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Theorem bloval 27976
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 6800 . . . 4 (𝑢 = 𝑈 → (𝑢 LnOp 𝑤) = (𝑈 LnOp 𝑤))
3 oveq1 6800 . . . . . 6 (𝑢 = 𝑈 → (𝑢 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑤))
43fveq1d 6334 . . . . 5 (𝑢 = 𝑈 → ((𝑢 normOpOLD 𝑤)‘𝑡) = ((𝑈 normOpOLD 𝑤)‘𝑡))
54breq1d 4796 . . . 4 (𝑢 = 𝑈 → (((𝑢 normOpOLD 𝑤)‘𝑡) < +∞ ↔ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞))
62, 5rabeqbidv 3345 . . 3 (𝑢 = 𝑈 → {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞})
7 oveq2 6801 . . . . 5 (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = (𝑈 LnOp 𝑊))
8 bloval.4 . . . . 5 𝐿 = (𝑈 LnOp 𝑊)
97, 8syl6eqr 2823 . . . 4 (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = 𝐿)
10 oveq2 6801 . . . . . . 7 (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑊))
11 bloval.3 . . . . . . 7 𝑁 = (𝑈 normOpOLD 𝑊)
1210, 11syl6eqr 2823 . . . . . 6 (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = 𝑁)
1312fveq1d 6334 . . . . 5 (𝑤 = 𝑊 → ((𝑈 normOpOLD 𝑤)‘𝑡) = (𝑁𝑡))
1413breq1d 4796 . . . 4 (𝑤 = 𝑊 → (((𝑈 normOpOLD 𝑤)‘𝑡) < +∞ ↔ (𝑁𝑡) < +∞))
159, 14rabeqbidv 3345 . . 3 (𝑤 = 𝑊 → {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
16 df-blo 27941 . . 3 BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
17 ovex 6823 . . . . 5 (𝑈 LnOp 𝑊) ∈ V
188, 17eqeltri 2846 . . . 4 𝐿 ∈ V
1918rabex 4946 . . 3 {𝑡𝐿 ∣ (𝑁𝑡) < +∞} ∈ V
206, 15, 16, 19ovmpt2 6943 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 BLnOp 𝑊) = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
211, 20syl5eq 2817 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351   class class class wbr 4786  cfv 6031  (class class class)co 6793  +∞cpnf 10273   < clt 10276  NrmCVeccnv 27779   LnOp clno 27935   normOpOLD cnmoo 27936   BLnOp cblo 27937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-blo 27941
This theorem is referenced by:  isblo  27977  hhbloi  29101
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