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Mirrors > Home > HSE Home > Th. List > pjhclii | Structured version Visualization version GIF version |
Description: Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjcli.1 | ⊢ 𝐻 ∈ Cℋ |
pjcli.2 | ⊢ 𝐴 ∈ ℋ |
Ref | Expression |
---|---|
pjhclii | ⊢ ((projℎ‘𝐻)‘𝐴) ∈ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjcli.2 | . 2 ⊢ 𝐴 ∈ ℋ | |
2 | pjcli.1 | . . 3 ⊢ 𝐻 ∈ Cℋ | |
3 | 2 | pjhcli 28853 | . 2 ⊢ (𝐴 ∈ ℋ → ((projℎ‘𝐻)‘𝐴) ∈ ℋ) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ((projℎ‘𝐻)‘𝐴) ∈ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ‘cfv 6137 ℋchba 28352 Cℋ cch 28362 projℎcpjh 28370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cc 9594 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 ax-hilex 28432 ax-hfvadd 28433 ax-hvcom 28434 ax-hvass 28435 ax-hv0cl 28436 ax-hvaddid 28437 ax-hfvmul 28438 ax-hvmulid 28439 ax-hvmulass 28440 ax-hvdistr1 28441 ax-hvdistr2 28442 ax-hvmul0 28443 ax-hfi 28512 ax-his1 28515 ax-his2 28516 ax-his3 28517 ax-his4 28518 ax-hcompl 28635 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-omul 7850 df-er 8028 df-map 8144 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-acn 9103 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-n0 11647 df-z 11733 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-ico 12497 df-icc 12498 df-fz 12648 df-fl 12916 df-seq 13124 df-exp 13183 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-clim 14631 df-rlim 14632 df-rest 16473 df-topgen 16494 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-fbas 20143 df-fg 20144 df-top 21110 df-topon 21127 df-bases 21162 df-cld 21235 df-ntr 21236 df-cls 21237 df-nei 21314 df-lm 21445 df-haus 21531 df-fil 22062 df-fm 22154 df-flim 22155 df-flf 22156 df-cfil 23465 df-cau 23466 df-cmet 23467 df-grpo 27924 df-gid 27925 df-ginv 27926 df-gdiv 27927 df-ablo 27976 df-vc 27990 df-nv 28023 df-va 28026 df-ba 28027 df-sm 28028 df-0v 28029 df-vs 28030 df-nmcv 28031 df-ims 28032 df-ssp 28153 df-ph 28244 df-cbn 28295 df-hnorm 28401 df-hba 28402 df-hvsub 28404 df-hlim 28405 df-hcau 28406 df-sh 28640 df-ch 28654 df-oc 28685 df-ch0 28686 df-shs 28743 df-pjh 28830 |
This theorem is referenced by: pjoc1i 28866 pjchi 28867 spansnpji 29013 spanunsni 29014 spansnji 29081 pjidmi 29108 pjadjii 29109 pjaddii 29110 pjinormii 29111 pjmulii 29112 pjsubii 29113 pjsslem 29114 pjss2i 29115 pjssmii 29116 pjssge0ii 29117 pjdifnormii 29118 pjcji 29119 pjopythi 29154 pjnormi 29156 pjneli 29158 |
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