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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 23934. (Contributed by AV, 8-Oct-2022.) |
Ref | Expression |
---|---|
cmslssbn.x | ⊢ 𝑋 = (𝑊 ↾s 𝑈) |
cmslssbn.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
cmslssbn | ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ Ban) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmslssbn.x | . . . 4 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
2 | cmslssbn.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssnvc 23306 | . . 3 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmVec) |
4 | 3 | ad2ant2rl 747 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ NrmVec) |
5 | simprl 769 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ CMetSp) | |
6 | eqid 2820 | . . . . . . . 8 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
7 | 1, 6 | resssca 16645 | . . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋)) |
8 | 7 | ad2antll 727 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → (Scalar‘𝑊) = (Scalar‘𝑋)) |
9 | 8 | eleq1d 2896 | . . . . 5 ⊢ ((𝑊 ∈ NrmVec ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → ((Scalar‘𝑊) ∈ CMetSp ↔ (Scalar‘𝑋) ∈ CMetSp)) |
10 | 9 | biimpd 231 | . . . 4 ⊢ ((𝑊 ∈ NrmVec ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → ((Scalar‘𝑊) ∈ CMetSp → (Scalar‘𝑋) ∈ CMetSp)) |
11 | 10 | impancom 454 | . . 3 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) → ((𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) ∈ CMetSp)) |
12 | 11 | imp 409 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → (Scalar‘𝑋) ∈ CMetSp) |
13 | eqid 2820 | . . 3 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋) | |
14 | 13 | isbn 23936 | . 2 ⊢ (𝑋 ∈ Ban ↔ (𝑋 ∈ NrmVec ∧ 𝑋 ∈ CMetSp ∧ (Scalar‘𝑋) ∈ CMetSp)) |
15 | 4, 5, 12, 14 | syl3anbrc 1338 | 1 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ Ban) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 ↾s cress 16479 Scalarcsca 16563 LSubSpclss 19698 NrmVeccnvc 23186 CMetSpccms 23930 Bancbn 23931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-sca 16576 df-vsca 16577 df-tset 16579 df-ds 16582 df-rest 16691 df-topn 16692 df-0g 16710 df-topgen 16712 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-mgp 19235 df-ur 19247 df-ring 19294 df-lmod 19631 df-lss 19699 df-lvec 19870 df-psmet 20532 df-xmet 20533 df-met 20534 df-bl 20535 df-mopn 20536 df-top 21497 df-topon 21514 df-topsp 21536 df-bases 21549 df-xms 22925 df-ms 22926 df-nm 23187 df-ngp 23188 df-nlm 23191 df-nvc 23192 df-bn 23934 |
This theorem is referenced by: bncssbn 23972 cssbn 23973 cmslsschl 23975 |
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