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Theorem matbas0 20035
Description: There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
Assertion
Ref Expression
matbas0 (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)

Proof of Theorem matbas0
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mat 20033 . . . 4 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
21mpt2ndm0 6773 . . 3 (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
32fveq2d 6107 . 2 (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = (Base‘∅))
4 base0 15740 . 2 ∅ = (Base‘∅)
53, 4syl6eqr 2662 1 (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  cop 4131  cotp 4133   × cxp 5036  cfv 5804  (class class class)co 6549  Fincfn 7841  ndxcnx 15692   sSet csts 15693  Basecbs 15695  .rcmulr 15769   freeLMod cfrlm 19909   maMul cmmul 20008   Mat cmat 20032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-slot 15699  df-base 15700  df-mat 20033
This theorem is referenced by:  nfimdetndef  20214  mdetfval1  20215
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