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Mirrors > Home > MPE Home > Th. List > cnfldex | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
cnfldex | ⊢ ℂfld ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnfld 21237 | . 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 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 | |
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 7730 | . . . 4 ⊢ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∈ V | |
6 | snex 5424 | . . . 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 2823 | 1 ⊢ ℂfld ∈ 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 ℂcc 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 ℂfldccnfld 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 |
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