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Mirrors > Home > HSE Home > Th. List > nmopge0 | Structured version Visualization version GIF version |
Description: The norm of any Hilbert space operator is nonnegative. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmopge0 | ⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (normop‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 30728 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | ffvelcdm 7074 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 0ℎ ∈ ℋ) → (𝑇‘0ℎ) ∈ ℋ) | |
3 | 1, 2 | mpan2 688 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇‘0ℎ) ∈ ℋ) |
4 | normge0 30851 | . . 3 ⊢ ((𝑇‘0ℎ) ∈ ℋ → 0 ≤ (normℎ‘(𝑇‘0ℎ))) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (normℎ‘(𝑇‘0ℎ))) |
6 | norm0 30853 | . . . 4 ⊢ (normℎ‘0ℎ) = 0 | |
7 | 0le1 11735 | . . . 4 ⊢ 0 ≤ 1 | |
8 | 6, 7 | eqbrtri 5160 | . . 3 ⊢ (normℎ‘0ℎ) ≤ 1 |
9 | nmoplb 31632 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 0ℎ ∈ ℋ ∧ (normℎ‘0ℎ) ≤ 1) → (normℎ‘(𝑇‘0ℎ)) ≤ (normop‘𝑇)) | |
10 | 1, 8, 9 | mp3an23 1449 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (normℎ‘(𝑇‘0ℎ)) ≤ (normop‘𝑇)) |
11 | normcl 30850 | . . . . 5 ⊢ ((𝑇‘0ℎ) ∈ ℋ → (normℎ‘(𝑇‘0ℎ)) ∈ ℝ) | |
12 | 3, 11 | syl 17 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → (normℎ‘(𝑇‘0ℎ)) ∈ ℝ) |
13 | 12 | rexrd 11262 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normℎ‘(𝑇‘0ℎ)) ∈ ℝ*) |
14 | nmopxr 31591 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) ∈ ℝ*) | |
15 | 0xr 11259 | . . . 4 ⊢ 0 ∈ ℝ* | |
16 | xrletr 13135 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ (normℎ‘(𝑇‘0ℎ)) ∈ ℝ* ∧ (normop‘𝑇) ∈ ℝ*) → ((0 ≤ (normℎ‘(𝑇‘0ℎ)) ∧ (normℎ‘(𝑇‘0ℎ)) ≤ (normop‘𝑇)) → 0 ≤ (normop‘𝑇))) | |
17 | 15, 16 | mp3an1 1444 | . . 3 ⊢ (((normℎ‘(𝑇‘0ℎ)) ∈ ℝ* ∧ (normop‘𝑇) ∈ ℝ*) → ((0 ≤ (normℎ‘(𝑇‘0ℎ)) ∧ (normℎ‘(𝑇‘0ℎ)) ≤ (normop‘𝑇)) → 0 ≤ (normop‘𝑇))) |
18 | 13, 14, 17 | syl2anc 583 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → ((0 ≤ (normℎ‘(𝑇‘0ℎ)) ∧ (normℎ‘(𝑇‘0ℎ)) ≤ (normop‘𝑇)) → 0 ≤ (normop‘𝑇))) |
19 | 5, 10, 18 | mp2and 696 | 1 ⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (normop‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 class class class wbr 5139 ⟶wf 6530 ‘cfv 6534 ℝcr 11106 0cc0 11107 1c1 11108 ℝ*cxr 11245 ≤ cle 11247 ℋchba 30644 normℎcno 30648 0ℎc0v 30649 normopcnop 30670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-hilex 30724 ax-hfvadd 30725 ax-hvcom 30726 ax-hvass 30727 ax-hv0cl 30728 ax-hvaddid 30729 ax-hfvmul 30730 ax-hvmulid 30731 ax-hvmulass 30732 ax-hvdistr1 30733 ax-hvdistr2 30734 ax-hvmul0 30735 ax-hfi 30804 ax-his1 30807 ax-his2 30808 ax-his3 30809 ax-his4 30810 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-n0 12471 df-z 12557 df-uz 12821 df-rp 12973 df-seq 13965 df-exp 14026 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-grpo 30218 df-gid 30219 df-ablo 30270 df-vc 30284 df-nv 30317 df-va 30320 df-ba 30321 df-sm 30322 df-0v 30323 df-nmcv 30325 df-hnorm 30693 df-hba 30694 df-hvsub 30696 df-nmop 31564 |
This theorem is referenced by: nmopgt0 31637 nmophmi 31756 cnlnadjlem7 31798 nmopadjlem 31814 nmopcoadji 31826 opsqrlem1 31865 |
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