Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > nmopub2tHIL | Structured version Visualization version GIF version |
Description: An upper bound for an operator norm. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmopub2tHIL | ⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normop‘𝑇) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hba 29022 | . 2 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
2 | eqid 2734 | . . 3 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
3 | 2 | hhnm 29224 | . 2 ⊢ normℎ = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
4 | eqid 2734 | . . 3 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 normOpOLD 〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 normOpOLD 〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
5 | 2, 4 | hhnmoi 29954 | . 2 ⊢ normop = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 normOpOLD 〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
6 | 2 | hhnv 29218 | . 2 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
7 | 1, 1, 3, 3, 5, 6, 6 | nmoub2i 28827 | 1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normop‘𝑇) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2110 ∀wral 3054 〈cop 4537 class class class wbr 5043 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ℝcr 10711 0cc0 10712 · cmul 10717 ≤ cle 10851 normOpOLD cnmoo 28794 ℋchba 28972 +ℎ cva 28973 ·ℎ csm 28974 normℎcno 28976 normopcnop 28998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-hilex 29052 ax-hfvadd 29053 ax-hvcom 29054 ax-hvass 29055 ax-hv0cl 29056 ax-hvaddid 29057 ax-hfvmul 29058 ax-hvmulid 29059 ax-hvmulass 29060 ax-hvdistr1 29061 ax-hvdistr2 29062 ax-hvmul0 29063 ax-hfi 29132 ax-his1 29135 ax-his2 29136 ax-his3 29137 ax-his4 29138 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 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 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-sup 9047 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-seq 13558 df-exp 13619 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-grpo 28546 df-gid 28547 df-ginv 28548 df-ablo 28598 df-vc 28612 df-nv 28645 df-va 28648 df-ba 28649 df-sm 28650 df-0v 28651 df-nmcv 28653 df-nmoo 28798 df-hnorm 29021 df-hba 29022 df-hvsub 29024 df-nmop 29892 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |