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Mirrors > Home > HSE Home > Th. List > normf | Structured version Visualization version GIF version |
Description: The norm function maps from Hilbert space to reals. (Contributed by NM, 6-Sep-2007.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normf | ⊢ normℎ: ℋ⟶ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfhnorm2 28681 | . 2 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) | |
2 | hiidrcl 28654 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑥 ·ih 𝑥) ∈ ℝ) | |
3 | hiidge0 28657 | . . 3 ⊢ (𝑥 ∈ ℋ → 0 ≤ (𝑥 ·ih 𝑥)) | |
4 | 2, 3 | resqrtcld 14641 | . 2 ⊢ (𝑥 ∈ ℋ → (√‘(𝑥 ·ih 𝑥)) ∈ ℝ) |
5 | 1, 4 | fmpti 6701 | 1 ⊢ normℎ: ℋ⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2050 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ℝcr 10336 √csqrt 14456 ℋchba 28478 ·ih csp 28481 normℎcno 28482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-hv0cl 28562 ax-hvmul0 28569 ax-hfi 28638 ax-his1 28641 ax-his3 28643 ax-his4 28644 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-sup 8703 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-n0 11711 df-z 11797 df-uz 12062 df-rp 12208 df-seq 13188 df-exp 13248 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-hnorm 28527 |
This theorem is referenced by: normcl 28684 hhnv 28724 hhph 28737 hhssva 28816 hhsssm 28817 hhssnm 28818 hhssnv 28823 hhshsslem1 28826 hhsssh 28828 |
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