![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > nmoplb | Structured version Visualization version GIF version |
Description: A lower bound for an operator norm. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoplb | ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ (normop‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmopsetretHIL 31892 | . . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ) | |
2 | ressxr 11302 | . . . . 5 ⊢ ℝ ⊆ ℝ* | |
3 | 1, 2 | sstrdi 4007 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*) |
4 | 3 | 3ad2ant1 1132 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*) |
5 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (normℎ‘𝑦) = (normℎ‘𝐴)) | |
6 | 5 | breq1d 5157 | . . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘𝐴) ≤ 1)) |
7 | 2fveq3 6911 | . . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (normℎ‘(𝑇‘𝑦)) = (normℎ‘(𝑇‘𝐴))) | |
8 | 7 | eqeq2d 2745 | . . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴)))) |
9 | 6, 8 | anbi12d 632 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝐴) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴))))) |
10 | eqid 2734 | . . . . . . . 8 ⊢ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴)) | |
11 | 10 | biantru 529 | . . . . . . 7 ⊢ ((normℎ‘𝐴) ≤ 1 ↔ ((normℎ‘𝐴) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴)))) |
12 | 9, 11 | bitr4di 289 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))) ↔ (normℎ‘𝐴) ≤ 1)) |
13 | 12 | rspcev 3621 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦)))) |
14 | fvex 6919 | . . . . . 6 ⊢ (normℎ‘(𝑇‘𝐴)) ∈ V | |
15 | eqeq1 2738 | . . . . . . . 8 ⊢ (𝑥 = (normℎ‘(𝑇‘𝐴)) → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦)))) | |
16 | 15 | anbi2d 630 | . . . . . . 7 ⊢ (𝑥 = (normℎ‘(𝑇‘𝐴)) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))))) |
17 | 16 | rexbidv 3176 | . . . . . 6 ⊢ (𝑥 = (normℎ‘(𝑇‘𝐴)) → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))))) |
18 | 14, 17 | elab 3680 | . . . . 5 ⊢ ((normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦)))) |
19 | 13, 18 | sylibr 234 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}) |
20 | 19 | 3adant1 1129 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}) |
21 | supxrub 13362 | . . 3 ⊢ (({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}) → (normℎ‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < )) | |
22 | 4, 20, 21 | syl2anc 584 | . 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < )) |
23 | nmopval 31884 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < )) | |
24 | 23 | 3ad2ant1 1132 | . 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < )) |
25 | 22, 24 | breqtrrd 5175 | 1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ (normop‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 {cab 2711 ∃wrex 3067 ⊆ wss 3962 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 supcsup 9477 ℝcr 11151 1c1 11153 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 ℋchba 30947 normℎcno 30951 normopcnop 30973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-hilex 31027 ax-hfvadd 31028 ax-hvcom 31029 ax-hvass 31030 ax-hv0cl 31031 ax-hvaddid 31032 ax-hfvmul 31033 ax-hvmulid 31034 ax-hvmulass 31035 ax-hvdistr1 31036 ax-hvdistr2 31037 ax-hvmul0 31038 ax-hfi 31107 ax-his1 31110 ax-his2 31111 ax-his3 31112 ax-his4 31113 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-grpo 30521 df-gid 30522 df-ablo 30573 df-vc 30587 df-nv 30620 df-va 30623 df-ba 30624 df-sm 30625 df-0v 30626 df-nmcv 30628 df-hnorm 30996 df-hba 30997 df-hvsub 30999 df-nmop 31867 |
This theorem is referenced by: nmopge0 31939 nmbdoplbi 32052 nmcoplbi 32056 nmophmi 32059 nmoptrii 32122 nmopcoi 32123 |
Copyright terms: Public domain | W3C validator |