|   | Hilbert Space Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > HSE Home > Th. List > h0elch | Structured version Visualization version GIF version | ||
| Description: The zero subspace is a closed subspace. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| h0elch | ⊢ 0ℋ ∈ Cℋ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ch0 31273 | . 2 ⊢ 0ℋ = {0ℎ} | |
| 2 | hsn0elch 31268 | . 2 ⊢ {0ℎ} ∈ Cℋ | |
| 3 | 1, 2 | eqeltri 2836 | 1 ⊢ 0ℋ ∈ Cℋ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 {csn 4625 0ℎc0v 30944 Cℋ cch 30949 0ℋc0h 30955 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 ax-hilex 31019 ax-hfvadd 31020 ax-hvcom 31021 ax-hvass 31022 ax-hv0cl 31023 ax-hvaddid 31024 ax-hfvmul 31025 ax-hvmulid 31026 ax-hvmulass 31027 ax-hvdistr1 31028 ax-hvdistr2 31029 ax-hvmul0 31030 ax-hfi 31099 ax-his1 31102 ax-his2 31103 ax-his3 31104 ax-his4 31105 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-icc 13395 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-topgen 17489 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-top 22901 df-topon 22918 df-bases 22954 df-lm 23238 df-haus 23324 df-grpo 30513 df-gid 30514 df-ginv 30515 df-gdiv 30516 df-ablo 30565 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-0v 30618 df-vs 30619 df-nmcv 30620 df-ims 30621 df-hnorm 30988 df-hvsub 30991 df-hlim 30992 df-sh 31227 df-ch 31241 df-ch0 31273 | 
| This theorem is referenced by: h0elsh 31276 chintcl 31352 omlsi 31424 pjoml 31456 pjoc2 31459 chj0i 31475 chj00i 31507 chm0 31511 chne0 31514 chocin 31515 chj0 31517 chlejb1 31532 chnle 31534 ledi 31560 chsup0 31568 h1datom 31602 cmbr3 31628 cm0 31629 pjoml2 31631 cmcm 31634 cmcm3 31635 lecm 31637 qlaxr3i 31656 nonbooli 31671 pjige0 31711 pjhfo 31726 pj11 31734 ho0f 31771 pjhmop 32170 pjidmco 32201 hst0 32253 largei 32287 mdslmd1lem3 32347 mdslmd1lem4 32348 csmdsymi 32354 elat2 32360 atcveq0 32368 hatomic 32380 atcv0eq 32399 atoml2i 32403 atordi 32404 atord 32408 atcvat2 32409 chirred 32415 | 
| Copyright terms: Public domain | W3C validator |