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Mirrors > Home > HSE Home > Th. List > nmfnge0 | Structured version Visualization version GIF version |
Description: The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmfnge0 | ⊢ (𝑇: ℋ⟶ℂ → 0 ≤ (normfn‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 28950 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | ffvelrn 6871 | . . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 0ℎ ∈ ℋ) → (𝑇‘0ℎ) ∈ ℂ) | |
3 | 1, 2 | mpan2 691 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (𝑇‘0ℎ) ∈ ℂ) |
4 | 3 | absge0d 14906 | . 2 ⊢ (𝑇: ℋ⟶ℂ → 0 ≤ (abs‘(𝑇‘0ℎ))) |
5 | norm0 29075 | . . . 4 ⊢ (normℎ‘0ℎ) = 0 | |
6 | 0le1 11253 | . . . 4 ⊢ 0 ≤ 1 | |
7 | 5, 6 | eqbrtri 5061 | . . 3 ⊢ (normℎ‘0ℎ) ≤ 1 |
8 | nmfnlb 29871 | . . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 0ℎ ∈ ℋ ∧ (normℎ‘0ℎ) ≤ 1) → (abs‘(𝑇‘0ℎ)) ≤ (normfn‘𝑇)) | |
9 | 1, 7, 8 | mp3an23 1454 | . 2 ⊢ (𝑇: ℋ⟶ℂ → (abs‘(𝑇‘0ℎ)) ≤ (normfn‘𝑇)) |
10 | 3 | abscld 14898 | . . . 4 ⊢ (𝑇: ℋ⟶ℂ → (abs‘(𝑇‘0ℎ)) ∈ ℝ) |
11 | 10 | rexrd 10781 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (abs‘(𝑇‘0ℎ)) ∈ ℝ*) |
12 | nmfnxr 29826 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) ∈ ℝ*) | |
13 | 0xr 10778 | . . . 4 ⊢ 0 ∈ ℝ* | |
14 | xrletr 12646 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ (abs‘(𝑇‘0ℎ)) ∈ ℝ* ∧ (normfn‘𝑇) ∈ ℝ*) → ((0 ≤ (abs‘(𝑇‘0ℎ)) ∧ (abs‘(𝑇‘0ℎ)) ≤ (normfn‘𝑇)) → 0 ≤ (normfn‘𝑇))) | |
15 | 13, 14 | mp3an1 1449 | . . 3 ⊢ (((abs‘(𝑇‘0ℎ)) ∈ ℝ* ∧ (normfn‘𝑇) ∈ ℝ*) → ((0 ≤ (abs‘(𝑇‘0ℎ)) ∧ (abs‘(𝑇‘0ℎ)) ≤ (normfn‘𝑇)) → 0 ≤ (normfn‘𝑇))) |
16 | 11, 12, 15 | syl2anc 587 | . 2 ⊢ (𝑇: ℋ⟶ℂ → ((0 ≤ (abs‘(𝑇‘0ℎ)) ∧ (abs‘(𝑇‘0ℎ)) ≤ (normfn‘𝑇)) → 0 ≤ (normfn‘𝑇))) |
17 | 4, 9, 16 | mp2and 699 | 1 ⊢ (𝑇: ℋ⟶ℂ → 0 ≤ (normfn‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2114 class class class wbr 5040 ⟶wf 6345 ‘cfv 6349 ℂcc 10625 0cc0 10627 1c1 10628 ℝ*cxr 10764 ≤ cle 10766 abscabs 14695 ℋchba 28866 normℎcno 28870 0ℎc0v 28871 normfncnmf 28898 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 ax-hilex 28946 ax-hv0cl 28950 ax-hvmul0 28957 ax-hfi 29026 ax-his3 29031 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-map 8451 df-en 8568 df-dom 8569 df-sdom 8570 df-sup 8991 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-n0 11989 df-z 12075 df-uz 12337 df-rp 12485 df-seq 13473 df-exp 13534 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-hnorm 28915 df-nmfn 29792 |
This theorem is referenced by: (None) |
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