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| Mirrors > Home > MPE Home > Th. List > cmslssbn | Structured version Visualization version GIF version | ||
| Description: A complete linear subspace of a normed vector space is a Banach space. We furthermore have to assume that the field of scalars is complete since this is a requirement in the current definition of Banach spaces df-bn 25219. (Contributed by AV, 8-Oct-2022.) |
| Ref | Expression |
|---|---|
| cmslssbn.x | β’ π = (π βΎs π) |
| cmslssbn.s | β’ π = (LSubSpβπ) |
| Ref | Expression |
|---|---|
| cmslssbn | β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ (π β CMetSp β§ π β π)) β π β Ban) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cmslssbn.x | . . . 4 β’ π = (π βΎs π) | |
| 2 | cmslssbn.s | . . . 4 β’ π = (LSubSpβπ) | |
| 3 | 1, 2 | lssnvc 24574 | . . 3 β’ ((π β NrmVec β§ π β π) β π β NrmVec) |
| 4 | 3 | ad2ant2rl 746 | . 2 β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ (π β CMetSp β§ π β π)) β π β NrmVec) |
| 5 | simprl 768 | . 2 β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ (π β CMetSp β§ π β π)) β π β CMetSp) | |
| 6 | eqid 2726 | . . . . . . . 8 β’ (Scalarβπ) = (Scalarβπ) | |
| 7 | 1, 6 | resssca 17297 | . . . . . . 7 β’ (π β π β (Scalarβπ) = (Scalarβπ)) |
| 8 | 7 | ad2antll 726 | . . . . . 6 β’ ((π β NrmVec β§ (π β CMetSp β§ π β π)) β (Scalarβπ) = (Scalarβπ)) |
| 9 | 8 | eleq1d 2812 | . . . . 5 β’ ((π β NrmVec β§ (π β CMetSp β§ π β π)) β ((Scalarβπ) β CMetSp β (Scalarβπ) β CMetSp)) |
| 10 | 9 | biimpd 228 | . . . 4 β’ ((π β NrmVec β§ (π β CMetSp β§ π β π)) β ((Scalarβπ) β CMetSp β (Scalarβπ) β CMetSp)) |
| 11 | 10 | impancom 451 | . . 3 β’ ((π β NrmVec β§ (Scalarβπ) β CMetSp) β ((π β CMetSp β§ π β π) β (Scalarβπ) β CMetSp)) |
| 12 | 11 | imp 406 | . 2 β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ (π β CMetSp β§ π β π)) β (Scalarβπ) β CMetSp) |
| 13 | eqid 2726 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
| 14 | 13 | isbn 25221 | . 2 β’ (π β Ban β (π β NrmVec β§ π β CMetSp β§ (Scalarβπ) β CMetSp)) |
| 15 | 4, 5, 12, 14 | syl3anbrc 1340 | 1 β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ (π β CMetSp β§ π β π)) β π β Ban) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 βΎs cress 17182 Scalarcsca 17209 LSubSpclss 20778 NrmVeccnvc 24445 CMetSpccms 25215 Bancbn 25216 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-sca 17222 df-vsca 17223 df-tset 17225 df-ds 17228 df-rest 17377 df-topn 17378 df-0g 17396 df-topgen 17398 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-mgp 20040 df-ur 20087 df-ring 20140 df-lmod 20708 df-lss 20779 df-lvec 20951 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-xms 24181 df-ms 24182 df-nm 24446 df-ngp 24447 df-nlm 24450 df-nvc 24451 df-bn 25219 |
| This theorem is referenced by: bncssbn 25257 cssbn 25258 cmslsschl 25260 |
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