Theorem cnfldex 21354

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 5297. (Revised by GG, 16-Mar-2025.)

Assertion

Ref	Expression
cnfldex	⊢ ℂfld ∈ V

Proof of Theorem cnfldex

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnfld 21352	. 2 ⊢ ℂfld = (({⟨(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 7693	. . . 4 ⊢ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∈ V
3		snex 5371	. . . 4 ⊢ {⟨(*𝑟‘ndx), ∗⟩} ∈ V
4	2, 3	unex 7691	. . 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 7693	. . . 4 ⊢ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∈ V
6		snex 5371	. . . 4 ⊢ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩} ∈ V
7	5, 6	unex 7691	. . 3 ⊢ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ∈ V
8	4, 7	unex 7691	. 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
9	1, 8	eqeltri 2837	1 ⊢ ℂfld ∈ V

Syntax hints:   ∈ wcel 2121  Vcvv 3433   ∪ cun 3883  {csn 4558  {ctp 4562  ⟨cop 4564   ∘ ccom 5625  ‘cfv 6489  (class class class)co 7360   ∈ cmpo 7362  ℂcc 11031   + caddc 11036   · cmul 11038   ≤ cle 11175   − cmin 11372  ∗ccj 15053  abscabs 15191  ndxcnx 17158  Basecbs 17174  +_gcplusg 17215  ._rcmulr 17216  *_𝑟cstv 17217  TopSetcts 17221  lecple 17222  distcds 17224  UnifSetcunif 17225  MetOpencmopn 21341  metUnifcmetu 21342  ℂ_fldccnfld 21351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-un 3890  df-ss 3902  df-sn 4559  df-pr 4561  df-tp 4563  df-uni 4842  df-cnfld 21352

This theorem is referenced by:  regsumfsum  21414  rge0srg  21417  cnlmodlem3  25127  cnstrcvs  25130  cncvs  25134  cnncvsmulassdemo  25153  gsumzrsum  33150  xrge0iifmhm  34135  xrge0pluscn  34136  xrge0tmd  34141  cnzh  34164  esumpfinvallem  34270  aacllem  50305