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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 23297 | . . 3 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
2 | nlmtlm 23295 | . . 3 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ TopMod) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ TopMod) |
4 | eqid 2819 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | 4 | nlmnrg 23280 | . . . 4 ⊢ (𝑊 ∈ NrmMod → (Scalar‘𝑊) ∈ NrmRing) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝑊 ∈ NrmVec → (Scalar‘𝑊) ∈ NrmRing) |
7 | nvclvec 23298 | . . . 4 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec) | |
8 | 4 | lvecdrng 19869 | . . . 4 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing) |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝑊 ∈ NrmVec → (Scalar‘𝑊) ∈ DivRing) |
10 | nrgtdrg 23294 | . . 3 ⊢ (((Scalar‘𝑊) ∈ NrmRing ∧ (Scalar‘𝑊) ∈ DivRing) → (Scalar‘𝑊) ∈ TopDRing) | |
11 | 6, 9, 10 | syl2anc 586 | . 2 ⊢ (𝑊 ∈ NrmVec → (Scalar‘𝑊) ∈ TopDRing) |
12 | 4 | istvc 22792 | . 2 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ (Scalar‘𝑊) ∈ TopDRing)) |
13 | 3, 11, 12 | sylanbrc 585 | 1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ TopVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 ‘cfv 6348 Scalarcsca 16560 DivRingcdr 19494 LVecclvec 19866 TopDRingctdrg 22757 TopModctlm 22758 TopVecctvc 22759 NrmRingcnrg 23181 NrmModcnlm 23182 NrmVeccnvc 23183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 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-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-tpos 7884 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 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 19232 df-ur 19244 df-ring 19291 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-subrg 19525 df-abv 19580 df-lmod 19628 df-scaf 19629 df-lvec 19867 df-sra 19936 df-rgmod 19937 df-nzr 20023 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cn 21827 df-cnp 21828 df-tx 22162 df-hmeo 22355 df-tmd 22672 df-tgp 22673 df-trg 22760 df-tdrg 22761 df-tlm 22762 df-tvc 22763 df-xms 22922 df-ms 22923 df-tms 22924 df-nm 23184 df-ngp 23185 df-nrg 23187 df-nlm 23188 df-nvc 23189 |
This theorem is referenced by: (None) |
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