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Theorem cpmatpmat 20736
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 20735 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
65eleq2d 2836 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)}))
7 elrabi 3511 . . 3 (𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)} → 𝑀𝐵)
86, 7syl6bi 243 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀𝐵))
983impia 1109 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝑀𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  {crab 3065  cfv 6032  (class class class)co 6794  Fincfn 8110  cn 11223  Basecbs 16065  0gc0g 16309  Poly1cpl1 19763  coe1cco1 19764   Mat cmat 20431   ConstPolyMat ccpmat 20729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5995  df-fun 6034  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-cpmat 20732
This theorem is referenced by:  cpmatelimp  20738  cpmatelimp2  20740  cpmatacl  20742  cpmatinvcl  20743  cpmatmcl  20745  cpm2mf  20778  m2cpminvid2lem  20780  m2cpminvid2  20781  m2cpmfo  20782  m2cpmrngiso  20784
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