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Mirrors > Home > MPE Home > Th. List > cnfldcusp | Structured version Visualization version GIF version |
Description: The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
Ref | Expression |
---|---|
cnfldcusp | ⊢ ℂfld ∈ CUnifSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 10318 | . . 3 ⊢ 0 ∈ ℂ | |
2 | 1 | ne0ii 4122 | . 2 ⊢ ℂ ≠ ∅ |
3 | cncms 23478 | . 2 ⊢ ℂfld ∈ CMetSp | |
4 | eqid 2797 | . . 3 ⊢ (UnifSt‘ℂfld) = (UnifSt‘ℂfld) | |
5 | 4 | cnflduss 23479 | . 2 ⊢ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − )) |
6 | cnfldbas 20069 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
7 | absf 14415 | . . . . . 6 ⊢ abs:ℂ⟶ℝ | |
8 | subf 10572 | . . . . . 6 ⊢ − :(ℂ × ℂ)⟶ℂ | |
9 | fco 6271 | . . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
10 | 7, 8, 9 | mp2an 684 | . . . . 5 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ |
11 | ffn 6254 | . . . . 5 ⊢ ((abs ∘ − ):(ℂ × ℂ)⟶ℝ → (abs ∘ − ) Fn (ℂ × ℂ)) | |
12 | fnresdm 6209 | . . . . 5 ⊢ ((abs ∘ − ) Fn (ℂ × ℂ) → ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − )) | |
13 | 10, 11, 12 | mp2b 10 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = (abs ∘ − ) |
14 | cnfldds 20075 | . . . . 5 ⊢ (abs ∘ − ) = (dist‘ℂfld) | |
15 | 14 | reseq1i 5594 | . . . 4 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
16 | 13, 15 | eqtr3i 2821 | . . 3 ⊢ (abs ∘ − ) = ((dist‘ℂfld) ↾ (ℂ × ℂ)) |
17 | 6, 16, 4 | cmetcusp1 23476 | . 2 ⊢ ((ℂ ≠ ∅ ∧ ℂfld ∈ CMetSp ∧ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − ))) → ℂfld ∈ CUnifSp) |
18 | 2, 3, 5, 17 | mp3an 1586 | 1 ⊢ ℂfld ∈ CUnifSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ∅c0 4113 × cxp 5308 ↾ cres 5312 ∘ ccom 5314 Fn wfn 6094 ⟶wf 6095 ‘cfv 6099 ℂcc 10220 ℝcr 10221 0cc0 10222 − cmin 10554 abscabs 14312 distcds 16273 metUnifcmetu 20056 ℂfldccnfld 20065 UnifStcuss 22382 CUnifSpccusp 22426 CMetSpccms 23455 |
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-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-inf2 8786 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 ax-pre-sup 10300 ax-addf 10301 ax-mulf 10302 |
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-rmo 3095 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-iin 4711 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-se 5270 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-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-supp 7531 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-2o 7798 df-oadd 7801 df-er 7980 df-map 8095 df-ixp 8147 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fsupp 8516 df-fi 8557 df-sup 8588 df-inf 8589 df-oi 8655 df-card 9049 df-cda 9276 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 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-q 12030 df-rp 12071 df-xneg 12189 df-xadd 12190 df-xmul 12191 df-ioo 12424 df-ico 12426 df-icc 12427 df-fz 12577 df-fzo 12717 df-seq 13052 df-exp 13111 df-hash 13367 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-hom 16288 df-cco 16289 df-rest 16395 df-topn 16396 df-0g 16414 df-gsum 16415 df-topgen 16416 df-pt 16417 df-prds 16420 df-xrs 16474 df-qtop 16479 df-imas 16480 df-xps 16482 df-mre 16558 df-mrc 16559 df-acs 16561 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-submnd 17648 df-mulg 17854 df-cntz 18059 df-cmn 18507 df-psmet 20057 df-xmet 20058 df-met 20059 df-bl 20060 df-mopn 20061 df-fbas 20062 df-fg 20063 df-metu 20064 df-cnfld 20066 df-top 21024 df-topon 21041 df-topsp 21063 df-bases 21076 df-cld 21149 df-ntr 21150 df-cls 21151 df-nei 21228 df-cn 21357 df-cnp 21358 df-haus 21445 df-cmp 21516 df-tx 21691 df-hmeo 21884 df-fil 21975 df-flim 22068 df-fcls 22070 df-ust 22329 df-utop 22360 df-uss 22385 df-usp 22386 df-cfilu 22416 df-cusp 22427 df-xms 22450 df-ms 22451 df-tms 22452 df-cncf 23006 df-cfil 23378 df-cmet 23380 df-cms 23458 |
This theorem is referenced by: cnrrext 30562 |
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