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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 28467 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | ffvelrn 6721 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 0ℎ ∈ ℋ) → (𝑇‘0ℎ) ∈ ℋ) | |
3 | 1, 2 | mpan2 687 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇‘0ℎ) ∈ ℋ) |
4 | normge0 28590 | . . 3 ⊢ ((𝑇‘0ℎ) ∈ ℋ → 0 ≤ (normℎ‘(𝑇‘0ℎ))) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (normℎ‘(𝑇‘0ℎ))) |
6 | norm0 28592 | . . . 4 ⊢ (normℎ‘0ℎ) = 0 | |
7 | 0le1 11017 | . . . 4 ⊢ 0 ≤ 1 | |
8 | 6, 7 | eqbrtri 4989 | . . 3 ⊢ (normℎ‘0ℎ) ≤ 1 |
9 | nmoplb 29371 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 0ℎ ∈ ℋ ∧ (normℎ‘0ℎ) ≤ 1) → (normℎ‘(𝑇‘0ℎ)) ≤ (normop‘𝑇)) | |
10 | 1, 8, 9 | mp3an23 1445 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (normℎ‘(𝑇‘0ℎ)) ≤ (normop‘𝑇)) |
11 | normcl 28589 | . . . . 5 ⊢ ((𝑇‘0ℎ) ∈ ℋ → (normℎ‘(𝑇‘0ℎ)) ∈ ℝ) | |
12 | 3, 11 | syl 17 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → (normℎ‘(𝑇‘0ℎ)) ∈ ℝ) |
13 | 12 | rexrd 10544 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normℎ‘(𝑇‘0ℎ)) ∈ ℝ*) |
14 | nmopxr 29330 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) ∈ ℝ*) | |
15 | 0xr 10541 | . . . 4 ⊢ 0 ∈ ℝ* | |
16 | xrletr 12405 | . . . 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 1440 | . . 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 584 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → ((0 ≤ (normℎ‘(𝑇‘0ℎ)) ∧ (normℎ‘(𝑇‘0ℎ)) ≤ (normop‘𝑇)) → 0 ≤ (normop‘𝑇))) |
19 | 5, 10, 18 | mp2and 695 | 1 ⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (normop‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2083 class class class wbr 4968 ⟶wf 6228 ‘cfv 6232 ℝcr 10389 0cc0 10390 1c1 10391 ℝ*cxr 10527 ≤ cle 10529 ℋchba 28383 normℎcno 28387 0ℎc0v 28388 normopcnop 28409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 ax-hilex 28463 ax-hfvadd 28464 ax-hvcom 28465 ax-hvass 28466 ax-hv0cl 28467 ax-hvaddid 28468 ax-hfvmul 28469 ax-hvmulid 28470 ax-hvmulass 28471 ax-hvdistr1 28472 ax-hvdistr2 28473 ax-hvmul0 28474 ax-hfi 28543 ax-his1 28546 ax-his2 28547 ax-his3 28548 ax-his4 28549 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-sup 8759 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-n0 11752 df-z 11836 df-uz 12098 df-rp 12244 df-seq 13224 df-exp 13284 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-grpo 27957 df-gid 27958 df-ablo 28009 df-vc 28023 df-nv 28056 df-va 28059 df-ba 28060 df-sm 28061 df-0v 28062 df-nmcv 28064 df-hnorm 28432 df-hba 28433 df-hvsub 28435 df-nmop 29303 |
This theorem is referenced by: nmopgt0 29376 nmophmi 29495 cnlnadjlem7 29537 nmopadjlem 29553 nmopcoadji 29565 opsqrlem1 29604 |
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