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Mirrors > Home > HSE Home > Th. List > h0elsh | Structured version Visualization version GIF version |
Description: The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
h0elsh | ⊢ 0ℋ ∈ Sℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | h0elch 28449 | . 2 ⊢ 0ℋ ∈ Cℋ | |
2 | 1 | chshii 28421 | 1 ⊢ 0ℋ ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 Sℋ csh 28122 0ℋc0h 28129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 ax-addf 10216 ax-mulf 10217 ax-hilex 28193 ax-hfvadd 28194 ax-hvcom 28195 ax-hvass 28196 ax-hv0cl 28197 ax-hvaddid 28198 ax-hfvmul 28199 ax-hvmulid 28200 ax-hvmulass 28201 ax-hvdistr1 28202 ax-hvdistr2 28203 ax-hvmul0 28204 ax-hfi 28273 ax-his1 28276 ax-his2 28277 ax-his3 28278 ax-his4 28279 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-map 8010 df-pm 8011 df-en 8109 df-dom 8110 df-sdom 8111 df-sup 8503 df-inf 8504 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-n0 11494 df-z 11579 df-uz 11888 df-q 11991 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-icc 12386 df-seq 13008 df-exp 13067 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-topgen 16311 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-top 20918 df-topon 20935 df-bases 20970 df-lm 21253 df-haus 21339 df-grpo 27684 df-gid 27685 df-ginv 27686 df-gdiv 27687 df-ablo 27736 df-vc 27751 df-nv 27784 df-va 27787 df-ba 27788 df-sm 27789 df-0v 27790 df-vs 27791 df-nmcv 27792 df-ims 27793 df-hnorm 28162 df-hvsub 28165 df-hlim 28166 df-sh 28401 df-ch 28415 df-ch0 28447 |
This theorem is referenced by: hhssnvt 28459 choc0 28522 choc1 28523 shintcl 28526 omlsi 28600 pjoml 28632 shs0i 28645 shs00i 28646 spansn0 28737 span0 28738 |
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