Theorem cnfldex 21429

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 5324. (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 21427	. 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 7731	. . . 4 ⊢ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∈ V
3		snex 5398	. . . 4 ⊢ {⟨(*𝑟‘ndx), ∗⟩} ∈ V
4	2, 3	unex 7729	. . 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 7731	. . . 4 ⊢ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∈ V
6		snex 5398	. . . 4 ⊢ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩} ∈ V
7	5, 6	unex 7729	. . 3 ⊢ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ∈ V
8	4, 7	unex 7729	. 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 2860	1 ⊢ ℂfld ∈ V

Syntax hints:   ∈ wcel 2144  Vcvv 3456   ∪ cun 3904  {csn 4584  {ctp 4588  ⟨cop 4590   ∘ ccom 5653  ‘cfv 6523  (class class class)co 7398   ∈ cmpo 7400  ℂcc 11073   + caddc 11078   · cmul 11080   ≤ cle 11219   − cmin 11416  ∗ccj 15125  abscabs 15263  ndxcnx 17231  Basecbs 17247  +_gcplusg 17288  ._rcmulr 17289  *_𝑟cstv 17290  TopSetcts 17294  lecple 17295  distcds 17297  UnifSetcunif 17298  MetOpencmopn 21416  metUnifcmetu 21417  ℂ_fldccnfld 21426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-un 3911  df-ss 3923  df-sn 4585  df-pr 4587  df-tp 4589  df-uni 4868  df-cnfld 21427

This theorem is referenced by:  regsumfsum  21489  rge0srg  21492  cnlmodlem3  25202  cnstrcvs  25205  cncvs  25209  cnncvsmulassdemo  25228  gsumzrsum  33247  xrge0iifmhm  34238  xrge0pluscn  34239  xrge0tmd  34244  cnzh  34267  esumpfinvallem  34373  aacllem  50427