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Theorem bloval 28552
Description: The class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (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
bloval ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
Distinct variable groups:   𝑡,𝐿   𝑡,𝑁   𝑡,𝑈   𝑡,𝑊
Allowed substitution hint:   𝐵(𝑡)

Proof of Theorem bloval
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bloval.5 . 2 𝐵 = (𝑈 BLnOp 𝑊)
2 oveq1 7157 . . . 4 (𝑢 = 𝑈 → (𝑢 LnOp 𝑤) = (𝑈 LnOp 𝑤))
3 oveq1 7157 . . . . . 6 (𝑢 = 𝑈 → (𝑢 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑤))
43fveq1d 6666 . . . . 5 (𝑢 = 𝑈 → ((𝑢 normOpOLD 𝑤)‘𝑡) = ((𝑈 normOpOLD 𝑤)‘𝑡))
54breq1d 5068 . . . 4 (𝑢 = 𝑈 → (((𝑢 normOpOLD 𝑤)‘𝑡) < +∞ ↔ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞))
62, 5rabeqbidv 3485 . . 3 (𝑢 = 𝑈 → {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞})
7 oveq2 7158 . . . . 5 (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = (𝑈 LnOp 𝑊))
8 bloval.4 . . . . 5 𝐿 = (𝑈 LnOp 𝑊)
97, 8syl6eqr 2874 . . . 4 (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = 𝐿)
10 oveq2 7158 . . . . . . 7 (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑊))
11 bloval.3 . . . . . . 7 𝑁 = (𝑈 normOpOLD 𝑊)
1210, 11syl6eqr 2874 . . . . . 6 (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = 𝑁)
1312fveq1d 6666 . . . . 5 (𝑤 = 𝑊 → ((𝑈 normOpOLD 𝑤)‘𝑡) = (𝑁𝑡))
1413breq1d 5068 . . . 4 (𝑤 = 𝑊 → (((𝑈 normOpOLD 𝑤)‘𝑡) < +∞ ↔ (𝑁𝑡) < +∞))
159, 14rabeqbidv 3485 . . 3 (𝑤 = 𝑊 → {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
16 df-blo 28517 . . 3 BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
178ovexi 7184 . . . 4 𝐿 ∈ V
1817rabex 5227 . . 3 {𝑡𝐿 ∣ (𝑁𝑡) < +∞} ∈ V
196, 15, 16, 18ovmpo 7304 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 BLnOp 𝑊) = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
201, 19syl5eq 2868 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  {crab 3142   class class class wbr 5058  cfv 6349  (class class class)co 7150  +∞cpnf 10666   < clt 10669  NrmCVeccnv 28355   LnOp clno 28511   normOpOLD cnmoo 28512   BLnOp cblo 28513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-blo 28517
This theorem is referenced by:  isblo  28553  hhbloi  29673
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