Nearby theorems
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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 | β’ π = (π βΎs π) |
| chlcsschl.s | β’ π = (ClSubSpβπ) |
| Ref | Expression |
|---|---|
| chlcsschl | β’ ((π β βHil β§ π β π) β π β βHil) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlbn 25307 | . . . 4 β’ (π β βHil β π β Ban) | |
| 2 | hlcph 25308 | . . . 4 β’ (π β βHil β π β βPreHil) | |
| 3 | 1, 2 | jca 510 | . . 3 β’ (π β βHil β (π β Ban β§ π β βPreHil)) |
| 4 | cmslsschl.x | . . . 4 β’ π = (π βΎs π) | |
| 5 | chlcsschl.s | . . . 4 β’ π = (ClSubSpβπ) | |
| 6 | 4, 5 | bncssbn 25318 | . . 3 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β Ban) |
| 7 | 3, 6 | sylan 578 | . 2 β’ ((π β βHil β§ π β π) β π β Ban) |
| 8 | hlphl 25309 | . . . 4 β’ (π β βHil β π β PreHil) | |
| 9 | eqid 2725 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 10 | 5, 9 | csslss 21625 | . . . 4 β’ ((π β PreHil β§ π β π) β π β (LSubSpβπ)) |
| 11 | 8, 10 | sylan 578 | . . 3 β’ ((π β βHil β§ π β π) β π β (LSubSpβπ)) |
| 12 | 4, 9 | cphsscph 25195 | . . 3 β’ ((π β βPreHil β§ π β (LSubSpβπ)) β π β βPreHil) |
| 13 | 2, 11, 12 | syl2an2r 683 | . 2 β’ ((π β βHil β§ π β π) β π β βPreHil) |
| 14 | ishl 25306 | . 2 β’ (π β βHil β (π β Ban β§ π β βPreHil)) | |
| 15 | 7, 13, 14 | sylanbrc 581 | 1 β’ ((π β βHil β§ π β π) β π β βHil) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6541 (class class class)co 7414 βΎs cress 17206 LSubSpclss 20817 PreHilcphl 21558 ClSubSpccss 21595 βPreHilccph 25110 Bancbn 25277 βHilchl 25278 |
| 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 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215 ax-mulf 11216 |
| 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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-rest 17401 df-topn 17402 df-0g 17420 df-gsum 17421 df-topgen 17422 df-pt 17423 df-prds 17426 df-xrs 17481 df-qtop 17486 df-imas 17487 df-xps 17489 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-cring 20178 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-drng 20628 df-staf 20727 df-srng 20728 df-lmod 20747 df-lss 20818 df-lsp 20858 df-lmhm 20909 df-lvec 20990 df-sra 21060 df-rgmod 21061 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-fbas 21278 df-fg 21279 df-cnfld 21282 df-phl 21560 df-ipf 21561 df-ocv 21597 df-css 21598 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-cn 23147 df-cnp 23148 df-t1 23234 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-flim 23859 df-xms 24242 df-ms 24243 df-tms 24244 df-nm 24507 df-ngp 24508 df-tng 24509 df-nlm 24511 df-nvc 24512 df-clm 25006 df-cph 25112 df-tcph 25113 df-cfil 25199 df-cmet 25201 df-cms 25279 df-bn 25280 df-hl 25281 |
| This theorem is referenced by: (None) |
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