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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 24880 | . . . 4 β’ (π β βHil β π β Ban) | |
| 2 | hlcph 24881 | . . . 4 β’ (π β βHil β π β βPreHil) | |
| 3 | 1, 2 | jca 513 | . . 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 24891 | . . 3 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β Ban) |
| 7 | 3, 6 | sylan 581 | . 2 β’ ((π β βHil β§ π β π) β π β Ban) |
| 8 | hlphl 24882 | . . . 4 β’ (π β βHil β π β PreHil) | |
| 9 | eqid 2733 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 10 | 5, 9 | csslss 21244 | . . . 4 β’ ((π β PreHil β§ π β π) β π β (LSubSpβπ)) |
| 11 | 8, 10 | sylan 581 | . . 3 β’ ((π β βHil β§ π β π) β π β (LSubSpβπ)) |
| 12 | 4, 9 | cphsscph 24768 | . . 3 β’ ((π β βPreHil β§ π β (LSubSpβπ)) β π β βPreHil) |
| 13 | 2, 11, 12 | syl2an2r 684 | . 2 β’ ((π β βHil β§ π β π) β π β βPreHil) |
| 14 | ishl 24879 | . 2 β’ (π β βHil β (π β Ban β§ π β βPreHil)) | |
| 15 | 7, 13, 14 | sylanbrc 584 | 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 397 = wceq 1542 β wcel 2107 βcfv 6544 (class class class)co 7409 βΎs cress 17173 LSubSpclss 20542 PreHilcphl 21177 ClSubSpccss 21214 βPreHilccph 24683 Bancbn 24850 βHilchl 24851 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-ghm 19090 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-rnghom 20251 df-subrg 20317 df-drng 20359 df-staf 20453 df-srng 20454 df-lmod 20473 df-lss 20543 df-lsp 20583 df-lmhm 20633 df-lvec 20714 df-sra 20785 df-rgmod 20786 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-phl 21179 df-ipf 21180 df-ocv 21216 df-css 21217 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-cn 22731 df-cnp 22732 df-t1 22818 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-flim 23443 df-xms 23826 df-ms 23827 df-tms 23828 df-nm 24091 df-ngp 24092 df-tng 24093 df-nlm 24095 df-nvc 24096 df-clm 24579 df-cph 24685 df-tcph 24686 df-cfil 24772 df-cmet 24774 df-cms 24852 df-bn 24853 df-hl 24854 |
| This theorem is referenced by: (None) |
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