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

Theorem cnfldex 21346
Description: The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Avoid complex number axioms and ax-pow 5369. (Revised by GG, 16-Mar-2025.)
Assertion
Ref Expression
cnfldex fld ∈ V

Proof of Theorem cnfldex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnfld 21344 . 2 fld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
2 tpex 7755 . . . 4 {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∈ V
3 snex 5437 . . . 4 {⟨(*𝑟‘ndx), ∗⟩} ∈ V
42, 3unex 7754 . . 3 ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∈ V
5 tpex 7755 . . . 4 {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∈ V
6 snex 5437 . . . 4 {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩} ∈ V
75, 6unex 7754 . . 3 ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ∈ V
84, 7unex 7754 . 2 (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) ∈ V
91, 8eqeltri 2822 1 fld ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  Vcvv 3462  cun 3945  {csn 4633  {ctp 4637  cop 4639  ccom 5686  cfv 6554  (class class class)co 7424  cmpo 7426  cc 11156   + caddc 11161   · cmul 11163  cle 11299  cmin 11494  ccj 15101  abscabs 15239  ndxcnx 17195  Basecbs 17213  +gcplusg 17266  .rcmulr 17267  *𝑟cstv 17268  TopSetcts 17272  lecple 17273  distcds 17275  UnifSetcunif 17276  MetOpencmopn 21333  metUnifcmetu 21334  fldccnfld 21343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-sn 4634  df-pr 4636  df-tp 4638  df-uni 4914  df-cnfld 21344
This theorem is referenced by:  regsumfsum  21432  rge0srg  21435  cnlmodlem3  25156  cnstrcvs  25159  cncvs  25163  cnncvsmulassdemo  25183  xrge0iifmhm  33754  xrge0pluscn  33755  xrge0tmd  33760  cnzh  33785  esumpfinvallem  33907  aacllem  48549
  Copyright terms: Public domain W3C validator