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Mirrors > Home > MPE Home > Th. List > cnncvs | Structured version Visualization version GIF version |
Description: The module of complex numbers (as a module over itself) is a normed subcomplex vector space. The vector operation is +, the scalar product is ·, and the norm is abs (see cnnm 24314) . (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 9-Oct-2021.) |
Ref | Expression |
---|---|
cnrnvc.c | ⊢ 𝐶 = (ringLMod‘ℂfld) |
Ref | Expression |
---|---|
cnncvs | ⊢ 𝐶 ∈ (NrmVec ∩ ℂVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnrnvc.c | . . 3 ⊢ 𝐶 = (ringLMod‘ℂfld) | |
2 | 1 | cnrnvc 24312 | . 2 ⊢ 𝐶 ∈ NrmVec |
3 | 1 | cncvs 24298 | . 2 ⊢ 𝐶 ∈ ℂVec |
4 | 2, 3 | elini 4132 | 1 ⊢ 𝐶 ∈ (NrmVec ∩ ℂVec) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 ∩ cin 3891 ‘cfv 6431 ringLModcrglmod 20421 ℂfldccnfld 20587 NrmVeccnvc 23727 ℂVecccvs 24276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 ax-pre-sup 10942 ax-addf 10943 ax-mulf 10944 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7702 df-1st 7818 df-2nd 7819 df-tpos 8027 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-1o 8282 df-er 8473 df-map 8592 df-en 8709 df-dom 8710 df-sdom 8711 df-fin 8712 df-sup 9171 df-inf 9172 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-div 11625 df-nn 11966 df-2 12028 df-3 12029 df-4 12030 df-5 12031 df-6 12032 df-7 12033 df-8 12034 df-9 12035 df-n0 12226 df-z 12312 df-dec 12429 df-uz 12574 df-q 12680 df-rp 12722 df-xneg 12839 df-xadd 12840 df-xmul 12841 df-ico 13076 df-fz 13231 df-seq 13712 df-exp 13773 df-cj 14800 df-re 14801 df-im 14802 df-sqrt 14936 df-abs 14937 df-struct 16838 df-sets 16855 df-slot 16873 df-ndx 16885 df-base 16903 df-ress 16932 df-plusg 16965 df-mulr 16966 df-starv 16967 df-sca 16968 df-vsca 16969 df-ip 16970 df-tset 16971 df-ple 16972 df-ds 16974 df-unif 16975 df-rest 17123 df-topn 17124 df-0g 17142 df-topgen 17144 df-mgm 18316 df-sgrp 18365 df-mnd 18376 df-grp 18570 df-minusg 18571 df-sbg 18572 df-subg 18742 df-cmn 19378 df-mgp 19711 df-ur 19728 df-ring 19775 df-cring 19776 df-oppr 19852 df-dvdsr 19873 df-unit 19874 df-invr 19904 df-dvr 19915 df-drng 19983 df-subrg 20012 df-abv 20067 df-lmod 20115 df-lvec 20355 df-sra 20424 df-rgmod 20425 df-psmet 20579 df-xmet 20580 df-met 20581 df-bl 20582 df-mopn 20583 df-cnfld 20588 df-top 22033 df-topon 22050 df-topsp 22072 df-bases 22086 df-xms 23463 df-ms 23464 df-nm 23728 df-ngp 23729 df-nrg 23731 df-nlm 23732 df-nvc 23733 df-clm 24216 df-cvs 24277 |
This theorem is referenced by: (None) |
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