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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 5371. (Revised by GG, 16-Mar-2025.) |
Ref | Expression |
---|---|
cnfldex | ⊢ ℂfld ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnfld 21383 | . 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 7765 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))〉, 〈(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))〉} ∈ V | |
3 | snex 5442 | . . . 4 ⊢ {〈(*𝑟‘ndx), ∗〉} ∈ V | |
4 | 2, 3 | unex 7763 | . . 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 7765 | . . . 4 ⊢ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∈ V | |
6 | snex 5442 | . . . 4 ⊢ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉} ∈ V | |
7 | 5, 6 | unex 7763 | . . 3 ⊢ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) ∈ V |
8 | 4, 7 | unex 7763 | . 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 2835 | 1 ⊢ ℂfld ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 {csn 4631 {ctp 4635 〈cop 4637 ∘ ccom 5693 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11151 + caddc 11156 · cmul 11158 ≤ cle 11294 − cmin 11490 ∗ccj 15132 abscabs 15270 ndxcnx 17227 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 *𝑟cstv 17300 TopSetcts 17304 lecple 17305 distcds 17307 UnifSetcunif 17308 MetOpencmopn 21372 metUnifcmetu 21373 ℂfldccnfld 21382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-sn 4632 df-pr 4634 df-tp 4636 df-uni 4913 df-cnfld 21383 |
This theorem is referenced by: regsumfsum 21471 rge0srg 21474 cnlmodlem3 25185 cnstrcvs 25188 cncvs 25192 cnncvsmulassdemo 25212 gsumzrsum 33045 xrge0iifmhm 33900 xrge0pluscn 33901 xrge0tmd 33906 cnzh 33931 esumpfinvallem 34055 aacllem 49032 |
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