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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 24703. (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 24069 | . . 3 β’ ((π β NrmVec β§ π β π) β π β NrmVec) |
4 | 3 | ad2ant2rl 748 | . 2 β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ (π β CMetSp β§ π β π)) β π β NrmVec) |
5 | simprl 770 | . 2 β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ (π β CMetSp β§ π β π)) β π β CMetSp) | |
6 | eqid 2737 | . . . . . . . 8 β’ (Scalarβπ) = (Scalarβπ) | |
7 | 1, 6 | resssca 17225 | . . . . . . 7 β’ (π β π β (Scalarβπ) = (Scalarβπ)) |
8 | 7 | ad2antll 728 | . . . . . 6 β’ ((π β NrmVec β§ (π β CMetSp β§ π β π)) β (Scalarβπ) = (Scalarβπ)) |
9 | 8 | eleq1d 2823 | . . . . 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 453 | . . 3 β’ ((π β NrmVec β§ (Scalarβπ) β CMetSp) β ((π β CMetSp β§ π β π) β (Scalarβπ) β CMetSp)) |
12 | 11 | imp 408 | . 2 β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ (π β CMetSp β§ π β π)) β (Scalarβπ) β CMetSp) |
13 | eqid 2737 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
14 | 13 | isbn 24705 | . 2 β’ (π β Ban β (π β NrmVec β§ π β CMetSp β§ (Scalarβπ) β CMetSp)) |
15 | 4, 5, 12, 14 | syl3anbrc 1344 | 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 397 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 βΎs cress 17113 Scalarcsca 17137 LSubSpclss 20395 NrmVeccnvc 23940 CMetSpccms 24699 Bancbn 24700 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-q 12875 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-sca 17150 df-vsca 17151 df-tset 17153 df-ds 17156 df-rest 17305 df-topn 17306 df-0g 17324 df-topgen 17326 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-minusg 18753 df-sbg 18754 df-subg 18926 df-mgp 19898 df-ur 19915 df-ring 19967 df-lmod 20327 df-lss 20396 df-lvec 20567 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 df-mopn 20795 df-top 22246 df-topon 22263 df-topsp 22285 df-bases 22299 df-xms 23676 df-ms 23677 df-nm 23941 df-ngp 23942 df-nlm 23945 df-nvc 23946 df-bn 24703 |
This theorem is referenced by: bncssbn 24741 cssbn 24742 cmslsschl 24744 |
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