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Mirrors > Home > HSE Home > Th. List > nmopnegi | Structured version Visualization version GIF version |
Description: Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 28890, the operator does not have to be bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmopneg.1 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
nmopnegi | ⊢ (normop‘(-1 ·op 𝑇)) = (normop‘𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 11124 | . . . . . . . . . 10 ⊢ -1 ∈ ℂ | |
2 | nmopneg.1 | . . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ | |
3 | homval 28600 | . . . . . . . . . 10 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((-1 ·op 𝑇)‘𝑦) = (-1 ·ℎ (𝑇‘𝑦))) | |
4 | 1, 2, 3 | mp3an12 1414 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → ((-1 ·op 𝑇)‘𝑦) = (-1 ·ℎ (𝑇‘𝑦))) |
5 | 4 | fveq2d 6195 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (normℎ‘((-1 ·op 𝑇)‘𝑦)) = (normℎ‘(-1 ·ℎ (𝑇‘𝑦)))) |
6 | 2 | ffvelrni 6358 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ) |
7 | normneg 28001 | . . . . . . . . 9 ⊢ ((𝑇‘𝑦) ∈ ℋ → (normℎ‘(-1 ·ℎ (𝑇‘𝑦))) = (normℎ‘(𝑇‘𝑦))) | |
8 | 6, 7 | syl 17 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (normℎ‘(-1 ·ℎ (𝑇‘𝑦))) = (normℎ‘(𝑇‘𝑦))) |
9 | 5, 8 | eqtrd 2656 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → (normℎ‘((-1 ·op 𝑇)‘𝑦)) = (normℎ‘(𝑇‘𝑦))) |
10 | 9 | eqeq2d 2632 | . . . . . 6 ⊢ (𝑦 ∈ ℋ → (𝑥 = (normℎ‘((-1 ·op 𝑇)‘𝑦)) ↔ 𝑥 = (normℎ‘(𝑇‘𝑦)))) |
11 | 10 | anbi2d 740 | . . . . 5 ⊢ (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘((-1 ·op 𝑇)‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))))) |
12 | 11 | rexbiia 3040 | . . . 4 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘((-1 ·op 𝑇)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))) |
13 | 12 | abbii 2739 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘((-1 ·op 𝑇)‘𝑦)))} = {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} |
14 | 13 | supeq1i 8353 | . 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘((-1 ·op 𝑇)‘𝑦)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) |
15 | homulcl 28618 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (-1 ·op 𝑇): ℋ⟶ ℋ) | |
16 | 1, 2, 15 | mp2an 708 | . . 3 ⊢ (-1 ·op 𝑇): ℋ⟶ ℋ |
17 | nmopval 28715 | . . 3 ⊢ ((-1 ·op 𝑇): ℋ⟶ ℋ → (normop‘(-1 ·op 𝑇)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘((-1 ·op 𝑇)‘𝑦)))}, ℝ*, < )) | |
18 | 16, 17 | ax-mp 5 | . 2 ⊢ (normop‘(-1 ·op 𝑇)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘((-1 ·op 𝑇)‘𝑦)))}, ℝ*, < ) |
19 | nmopval 28715 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < )) | |
20 | 2, 19 | ax-mp 5 | . 2 ⊢ (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) |
21 | 14, 18, 20 | 3eqtr4i 2654 | 1 ⊢ (normop‘(-1 ·op 𝑇)) = (normop‘𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supcsup 8346 ℂcc 9934 1c1 9937 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 -cneg 10267 ℋchil 27776 ·ℎ csm 27778 normℎcno 27780 ·op chot 27796 normopcnop 27802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his3 27941 ax-his4 27942 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-hnorm 27825 df-hvsub 27828 df-homul 28590 df-nmop 28698 |
This theorem is referenced by: nmoptri2i 28958 |
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