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Mirrors > Home > MPE Home > Th. List > hlhil | Structured version Visualization version GIF version |
Description: Corollary of the Projection Theorem: A subcomplex Hilbert space is a Hilbert space (in the algebraic sense, meaning that all algebraically closed subspaces have a projection decomposition). (Contributed by Mario Carneiro, 17-Oct-2015.) |
Ref | Expression |
---|---|
hlhil | ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Hil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlphl 25286 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) | |
2 | eqid 2727 | . . . . 5 ⊢ (proj‘𝑊) = (proj‘𝑊) | |
3 | eqid 2727 | . . . . 5 ⊢ (ClSubSp‘𝑊) = (ClSubSp‘𝑊) | |
4 | 2, 3 | pjcss 21643 | . . . 4 ⊢ (𝑊 ∈ PreHil → dom (proj‘𝑊) ⊆ (ClSubSp‘𝑊)) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝑊 ∈ ℂHil → dom (proj‘𝑊) ⊆ (ClSubSp‘𝑊)) |
6 | eqid 2727 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | eqid 2727 | . . . . 5 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊) | |
8 | eqid 2727 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
9 | 6, 7, 8, 3 | cldcss2 25363 | . . . 4 ⊢ (𝑊 ∈ ℂHil → (ClSubSp‘𝑊) = ((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) |
10 | elin 3960 | . . . . . 6 ⊢ (𝑥 ∈ ((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑥 ∈ (Clsd‘(TopOpen‘𝑊)))) | |
11 | 7, 8, 2 | pjth2 25361 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑥 ∈ (Clsd‘(TopOpen‘𝑊))) → 𝑥 ∈ dom (proj‘𝑊)) |
12 | 11 | 3expib 1120 | . . . . . 6 ⊢ (𝑊 ∈ ℂHil → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑥 ∈ (Clsd‘(TopOpen‘𝑊))) → 𝑥 ∈ dom (proj‘𝑊))) |
13 | 10, 12 | biimtrid 241 | . . . . 5 ⊢ (𝑊 ∈ ℂHil → (𝑥 ∈ ((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) → 𝑥 ∈ dom (proj‘𝑊))) |
14 | 13 | ssrdv 3984 | . . . 4 ⊢ (𝑊 ∈ ℂHil → ((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ⊆ dom (proj‘𝑊)) |
15 | 9, 14 | eqsstrd 4016 | . . 3 ⊢ (𝑊 ∈ ℂHil → (ClSubSp‘𝑊) ⊆ dom (proj‘𝑊)) |
16 | 5, 15 | eqssd 3995 | . 2 ⊢ (𝑊 ∈ ℂHil → dom (proj‘𝑊) = (ClSubSp‘𝑊)) |
17 | 2, 3 | ishil 21645 | . 2 ⊢ (𝑊 ∈ Hil ↔ (𝑊 ∈ PreHil ∧ dom (proj‘𝑊) = (ClSubSp‘𝑊))) |
18 | 1, 16, 17 | sylanbrc 582 | 1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Hil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3943 ⊆ wss 3944 dom cdm 5672 ‘cfv 6542 Basecbs 17173 TopOpenctopn 17396 LSubSpclss 20808 PreHilcphl 21549 ClSubSpccss 21586 projcpj 21627 Hilchil 21628 Clsdccld 22913 ℂHilchl 25255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 ax-mulf 11212 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-ghm 19161 df-cntz 19261 df-lsm 19584 df-pj1 19585 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-rhm 20404 df-subrng 20476 df-subrg 20501 df-drng 20619 df-staf 20718 df-srng 20719 df-lmod 20738 df-lss 20809 df-lmhm 20900 df-lvec 20981 df-sra 21051 df-rgmod 21052 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-fbas 21269 df-fg 21270 df-cnfld 21273 df-phl 21551 df-ipf 21552 df-ocv 21588 df-css 21589 df-pj 21630 df-hil 21631 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-nei 22995 df-cn 23124 df-cnp 23125 df-t1 23211 df-haus 23212 df-cmp 23284 df-tx 23459 df-hmeo 23652 df-fil 23743 df-flim 23836 df-fcls 23838 df-xms 24219 df-ms 24220 df-tms 24221 df-nm 24484 df-ngp 24485 df-tng 24486 df-nlm 24488 df-cncf 24791 df-clm 24983 df-cph 25089 df-tcph 25090 df-cfil 25176 df-cmet 25178 df-cms 25256 df-bn 25257 df-hl 25258 |
This theorem is referenced by: (None) |
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