![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3199 | . 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 24595 | . 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 1343 | 1 ⊢ (𝜑 → 𝑊 ∈ (NrmVec ∩ ℂVec)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 ∩ cin 3943 ‘cfv 6532 (class class class)co 7393 · cmul 11097 abscabs 15163 Basecbs 17126 Scalarcsca 17182 ·𝑠 cvsca 17183 normcnm 24014 NrmGrpcngp 24015 NrmVeccnvc 24019 ℂVecccvs 24568 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-ico 13312 df-fz 13467 df-seq 13949 df-exp 14010 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-rest 17350 df-topn 17351 df-0g 17369 df-topgen 17371 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 df-sbg 18799 df-subg 18975 df-cmn 19614 df-mgp 19947 df-ring 20016 df-cring 20017 df-subrg 20310 df-abv 20374 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-cnfld 20879 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-xms 23755 df-ms 23756 df-nm 24020 df-ngp 24021 df-nrg 24023 df-nlm 24024 df-nvc 24025 df-clm 24508 df-cvs 24569 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |