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Mirrors > Home > MPE Home > Th. List > nvctvc | Structured version Visualization version GIF version |
Description: A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nvctvc | ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ TopVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvcnlm 23302 | . . 3 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
2 | nlmtlm 23300 | . . 3 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ TopMod) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ TopMod) |
4 | eqid 2798 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | 4 | nlmnrg 23285 | . . . 4 ⊢ (𝑊 ∈ NrmMod → (Scalar‘𝑊) ∈ NrmRing) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝑊 ∈ NrmVec → (Scalar‘𝑊) ∈ NrmRing) |
7 | nvclvec 23303 | . . . 4 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec) | |
8 | 4 | lvecdrng 19870 | . . . 4 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing) |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝑊 ∈ NrmVec → (Scalar‘𝑊) ∈ DivRing) |
10 | nrgtdrg 23299 | . . 3 ⊢ (((Scalar‘𝑊) ∈ NrmRing ∧ (Scalar‘𝑊) ∈ DivRing) → (Scalar‘𝑊) ∈ TopDRing) | |
11 | 6, 9, 10 | syl2anc 587 | . 2 ⊢ (𝑊 ∈ NrmVec → (Scalar‘𝑊) ∈ TopDRing) |
12 | 4 | istvc 22797 | . 2 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ (Scalar‘𝑊) ∈ TopDRing)) |
13 | 3, 11, 12 | sylanbrc 586 | 1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ TopVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ‘cfv 6324 Scalarcsca 16560 DivRingcdr 19495 LVecclvec 19867 TopDRingctdrg 22762 TopModctlm 22763 TopVecctvc 22764 NrmRingcnrg 23186 NrmModcnlm 23187 NrmVeccnvc 23188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-plusf 17843 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-subrg 19526 df-abv 19581 df-lmod 19629 df-scaf 19630 df-lvec 19868 df-sra 19937 df-rgmod 19938 df-nzr 20024 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cn 21832 df-cnp 21833 df-tx 22167 df-hmeo 22360 df-tmd 22677 df-tgp 22678 df-trg 22765 df-tdrg 22766 df-tlm 22767 df-tvc 22768 df-xms 22927 df-ms 22928 df-tms 22929 df-nm 23189 df-ngp 23190 df-nrg 23192 df-nlm 23193 df-nvc 23194 |
This theorem is referenced by: (None) |
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