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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 30809 | . 2 ⊢ normℎ: ℋ⟶ℝ | |
2 | 1 | ffvelcdmi 7085 | 1 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ‘cfv 6543 ℝcr 11115 ℋchba 30605 normℎcno 30609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-hv0cl 30689 ax-hvmul0 30696 ax-hfi 30765 ax-his1 30768 ax-his3 30770 ax-his4 30771 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-hnorm 30654 |
This theorem is referenced by: norm-i 30815 normcli 30817 normpyc 30832 hhph 30864 bcs2 30868 norm1 30935 norm1exi 30936 pjhthlem1 31077 chscllem2 31324 pjige0i 31376 pjnorm2 31413 nmopsetretALT 31549 nmopub2tALT 31595 nmopge0 31597 unopnorm 31603 nmfnleub2 31612 eigvalcl 31647 nmlnop0iALT 31681 nmbdoplbi 31710 nmcexi 31712 nmcopexi 31713 nmcoplbi 31714 nmophmi 31717 lnconi 31719 lnopconi 31720 nmbdfnlbi 31735 nmcfnlbi 31738 riesz4i 31749 riesz1 31751 cnlnadjlem2 31754 cnlnadjlem7 31759 nmopadjlem 31775 nmoptrii 31780 nmopcoi 31781 nmopcoadji 31787 branmfn 31791 brabn 31792 leopnmid 31824 pjnmopi 31834 pjnormssi 31854 pjssposi 31858 hstle1 31912 hst1h 31913 hstle 31916 hstles 31917 hstoh 31918 strlem1 31936 strlem3a 31938 strlem5 31941 hstrlem6 31950 jplem1 31954 cdj1i 32119 cdj3lem1 32120 cdj3lem2b 32123 cdj3lem3b 32126 cdj3i 32127 |
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