Theorem matbas0pc 20034

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.

Step	Hyp	Ref	Expression
1		df-mat 20033	. . . . 5 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
2	1	reldmmpt2 6669	. . . 4 ⊢ Rel dom Mat
3	2	ovprc 6581	. . 3 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = ∅)
4	3	fveq2d 6107	. 2 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = (Base‘∅))
5		base0 15740	. 2 ⊢ ∅ = (Base‘∅)
6	4, 5	syl6eqr 2662	1 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)

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:  marrepfval  20185  marepvfval  20190  submafval  20204  minmar1fval  20271