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Mirrors > Home > MPE Home > Th. List > bnsscmcl | Structured version Unicode version |
Description: A subspace of a Banach space is a Banach space iff it is closed in the norm-induced metric of the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnsscmcl.x |
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bnsscmcl.d |
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bnsscmcl.j |
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bnsscmcl.h |
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bnsscmcl.y |
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Ref | Expression |
---|---|
bnsscmcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnnv 24412 |
. . . 4
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2 | bnsscmcl.h |
. . . . 5
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3 | 2 | sspnv 24269 |
. . . 4
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4 | 1, 3 | sylan 471 |
. . 3
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5 | bnsscmcl.y |
. . . . 5
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6 | eqid 2451 |
. . . . 5
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7 | 5, 6 | iscbn 24410 |
. . . 4
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8 | 7 | baib 896 |
. . 3
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9 | 4, 8 | syl 16 |
. 2
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10 | bnsscmcl.d |
. . . . 5
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11 | 5, 10, 6, 2 | sspims 24284 |
. . . 4
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12 | 1, 11 | sylan 471 |
. . 3
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13 | 12 | eleq1d 2520 |
. 2
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14 | bnsscmcl.x |
. . . . 5
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15 | 14, 10 | cbncms 24411 |
. . . 4
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16 | 15 | adantr 465 |
. . 3
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17 | bnsscmcl.j |
. . . 4
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18 | 17 | cmetss 20950 |
. . 3
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19 | 16, 18 | syl 16 |
. 2
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20 | 9, 13, 19 | 3bitrd 279 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-cnex 9442 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-mulcom 9450 ax-addass 9451 ax-mulass 9452 ax-distr 9453 ax-i2m1 9454 ax-1ne0 9455 ax-1rid 9456 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 ax-pre-lttri 9460 ax-pre-lttrn 9461 ax-pre-ltadd 9462 ax-pre-mulgt0 9463 ax-pre-sup 9464 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-tp 3983 df-op 3985 df-uni 4193 df-int 4230 df-iun 4274 df-iin 4275 df-br 4394 df-opab 4452 df-mpt 4453 df-tr 4487 df-eprel 4733 df-id 4737 df-po 4742 df-so 4743 df-fr 4780 df-we 4782 df-ord 4823 df-on 4824 df-lim 4825 df-suc 4826 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-om 6580 df-1st 6680 df-2nd 6681 df-recs 6935 df-rdg 6969 df-1o 7023 df-oadd 7027 df-er 7204 df-map 7319 df-en 7414 df-dom 7415 df-sdom 7416 df-fin 7417 df-fi 7765 df-sup 7795 df-pnf 9524 df-mnf 9525 df-xr 9526 df-ltxr 9527 df-le 9528 df-sub 9701 df-neg 9702 df-div 10098 df-nn 10427 df-2 10484 df-n0 10684 df-z 10751 df-uz 10966 df-q 11058 df-rp 11096 df-xneg 11193 df-xadd 11194 df-xmul 11195 df-ico 11410 df-icc 11411 df-rest 14472 df-topgen 14493 df-psmet 17927 df-xmet 17928 df-met 17929 df-bl 17930 df-mopn 17931 df-fbas 17932 df-fg 17933 df-top 18628 df-bases 18630 df-topon 18631 df-cld 18748 df-ntr 18749 df-cls 18750 df-nei 18827 df-haus 19044 df-fil 19544 df-flim 19637 df-cfil 20891 df-cmet 20893 df-grpo 23823 df-gid 23824 df-ginv 23825 df-gdiv 23826 df-ablo 23914 df-vc 24069 df-nv 24115 df-va 24118 df-ba 24119 df-sm 24120 df-0v 24121 df-vs 24122 df-nmcv 24123 df-ims 24124 df-ssp 24265 df-cbn 24409 |
This theorem is referenced by: (None) |
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