polynomial matrix is a polynomial matrix. (Contributed by AV, 16-Nov-2019.)
Hypotheses
Ref Expression
cpmat.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmat.p 𝑃 = (Poly1𝑅)
cpmat.c 𝐶 = (𝑁 Mat 𝑃)
cpmat.b 𝐵 = (Base‘𝐶)
Assertion
Ref Expression
cpmatpmat ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝑀𝐵)

Proof of Theorem cpmatpmat
Dummy variables 𝑚 𝑖 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cpmat.s . . . . 5 𝑆 = (𝑁 ConstPolyMat 𝑅)
2 cpmat.p . . . . 5 𝑃 = (Poly1𝑅)
3 cpmat.c . . . . 5 𝐶 = (𝑁 Mat 𝑃)
4 cpmat.b . . . . 5 𝐵 = (Base‘𝐶)
51, 2, 3, 4cpmat 21245 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
65eleq2d 2895 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)}))
7 elrabi 3672 . . 3 (𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)} → 𝑀𝐵)
86, 7syl6bi 254 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀𝐵))
983impia 1109 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝑀𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  cfv 6348  (class class class)co 7145  Fincfn 8497  cn 11626  Basecbs 16471  0gc0g 16701  Poly1cpl1 20273  coe1cco1 20274   Mat cmat 20944   ConstPolyMat ccpmat 21239
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-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-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  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-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-cpmat 21242
This theorem is referenced by:  cpmatelimp  21248  cpmatelimp2  21250  cpmatacl  21252  cpmatinvcl  21253  cpmatmcl  21255  cpm2mf  21288  m2cpminvid2lem  21290  m2cpminvid2  21291  m2cpmfo  21292  m2cpmrngiso  21294
  Copyright terms: Public domain W3C validator