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Theorem cpmatpmat 20922
 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 20921 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
65eleq2d 2844 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)}))
7 elrabi 3566 . . 3 (𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)} → 𝑀𝐵)
86, 7syl6bi 245 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀𝐵))
983impia 1106 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝑀𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106  ∀wral 3089  {crab 3093  ‘cfv 6135  (class class class)co 6922  Fincfn 8241  ℕcn 11374  Basecbs 16255  0gc0g 16486  Poly1cpl1 19943  coe1cco1 19944   Mat cmat 20617   ConstPolyMat ccpmat 20915 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-cpmat 20918 This theorem is referenced by:  cpmatelimp  20924  cpmatelimp2  20926  cpmatacl  20928  cpmatinvcl  20929  cpmatmcl  20931  cpm2mf  20964  m2cpminvid2lem  20966  m2cpminvid2  20967  m2cpmfo  20968  m2cpmrngiso  20970
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