HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  nmopub Structured version   Visualization version   GIF version

Theorem nmopub 29076
Description: An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nmopub ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem nmopub
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmopval 29024 . . . 4 (𝑇: ℋ⟶ ℋ → (normop𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ))
21adantr 472 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (normop𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ))
32breq1d 4814 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
4 nmopsetretALT 29031 . . . . 5 (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ)
5 ressxr 10275 . . . . 5 ℝ ⊆ ℝ*
64, 5syl6ss 3756 . . . 4 (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ*)
7 supxrleub 12349 . . . 4 (({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ*𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴))
86, 7sylan 489 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴))
9 ancom 465 . . . . . . 7 (((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ (𝑦 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1))
10 eqeq1 2764 . . . . . . . 8 (𝑦 = 𝑧 → (𝑦 = (norm‘(𝑇𝑥)) ↔ 𝑧 = (norm‘(𝑇𝑥))))
1110anbi1d 743 . . . . . . 7 (𝑦 = 𝑧 → ((𝑦 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) ↔ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
129, 11syl5bb 272 . . . . . 6 (𝑦 = 𝑧 → (((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1312rexbidv 3190 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1413ralab 3508 . . . 4 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
15 ralcom4 3364 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
16 impexp 461 . . . . . . . 8 (((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
1716albii 1896 . . . . . . 7 (∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
18 fvex 6362 . . . . . . . 8 (norm‘(𝑇𝑥)) ∈ V
19 breq1 4807 . . . . . . . . 9 (𝑧 = (norm‘(𝑇𝑥)) → (𝑧𝐴 ↔ (norm‘(𝑇𝑥)) ≤ 𝐴))
2019imbi2d 329 . . . . . . . 8 (𝑧 = (norm‘(𝑇𝑥)) → (((norm𝑥) ≤ 1 → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
2118, 20ceqsalv 3373 . . . . . . 7 (∀𝑧(𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
2217, 21bitri 264 . . . . . 6 (∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
2322ralbii 3118 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
24 r19.23v 3161 . . . . . 6 (∀𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2524albii 1896 . . . . 5 (∀𝑧𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2615, 23, 253bitr3i 290 . . . 4 (∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2714, 26bitr4i 267 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
288, 27syl6bb 276 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
293, 28bitrd 268 1 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  wss 3715   class class class wbr 4804  wf 6045  cfv 6049  supcsup 8511  cr 10127  1c1 10129  *cxr 10265   < clt 10266  cle 10267  chil 28085  normcno 28089  normopcnop 28111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-hilex 28165  ax-hv0cl 28169  ax-hvmul0 28176  ax-hfi 28245  ax-his1 28248  ax-his3 28250  ax-his4 28251
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-sup 8513  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-seq 12996  df-exp 13055  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-hnorm 28134  df-nmop 29007
This theorem is referenced by:  nmopub2tALT  29077  nmophmi  29199  nmopadjlem  29257  nmoptrii  29262  nmopcoi  29263  nmopcoadji  29269
  Copyright terms: Public domain W3C validator