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

Theorem matbas0pc 19972
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 19971 . . . . 5 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
21reldmmpt2 6643 . . . 4 Rel dom Mat
32ovprc 6555 . . 3 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
43fveq2d 6088 . 2 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = (Base‘∅))
5 base0 15682 . 2 ∅ = (Base‘∅)
64, 5syl6eqr 2657 1 (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1975  Vcvv 3168  c0 3869  cop 4126  cotp 4128   × cxp 5022  cfv 5786  (class class class)co 6523  Fincfn 7814  ndxcnx 15634   sSet csts 15635  Basecbs 15637  .rcmulr 15711   freeLMod cfrlm 19847   maMul cmmul 19946   Mat cmat 19970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-iota 5750  df-fun 5788  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-slot 15641  df-base 15642  df-mat 19971
This theorem is referenced by:  marrepfval  20123  marepvfval  20128  submafval  20142  minmar1fval  20209
  Copyright terms: Public domain W3C validator