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Theorem isblo 27021
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 27020 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
54eleq2d 2673 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐵𝑇 ∈ {𝑡𝐿 ∣ (𝑁𝑡) < +∞}))
6 fveq2 6103 . . . 4 (𝑡 = 𝑇 → (𝑁𝑡) = (𝑁𝑇))
76breq1d 4593 . . 3 (𝑡 = 𝑇 → ((𝑁𝑡) < +∞ ↔ (𝑁𝑇) < +∞))
87elrab 3331 . 2 (𝑇 ∈ {𝑡𝐿 ∣ (𝑁𝑡) < +∞} ↔ (𝑇𝐿 ∧ (𝑁𝑇) < +∞))
95, 8syl6bb 275 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐵 ↔ (𝑇𝐿 ∧ (𝑁𝑇) < +∞)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {crab 2900   class class class wbr 4583  cfv 5804  (class class class)co 6549  +∞cpnf 9950   < clt 9953  NrmCVeccnv 26823   LnOp clno 26979   normOpOLD cnmoo 26980   BLnOp cblo 26981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-blo 26985
This theorem is referenced by:  isblo2  27022  bloln  27023  nmblore  27025  isblo3i  27040  htthlem  27158
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