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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 28774 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | ffvelrn 6844 | . . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ 0ℎ ∈ ℋ) → (𝑇‘0ℎ) ∈ ℂ) | |
3 | 1, 2 | mpan2 689 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (𝑇‘0ℎ) ∈ ℂ) |
4 | 3 | absge0d 14798 | . 2 ⊢ (𝑇: ℋ⟶ℂ → 0 ≤ (abs‘(𝑇‘0ℎ))) |
5 | norm0 28899 | . . . 4 ⊢ (normℎ‘0ℎ) = 0 | |
6 | 0le1 11157 | . . . 4 ⊢ 0 ≤ 1 | |
7 | 5, 6 | eqbrtri 5080 | . . 3 ⊢ (normℎ‘0ℎ) ≤ 1 |
8 | nmfnlb 29695 | . . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 0ℎ ∈ ℋ ∧ (normℎ‘0ℎ) ≤ 1) → (abs‘(𝑇‘0ℎ)) ≤ (normfn‘𝑇)) | |
9 | 1, 7, 8 | mp3an23 1449 | . 2 ⊢ (𝑇: ℋ⟶ℂ → (abs‘(𝑇‘0ℎ)) ≤ (normfn‘𝑇)) |
10 | 3 | abscld 14790 | . . . 4 ⊢ (𝑇: ℋ⟶ℂ → (abs‘(𝑇‘0ℎ)) ∈ ℝ) |
11 | 10 | rexrd 10685 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (abs‘(𝑇‘0ℎ)) ∈ ℝ*) |
12 | nmfnxr 29650 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) ∈ ℝ*) | |
13 | 0xr 10682 | . . . 4 ⊢ 0 ∈ ℝ* | |
14 | xrletr 12545 | . . . 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 1444 | . . 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 586 | . 2 ⊢ (𝑇: ℋ⟶ℂ → ((0 ≤ (abs‘(𝑇‘0ℎ)) ∧ (abs‘(𝑇‘0ℎ)) ≤ (normfn‘𝑇)) → 0 ≤ (normfn‘𝑇))) |
17 | 4, 9, 16 | mp2and 697 | 1 ⊢ (𝑇: ℋ⟶ℂ → 0 ≤ (normfn‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 class class class wbr 5059 ⟶wf 6346 ‘cfv 6350 ℂcc 10529 0cc0 10531 1c1 10532 ℝ*cxr 10668 ≤ cle 10670 abscabs 14587 ℋchba 28690 normℎcno 28694 0ℎc0v 28695 normfncnmf 28722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-hilex 28770 ax-hv0cl 28774 ax-hvmul0 28781 ax-hfi 28850 ax-his3 28855 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-hnorm 28739 df-nmfn 29616 |
This theorem is referenced by: (None) |
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