Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  matbas0pc Structured version   Visualization version   GIF version

Theorem matbas0pc 21018
 Description: There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
Assertion
Ref Expression
matbas0pc (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)

Proof of Theorem matbas0pc
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mat 21017 . . . . 5 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
21reldmmpo 7268 . . . 4 Rel dom Mat
32ovprc 7177 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
43fveq2d 6653 . 2 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = (Base‘∅))
5 base0 16532 . 2 ∅ = (Base‘∅)
64, 5eqtr4di 2854 1 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ∅c0 4246  ⟨cop 4534  ⟨cotp 4536   × cxp 5521  ‘cfv 6328  (class class class)co 7139  Fincfn 8496  ndxcnx 16476   sSet csts 16477  Basecbs 16479  .rcmulr 16562   freeLMod cfrlm 20439   maMul cmmul 20994   Mat cmat 21016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-slot 16483  df-base 16485  df-mat 21017 This theorem is referenced by:  marrepfval  21169  marepvfval  21174  submafval  21188  minmar1fval  21255
 Copyright terms: Public domain W3C validator