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Mirrors > Home > MPE Home > Th. List > cnbn | Structured version Visualization version GIF version |
Description: The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnbn.6 | ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 |
Ref | Expression |
---|---|
cnbn | ⊢ 𝑈 ∈ CBan |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnbn.6 | . . 3 ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 | |
2 | 1 | cnnv 30602 | . 2 ⊢ 𝑈 ∈ NrmCVec |
3 | eqid 2725 | . . . . 5 ⊢ 〈〈 + , · 〉, abs〉 = 〈〈 + , · 〉, abs〉 | |
4 | eqid 2725 | . . . . 5 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
5 | 3, 4 | cnims 30618 | . . . 4 ⊢ (abs ∘ − ) = (IndMet‘〈〈 + , · 〉, abs〉) |
6 | 5 | eqcomi 2734 | . . 3 ⊢ (IndMet‘〈〈 + , · 〉, abs〉) = (abs ∘ − ) |
7 | 6 | cncmet 25333 | . 2 ⊢ (IndMet‘〈〈 + , · 〉, abs〉) ∈ (CMet‘ℂ) |
8 | 1 | cnnvba 30604 | . . 3 ⊢ ℂ = (BaseSet‘𝑈) |
9 | 1 | fveq2i 6903 | . . . 4 ⊢ (IndMet‘𝑈) = (IndMet‘〈〈 + , · 〉, abs〉) |
10 | 9 | eqcomi 2734 | . . 3 ⊢ (IndMet‘〈〈 + , · 〉, abs〉) = (IndMet‘𝑈) |
11 | 8, 10 | iscbn 30789 | . 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ (IndMet‘〈〈 + , · 〉, abs〉) ∈ (CMet‘ℂ))) |
12 | 2, 7, 11 | mpbir2an 709 | 1 ⊢ 𝑈 ∈ CBan |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 〈cop 4638 ∘ ccom 5685 ‘cfv 6553 ℂcc 11152 + caddc 11157 · cmul 11159 − cmin 11490 abscabs 15234 CMetccmet 25265 NrmCVeccnv 30509 IndMetcims 30516 CBanccbn 30787 |
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-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 ax-addf 11233 ax-mulf 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-om 7876 df-1st 8002 df-2nd 8003 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8856 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9402 df-fi 9450 df-sup 9481 df-inf 9482 df-oi 9549 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-q 12980 df-rp 13024 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ioo 13377 df-ico 13379 df-icc 13380 df-fz 13534 df-fzo 13677 df-seq 14017 df-exp 14077 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19057 df-cntz 19306 df-cmn 19775 df-psmet 21327 df-xmet 21328 df-met 21329 df-bl 21330 df-mopn 21331 df-fbas 21332 df-fg 21333 df-cnfld 21336 df-top 22879 df-topon 22896 df-topsp 22918 df-bases 22932 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-cn 23214 df-cnp 23215 df-haus 23302 df-cmp 23374 df-tx 23549 df-hmeo 23742 df-fil 23833 df-flim 23926 df-fcls 23928 df-xms 24309 df-ms 24310 df-tms 24311 df-cncf 24881 df-cfil 25266 df-cmet 25268 df-grpo 30418 df-gid 30419 df-ginv 30420 df-gdiv 30421 df-ablo 30470 df-vc 30484 df-nv 30517 df-va 30520 df-ba 30521 df-sm 30522 df-0v 30523 df-vs 30524 df-nmcv 30525 df-ims 30526 df-cbn 30788 |
This theorem is referenced by: ubth 30798 cnchl 30841 |
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