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| Mirrors > Home > MPE Home > Th. List > cphssphl | Structured version Visualization version GIF version | ||
| Description: A Banach subspace of a subcomplex pre-Hilbert space is a subcomplex Hilbert space. (Contributed by NM, 11-Apr-2008.) (Revised by AV, 25-Sep-2022.) |
| Ref | Expression |
|---|---|
| cphssphl.x | β’ π = (π βΎs π) |
| cphssphl.s | β’ π = (LSubSpβπ) |
| Ref | Expression |
|---|---|
| cphssphl | β’ ((π β βPreHil β§ π β π β§ π β Ban) β π β βHil) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp3 1137 | . 2 β’ ((π β βPreHil β§ π β π β§ π β Ban) β π β Ban) | |
| 2 | cphssphl.x | . . . 4 β’ π = (π βΎs π) | |
| 3 | cphssphl.s | . . . 4 β’ π = (LSubSpβπ) | |
| 4 | 2, 3 | cphsscph 25000 | . . 3 β’ ((π β βPreHil β§ π β π) β π β βPreHil) |
| 5 | 4 | 3adant3 1131 | . 2 β’ ((π β βPreHil β§ π β π β§ π β Ban) β π β βPreHil) |
| 6 | ishl 25111 | . 2 β’ (π β βHil β (π β Ban β§ π β βPreHil)) | |
| 7 | 1, 5, 6 | sylanbrc 582 | 1 β’ ((π β βPreHil β§ π β π β§ π β Ban) β π β βHil) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1086 = wceq 1540 β wcel 2105 βcfv 6543 (class class class)co 7412 βΎs cress 17178 LSubSpclss 20687 βPreHilccph 24915 Bancbn 25082 βHilchl 25083 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9440 df-inf 9441 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ico 13335 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ds 17224 df-rest 17373 df-topn 17374 df-0g 17392 df-topgen 17394 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-ghm 19129 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-subrg 20460 df-lmod 20617 df-lss 20688 df-lsp 20728 df-lmhm 20778 df-lvec 20859 df-sra 20931 df-rgmod 20932 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-phl 21399 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-xms 24047 df-ms 24048 df-nm 24312 df-ngp 24313 df-nlm 24316 df-cph 24917 df-hl 25086 |
| This theorem is referenced by: csschl 25125 |
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