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

Theorem cnfldex 21239
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 5356. (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 21237 . 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 7730 . . . 4 {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∈ V
3 snex 5424 . . . 4 {⟨(*𝑟‘ndx), ∗⟩} ∈ V
42, 3unex 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 7730 . . . 4 {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∈ V
6 snex 5424 . . . 4 {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩} ∈ V
75, 6unex 7729 . . 3 ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ∈ V
84, 7unex 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
91, 8eqeltri 2823 1 fld ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3468  cun 3941  {csn 4623  {ctp 4627  cop 4629  ccom 5673  cfv 6536  (class class class)co 7404  cmpo 7406  cc 11107   + caddc 11112   · cmul 11114  cle 11250  cmin 11445  ccj 15047  abscabs 15185  ndxcnx 17133  Basecbs 17151  +gcplusg 17204  .rcmulr 17205  *𝑟cstv 17206  TopSetcts 17210  lecple 17211  distcds 17213  UnifSetcunif 17214  MetOpencmopn 21226  metUnifcmetu 21227  fldccnfld 21236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-tp 4628  df-uni 4903  df-cnfld 21237
This theorem is referenced by:  regsumfsum  21325  rge0srg  21328  cnlmodlem3  25016  cnstrcvs  25019  cncvs  25023  cnncvsmulassdemo  25043  xrge0iifmhm  33449  xrge0pluscn  33450  xrge0tmd  33455  cnzh  33480  esumpfinvallem  33602  aacllem  48103
  Copyright terms: Public domain W3C validator