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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 25306 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) | |
2 | eqid 2728 | . . . . 5 ⊢ (proj‘𝑊) = (proj‘𝑊) | |
3 | eqid 2728 | . . . . 5 ⊢ (ClSubSp‘𝑊) = (ClSubSp‘𝑊) | |
4 | 2, 3 | pjcss 21650 | . . . 4 ⊢ (𝑊 ∈ PreHil → dom (proj‘𝑊) ⊆ (ClSubSp‘𝑊)) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝑊 ∈ ℂHil → dom (proj‘𝑊) ⊆ (ClSubSp‘𝑊)) |
6 | eqid 2728 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | eqid 2728 | . . . . 5 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊) | |
8 | eqid 2728 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
9 | 6, 7, 8, 3 | cldcss2 25383 | . . . 4 ⊢ (𝑊 ∈ ℂHil → (ClSubSp‘𝑊) = ((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) |
10 | elin 3963 | . . . . . 6 ⊢ (𝑥 ∈ ((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑥 ∈ (Clsd‘(TopOpen‘𝑊)))) | |
11 | 7, 8, 2 | pjth2 25381 | . . . . . . 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 3986 | . . . 4 ⊢ (𝑊 ∈ ℂHil → ((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ⊆ dom (proj‘𝑊)) |
15 | 9, 14 | eqsstrd 4018 | . . 3 ⊢ (𝑊 ∈ ℂHil → (ClSubSp‘𝑊) ⊆ dom (proj‘𝑊)) |
16 | 5, 15 | eqssd 3997 | . 2 ⊢ (𝑊 ∈ ℂHil → dom (proj‘𝑊) = (ClSubSp‘𝑊)) |
17 | 2, 3 | ishil 21652 | . 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 3946 ⊆ wss 3947 dom cdm 5678 ‘cfv 6548 Basecbs 17180 TopOpenctopn 17403 LSubSpclss 20815 PreHilcphl 21556 ClSubSpccss 21593 projcpj 21634 Hilchil 21635 Clsdccld 22933 ℂHilchl 25275 |
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 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-lsm 19591 df-pj1 19592 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-rhm 20411 df-subrng 20483 df-subrg 20508 df-drng 20626 df-staf 20725 df-srng 20726 df-lmod 20745 df-lss 20816 df-lmhm 20907 df-lvec 20988 df-sra 21058 df-rgmod 21059 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-phl 21558 df-ipf 21559 df-ocv 21595 df-css 21596 df-pj 21637 df-hil 21638 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-cn 23144 df-cnp 23145 df-t1 23231 df-haus 23232 df-cmp 23304 df-tx 23479 df-hmeo 23672 df-fil 23763 df-flim 23856 df-fcls 23858 df-xms 24239 df-ms 24240 df-tms 24241 df-nm 24504 df-ngp 24505 df-tng 24506 df-nlm 24508 df-cncf 24811 df-clm 25003 df-cph 25109 df-tcph 25110 df-cfil 25196 df-cmet 25198 df-cms 25276 df-bn 25277 df-hl 25278 |
This theorem is referenced by: (None) |
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