MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isblo Structured version   Visualization version   GIF version

Theorem isblo 28894
Description: The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
bloval.3 𝑁 = (𝑈 normOpOLD 𝑊)
bloval.4 𝐿 = (𝑈 LnOp 𝑊)
bloval.5 𝐵 = (𝑈 BLnOp 𝑊)
Assertion
Ref Expression
isblo ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐵 ↔ (𝑇𝐿 ∧ (𝑁𝑇) < +∞)))

Proof of Theorem isblo
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 bloval.3 . . . 4 𝑁 = (𝑈 normOpOLD 𝑊)
2 bloval.4 . . . 4 𝐿 = (𝑈 LnOp 𝑊)
3 bloval.5 . . . 4 𝐵 = (𝑈 BLnOp 𝑊)
41, 2, 3bloval 28893 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
54eleq2d 2825 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐵𝑇 ∈ {𝑡𝐿 ∣ (𝑁𝑡) < +∞}))
6 fveq2 6738 . . . 4 (𝑡 = 𝑇 → (𝑁𝑡) = (𝑁𝑇))
76breq1d 5079 . . 3 (𝑡 = 𝑇 → ((𝑁𝑡) < +∞ ↔ (𝑁𝑇) < +∞))
87elrab 3617 . 2 (𝑇 ∈ {𝑡𝐿 ∣ (𝑁𝑡) < +∞} ↔ (𝑇𝐿 ∧ (𝑁𝑇) < +∞))
95, 8bitrdi 290 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐵 ↔ (𝑇𝐿 ∧ (𝑁𝑇) < +∞)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2112  {crab 3068   class class class wbr 5069  cfv 6400  (class class class)co 7234  +∞cpnf 10893   < clt 10896  NrmCVeccnv 28696   LnOp clno 28852   normOpOLD cnmoo 28853   BLnOp cblo 28854
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 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-iota 6358  df-fun 6402  df-fv 6408  df-ov 7237  df-oprab 7238  df-mpo 7239  df-blo 28858
This theorem is referenced by:  isblo2  28895  bloln  28896  nmblore  28898  isblo3i  28913  htthlem  29029
  Copyright terms: Public domain W3C validator