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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 29626 | . 2 ⊢ 0ℋ ∈ Cℋ | |
2 | 1 | chshii 29598 | 1 ⊢ 0ℋ ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Sℋ csh 29299 0ℋc0h 29306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 ax-addf 10961 ax-mulf 10962 ax-hilex 29370 ax-hfvadd 29371 ax-hvcom 29372 ax-hvass 29373 ax-hv0cl 29374 ax-hvaddid 29375 ax-hfvmul 29376 ax-hvmulid 29377 ax-hvmulass 29378 ax-hvdistr1 29379 ax-hvdistr2 29380 ax-hvmul0 29381 ax-hfi 29450 ax-his1 29453 ax-his2 29454 ax-his3 29455 ax-his4 29456 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-er 8490 df-map 8609 df-pm 8610 df-en 8726 df-dom 8727 df-sdom 8728 df-sup 9189 df-inf 9190 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-n0 12245 df-z 12331 df-uz 12594 df-q 12700 df-rp 12742 df-xneg 12859 df-xadd 12860 df-xmul 12861 df-icc 13097 df-seq 13733 df-exp 13794 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-topgen 17165 df-psmet 20600 df-xmet 20601 df-met 20602 df-bl 20603 df-mopn 20604 df-top 22054 df-topon 22071 df-bases 22107 df-lm 22391 df-haus 22477 df-grpo 28864 df-gid 28865 df-ginv 28866 df-gdiv 28867 df-ablo 28916 df-vc 28930 df-nv 28963 df-va 28966 df-ba 28967 df-sm 28968 df-0v 28969 df-vs 28970 df-nmcv 28971 df-ims 28972 df-hnorm 29339 df-hvsub 29342 df-hlim 29343 df-sh 29578 df-ch 29592 df-ch0 29624 |
This theorem is referenced by: hhssnvt 29636 choc0 29697 choc1 29698 shintcl 29701 omlsi 29775 pjoml 29807 shs0i 29820 shs00i 29821 spansn0 29912 span0 29913 |
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