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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 5365. (Revised by GG, 16-Mar-2025.) |
| Ref | Expression |
|---|---|
| cnfldex | ⊢ ℂfld ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-cnfld 21365 | . 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 7766 | . . . 4 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))〉, 〈(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))〉} ∈ V | |
| 3 | snex 5436 | . . . 4 ⊢ {〈(*𝑟‘ndx), ∗〉} ∈ V | |
| 4 | 2, 3 | unex 7764 | . . 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 7766 | . . . 4 ⊢ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∈ V | |
| 6 | snex 5436 | . . . 4 ⊢ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉} ∈ V | |
| 7 | 5, 6 | unex 7764 | . . 3 ⊢ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) ∈ V |
| 8 | 4, 7 | unex 7764 | . 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 2837 | 1 ⊢ ℂfld ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 4626 {ctp 4630 〈cop 4632 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11153 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 ∗ccj 15135 abscabs 15273 ndxcnx 17230 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 *𝑟cstv 17299 TopSetcts 17303 lecple 17304 distcds 17306 UnifSetcunif 17307 MetOpencmopn 21354 metUnifcmetu 21355 ℂfldccnfld 21364 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-tp 4631 df-uni 4908 df-cnfld 21365 |
| This theorem is referenced by: regsumfsum 21453 rge0srg 21456 cnlmodlem3 25171 cnstrcvs 25174 cncvs 25178 cnncvsmulassdemo 25198 gsumzrsum 33062 xrge0iifmhm 33938 xrge0pluscn 33939 xrge0tmd 33944 cnzh 33969 esumpfinvallem 34075 aacllem 49320 |
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