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Mirrors > Home > MPE Home > Th. List > cnfldstr | Structured version Visualization version GIF version |
Description: The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) |
Ref | Expression |
---|---|
cnfldstr | ⊢ ℂfld Struct 〈1, ;13〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnfld 20066 | . 2 ⊢ ℂfld = (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | |
2 | eqid 2797 | . . . 4 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
3 | 2 | srngfn 16326 | . . 3 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) Struct 〈1, 4〉 |
4 | 9nn 11413 | . . . . 5 ⊢ 9 ∈ ℕ | |
5 | tsetndx 16358 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 11912 | . . . . 5 ⊢ 9 < ;10 | |
7 | 10nn 11795 | . . . . 5 ⊢ ;10 ∈ ℕ | |
8 | plendx 16365 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 11594 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 11593 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 11382 | . . . . . 6 ⊢ 2 ∈ ℕ | |
12 | 2pos 11419 | . . . . . 6 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 11808 | . . . . 5 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 11800 | . . . . 5 ⊢ ;12 ∈ ℕ |
15 | dsndx 16374 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 16293 | . . . 4 ⊢ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} Struct 〈9, ;12〉 |
17 | 3nn 11388 | . . . . . 6 ⊢ 3 ∈ ℕ | |
18 | 9, 17 | decnncl 11800 | . . . . 5 ⊢ ;13 ∈ ℕ |
19 | unifndx 16376 | . . . . 5 ⊢ (UnifSet‘ndx) = ;13 | |
20 | 18, 19 | strle1 16291 | . . . 4 ⊢ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉} Struct 〈;13, ;13〉 |
21 | 2nn0 11595 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
22 | 2lt3 11488 | . . . . 5 ⊢ 2 < 3 | |
23 | 9, 21, 17, 22 | declt 11808 | . . . 4 ⊢ ;12 < ;13 |
24 | 16, 20, 23 | strleun 16290 | . . 3 ⊢ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) Struct 〈9, ;13〉 |
25 | 4lt9 11519 | . . 3 ⊢ 4 < 9 | |
26 | 3, 24, 25 | strleun 16290 | . 2 ⊢ (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) Struct 〈1, ;13〉 |
27 | 1, 26 | eqbrtri 4862 | 1 ⊢ ℂfld Struct 〈1, ;13〉 |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3765 {csn 4366 {ctp 4370 〈cop 4372 class class class wbr 4841 ∘ ccom 5314 ‘cfv 6099 ℂcc 10220 0cc0 10222 1c1 10223 + caddc 10225 · cmul 10227 ≤ cle 10362 − cmin 10554 2c2 11364 3c3 11365 4c4 11366 9c9 11371 ;cdc 11779 ∗ccj 14174 abscabs 14312 Struct cstr 16177 ndxcnx 16178 Basecbs 16181 +gcplusg 16264 .rcmulr 16265 *𝑟cstv 16266 TopSetcts 16270 lecple 16271 distcds 16273 UnifSetcunif 16274 MetOpencmopn 20055 metUnifcmetu 20056 ℂfldccnfld 20065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-fz 12577 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-plusg 16277 df-mulr 16278 df-starv 16279 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-cnfld 20066 |
This theorem is referenced by: cnfldex 20068 cnfldbas 20069 cnfldadd 20070 cnfldmul 20071 cnfldcj 20072 cnfldtset 20073 cnfldle 20074 cnfldds 20075 cnfldunif 20076 cnfldfunALT 20078 cffldtocusgr 26689 |
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