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Mirrors > Home > MPE Home > Th. List > chlcsschl | Structured version Visualization version GIF version |
Description: A closed subspace of a subcomplex Hilbert space is a subcomplex Hilbert space. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 8-Oct-2022.) |
Ref | Expression |
---|---|
cmslsschl.x | ⊢ 𝑋 = (𝑊 ↾s 𝑈) |
chlcsschl.s | ⊢ 𝑆 = (ClSubSp‘𝑊) |
Ref | Expression |
---|---|
chlcsschl | ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂHil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlbn 25352 | . . . 4 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) | |
2 | hlcph 25353 | . . . 4 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) | |
3 | 1, 2 | jca 510 | . . 3 ⊢ (𝑊 ∈ ℂHil → (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) |
4 | cmslsschl.x | . . . 4 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
5 | chlcsschl.s | . . . 4 ⊢ 𝑆 = (ClSubSp‘𝑊) | |
6 | 4, 5 | bncssbn 25363 | . . 3 ⊢ (((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ Ban) |
7 | 3, 6 | sylan 578 | . 2 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ Ban) |
8 | hlphl 25354 | . . . 4 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) | |
9 | eqid 2725 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
10 | 5, 9 | csslss 21657 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (LSubSp‘𝑊)) |
11 | 8, 10 | sylan 578 | . . 3 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (LSubSp‘𝑊)) |
12 | 4, 9 | cphsscph 25240 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑋 ∈ ℂPreHil) |
13 | 2, 11, 12 | syl2an2r 683 | . 2 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂPreHil) |
14 | ishl 25351 | . 2 ⊢ (𝑋 ∈ ℂHil ↔ (𝑋 ∈ Ban ∧ 𝑋 ∈ ℂPreHil)) | |
15 | 7, 13, 14 | sylanbrc 581 | 1 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 ↾s cress 17228 LSubSpclss 20844 PreHilcphl 21590 ClSubSpccss 21627 ℂPreHilccph 25155 Bancbn 25322 ℂHilchl 25323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 ax-mulf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 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 9393 df-fi 9441 df-sup 9472 df-inf 9473 df-oi 9540 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-seq 14008 df-exp 14068 df-hash 14334 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-starv 17267 df-sca 17268 df-vsca 17269 df-ip 17270 df-tset 17271 df-ple 17272 df-ds 17274 df-unif 17275 df-hom 17276 df-cco 17277 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17503 df-qtop 17508 df-imas 17509 df-xps 17511 df-mre 17585 df-mrc 17586 df-acs 17588 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-mhm 18759 df-submnd 18760 df-grp 18917 df-minusg 18918 df-sbg 18919 df-mulg 19048 df-subg 19103 df-ghm 19193 df-cntz 19297 df-cmn 19766 df-abl 19767 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 df-cring 20205 df-oppr 20302 df-dvdsr 20325 df-unit 20326 df-invr 20356 df-dvr 20369 df-rhm 20440 df-subrng 20512 df-subrg 20537 df-drng 20655 df-staf 20754 df-srng 20755 df-lmod 20774 df-lss 20845 df-lsp 20885 df-lmhm 20936 df-lvec 21017 df-sra 21087 df-rgmod 21088 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-phl 21592 df-ipf 21593 df-ocv 21629 df-css 21630 df-top 22857 df-topon 22874 df-topsp 22896 df-bases 22910 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-cn 23192 df-cnp 23193 df-t1 23279 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-flim 23904 df-xms 24287 df-ms 24288 df-tms 24289 df-nm 24552 df-ngp 24553 df-tng 24554 df-nlm 24556 df-nvc 24557 df-clm 25051 df-cph 25157 df-tcph 25158 df-cfil 25244 df-cmet 25246 df-cms 25324 df-bn 25325 df-hl 25326 |
This theorem is referenced by: (None) |
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