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Theorem bloval 28867
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 7225 . . . 4 (𝑢 = 𝑈 → (𝑢 LnOp 𝑤) = (𝑈 LnOp 𝑤))
3 oveq1 7225 . . . . . 6 (𝑢 = 𝑈 → (𝑢 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑤))
43fveq1d 6724 . . . . 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 3401 . . 3 (𝑢 = 𝑈 → {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞})
7 oveq2 7226 . . . . 5 (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = (𝑈 LnOp 𝑊))
8 bloval.4 . . . . 5 𝐿 = (𝑈 LnOp 𝑊)
97, 8eqtr4di 2796 . . . 4 (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = 𝐿)
10 oveq2 7226 . . . . . . 7 (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑊))
11 bloval.3 . . . . . . 7 𝑁 = (𝑈 normOpOLD 𝑊)
1210, 11eqtr4di 2796 . . . . . 6 (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = 𝑁)
1312fveq1d 6724 . . . . 5 (𝑤 = 𝑊 → ((𝑈 normOpOLD 𝑤)‘𝑡) = (𝑁𝑡))
1413breq1d 5068 . . . 4 (𝑤 = 𝑊 → (((𝑈 normOpOLD 𝑤)‘𝑡) < +∞ ↔ (𝑁𝑡) < +∞))
159, 14rabeqbidv 3401 . . 3 (𝑤 = 𝑊 → {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
16 df-blo 28832 . . 3 BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
178ovexi 7252 . . . 4 𝐿 ∈ V
1817rabex 5230 . . 3 {𝑡𝐿 ∣ (𝑁𝑡) < +∞} ∈ V
196, 15, 16, 18ovmpo 7374 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 BLnOp 𝑊) = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
201, 19syl5eq 2790 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  {crab 3065   class class class wbr 5058  cfv 6385  (class class class)co 7218  +∞cpnf 10869   < clt 10872  NrmCVeccnv 28670   LnOp clno 28826   normOpOLD cnmoo 28827   BLnOp cblo 28828
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 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pr 5327
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 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-sbc 3700  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-br 5059  df-opab 5121  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-iota 6343  df-fun 6387  df-fv 6393  df-ov 7221  df-oprab 7222  df-mpo 7223  df-blo 28832
This theorem is referenced by:  isblo  28868  hhbloi  29988
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