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Mirrors > Home > HSE Home > Th. List > normcl | Structured version Visualization version GIF version |
Description: Real closure of the norm of a vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normcl | ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | normf 28108 | . 2 ⊢ normℎ: ℋ⟶ℝ | |
2 | 1 | ffvelrni 6398 | 1 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 ‘cfv 5926 ℝcr 9973 ℋchil 27904 normℎcno 27908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-hv0cl 27988 ax-hvmul0 27995 ax-hfi 28064 ax-his1 28067 ax-his3 28069 ax-his4 28070 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-hnorm 27953 |
This theorem is referenced by: norm-i 28114 normcli 28116 normpyc 28131 hhph 28163 bcs2 28167 norm1 28234 norm1exi 28235 pjhthlem1 28378 chscllem2 28625 pjige0i 28677 pjnorm2 28714 nmopsetretALT 28850 nmopub2tALT 28896 nmopge0 28898 unopnorm 28904 nmfnleub2 28913 eigvalcl 28948 nmlnop0iALT 28982 nmbdoplbi 29011 nmcexi 29013 nmcopexi 29014 nmcoplbi 29015 nmophmi 29018 lnconi 29020 lnopconi 29021 nmbdfnlbi 29036 nmcfnlbi 29039 riesz4i 29050 riesz1 29052 cnlnadjlem2 29055 cnlnadjlem7 29060 nmopadjlem 29076 nmoptrii 29081 nmopcoi 29082 nmopcoadji 29088 branmfn 29092 brabn 29093 leopnmid 29125 pjnmopi 29135 pjnormssi 29155 pjssposi 29159 hstle1 29213 hst1h 29214 hstle 29217 hstles 29218 hstoh 29219 strlem1 29237 strlem3a 29239 strlem5 29242 hstrlem6 29251 jplem1 29255 cdj1i 29420 cdj3lem1 29421 cdj3lem2b 29424 cdj3lem3b 29427 cdj3i 29428 |
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