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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 | β’ π = (Baseβπ) |
isncvsngp.n | β’ π = (normβπ) |
isncvsngp.s | β’ Β· = ( Β·π βπ) |
isncvsngp.f | β’ πΉ = (Scalarβπ) |
isncvsngp.k | β’ πΎ = (BaseβπΉ) |
isncvsngpd.v | β’ (π β π β βVec) |
isncvsngpd.g | β’ (π β π β NrmGrp) |
isncvsngpd.t | β’ ((π β§ (π₯ β π β§ π β πΎ)) β (πβ(π Β· π₯)) = ((absβπ) Β· (πβπ₯))) |
Ref | Expression |
---|---|
isncvsngpd | β’ (π β π β (NrmVec β© βVec)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isncvsngpd.v | . 2 β’ (π β π β βVec) | |
2 | isncvsngpd.g | . 2 β’ (π β π β NrmGrp) | |
3 | isncvsngpd.t | . . 3 β’ ((π β§ (π₯ β π β§ π β πΎ)) β (πβ(π Β· π₯)) = ((absβπ) Β· (πβπ₯))) | |
4 | 3 | ralrimivva 3201 | . 2 β’ (π β βπ₯ β π βπ β πΎ (πβ(π Β· π₯)) = ((absβπ) Β· (πβπ₯))) |
5 | isncvsngp.v | . . 3 β’ π = (Baseβπ) | |
6 | isncvsngp.n | . . 3 β’ π = (normβπ) | |
7 | isncvsngp.s | . . 3 β’ Β· = ( Β·π βπ) | |
8 | isncvsngp.f | . . 3 β’ πΉ = (Scalarβπ) | |
9 | isncvsngp.k | . . 3 β’ πΎ = (BaseβπΉ) | |
10 | 5, 6, 7, 8, 9 | isncvsngp 24666 | . 2 β’ (π β (NrmVec β© βVec) β (π β βVec β§ π β NrmGrp β§ βπ₯ β π βπ β πΎ (πβ(π Β· π₯)) = ((absβπ) Β· (πβπ₯)))) |
11 | 1, 2, 4, 10 | syl3anbrc 1344 | 1 β’ (π β π β (NrmVec β© βVec)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 β© cin 3948 βcfv 6544 (class class class)co 7409 Β· cmul 11115 abscabs 15181 Basecbs 17144 Scalarcsca 17200 Β·π cvsca 17201 normcnm 24085 NrmGrpcngp 24086 NrmVeccnvc 24090 βVecccvs 24639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ico 13330 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-topn 17369 df-0g 17387 df-topgen 17389 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-cmn 19650 df-mgp 19988 df-ring 20058 df-cring 20059 df-subrg 20317 df-abv 20425 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-xms 23826 df-ms 23827 df-nm 24091 df-ngp 24092 df-nrg 24094 df-nlm 24095 df-nvc 24096 df-clm 24579 df-cvs 24640 |
This theorem is referenced by: (None) |
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