Theorem cnfldex 21587

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 5334. (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 21585	. 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 7750	. . . 4 ⊢ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∈ V
3		snex 5408	. . . 4 ⊢ {⟨(*𝑟‘ndx), ∗⟩} ∈ V
4	2, 3	unex 7749	. . 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 7750	. . . 4 ⊢ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∈ V
6		snex 5408	. . . 4 ⊢ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩} ∈ V
7	5, 6	unex 7749	. . 3 ⊢ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ∈ V
8	4, 7	unex 7749	. 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 2858	1 ⊢ ℂfld ∈ V

This proof depends on syntax axioms:   ∈ wcel 2145  Vcvv 3453   ∪ cun 3900  {csn 4587  {ctp 4591  ⟨cop 4593   ∘ ccom 5663  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418  ℂcc 11123   + caddc 11128   · cmul 11130   ≤ cle 11269   − cmin 11466  ∗ccj 15183  abscabs 15321  ndxcnx 17287  Basecbs 17303  +_gcplusg 17344  ._rcmulr 17345  *_𝑟cstv 17346  TopSetcts 17350  lecple 17351  distcds 17353  UnifSetcunif 17354  MetOpencmopn 21574  metUnifcmetu 21575  ℂ_fldccnfld 21584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-ss 3919  df-sn 4588  df-pr 4590  df-tp 4592  df-uni 4871  df-cnfld 21585

This theorem is used by:  regsumfsum  21647  rge0srg  21650  cnlmodlem3  25365  cnstrcvs  25368  cncvs  25372  cnncvsmulassdemo  25391  gsumzrsum  33490  xrge0iifmhm  34434  xrge0pluscn  34435  xrge0tmd  34440  cnzh  34463  esumpfinvallem  34569  aacllem  50754