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Mirrors > Home > MPE Home > Th. List > isncvsngpd | Structured version Visualization version GIF version |
Description: Properties that determine a normed subcomplex vector space. (Contributed by NM, 15-Apr-2007.) (Revised by AV, 7-Oct-2021.) |
Ref | Expression |
---|---|
isncvsngp.v | ⊢ 𝑉 = (Base‘𝑊) |
isncvsngp.n | ⊢ 𝑁 = (norm‘𝑊) |
isncvsngp.s | ⊢ · = ( ·𝑠 ‘𝑊) |
isncvsngp.f | ⊢ 𝐹 = (Scalar‘𝑊) |
isncvsngp.k | ⊢ 𝐾 = (Base‘𝐹) |
isncvsngpd.v | ⊢ (𝜑 → 𝑊 ∈ ℂVec) |
isncvsngpd.g | ⊢ (𝜑 → 𝑊 ∈ NrmGrp) |
isncvsngpd.t | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑘 ∈ 𝐾)) → (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) |
Ref | Expression |
---|---|
isncvsngpd | ⊢ (𝜑 → 𝑊 ∈ (NrmVec ∩ ℂVec)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isncvsngpd.v | . 2 ⊢ (𝜑 → 𝑊 ∈ ℂVec) | |
2 | isncvsngpd.g | . 2 ⊢ (𝜑 → 𝑊 ∈ NrmGrp) | |
3 | isncvsngpd.t | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑘 ∈ 𝐾)) → (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) | |
4 | 3 | ralrimivva 3105 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) |
5 | isncvsngp.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
6 | isncvsngp.n | . . 3 ⊢ 𝑁 = (norm‘𝑊) | |
7 | isncvsngp.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
8 | isncvsngp.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
9 | isncvsngp.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
10 | 5, 6, 7, 8, 9 | isncvsngp 24018 | . 2 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))) |
11 | 1, 2, 4, 10 | syl3anbrc 1345 | 1 ⊢ (𝜑 → 𝑊 ∈ (NrmVec ∩ ℂVec)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ∩ cin 3856 ‘cfv 6369 (class class class)co 7202 · cmul 10717 abscabs 14780 Basecbs 16684 Scalarcsca 16770 ·𝑠 cvsca 16771 normcnm 23446 NrmGrpcngp 23447 NrmVeccnvc 23451 ℂVecccvs 23992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ico 12924 df-fz 13079 df-seq 13558 df-exp 13619 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-rest 16899 df-topn 16900 df-0g 16918 df-topgen 16920 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-cmn 19144 df-mgp 19477 df-ring 19536 df-cring 19537 df-subrg 19770 df-abv 19825 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-xms 23190 df-ms 23191 df-nm 23452 df-ngp 23453 df-nrg 23455 df-nlm 23456 df-nvc 23457 df-clm 23932 df-cvs 23993 |
This theorem is referenced by: (None) |
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