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Theorem isblo 27765
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 27764 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
54eleq2d 2716 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐵𝑇 ∈ {𝑡𝐿 ∣ (𝑁𝑡) < +∞}))
6 fveq2 6229 . . . 4 (𝑡 = 𝑇 → (𝑁𝑡) = (𝑁𝑇))
76breq1d 4695 . . 3 (𝑡 = 𝑇 → ((𝑁𝑡) < +∞ ↔ (𝑁𝑇) < +∞))
87elrab 3396 . 2 (𝑇 ∈ {𝑡𝐿 ∣ (𝑁𝑡) < +∞} ↔ (𝑇𝐿 ∧ (𝑁𝑇) < +∞))
95, 8syl6bb 276 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐵 ↔ (𝑇𝐿 ∧ (𝑁𝑇) < +∞)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {crab 2945   class class class wbr 4685  cfv 5926  (class class class)co 6690  +∞cpnf 10109   < clt 10112  NrmCVeccnv 27567   LnOp clno 27723   normOpOLD cnmoo 27724   BLnOp cblo 27725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-blo 27729
This theorem is referenced by:  isblo2  27766  bloln  27767  nmblore  27769  isblo3i  27784  htthlem  27902
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