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Theorem cnfldex 21385
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 5371. (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 21383 . 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 7765 . . . 4 {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∈ V
3 snex 5442 . . . 4 {⟨(*𝑟‘ndx), ∗⟩} ∈ V
42, 3unex 7763 . . 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 7765 . . . 4 {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∈ V
6 snex 5442 . . . 4 {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩} ∈ V
75, 6unex 7763 . . 3 ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ∈ V
84, 7unex 7763 . 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 2835 1 fld ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3478  cun 3961  {csn 4631  {ctp 4635  cop 4637  ccom 5693  cfv 6563  (class class class)co 7431  cmpo 7433  cc 11151   + caddc 11156   · cmul 11158  cle 11294  cmin 11490  ccj 15132  abscabs 15270  ndxcnx 17227  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  *𝑟cstv 17300  TopSetcts 17304  lecple 17305  distcds 17307  UnifSetcunif 17308  MetOpencmopn 21372  metUnifcmetu 21373  fldccnfld 21382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-tp 4636  df-uni 4913  df-cnfld 21383
This theorem is referenced by:  regsumfsum  21471  rge0srg  21474  cnlmodlem3  25185  cnstrcvs  25188  cncvs  25192  cnncvsmulassdemo  25212  gsumzrsum  33045  xrge0iifmhm  33900  xrge0pluscn  33901  xrge0tmd  33906  cnzh  33931  esumpfinvallem  34055  aacllem  49032
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