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Theorem bloval 27945
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 6820 . . . 4 (𝑢 = 𝑈 → (𝑢 LnOp 𝑤) = (𝑈 LnOp 𝑤))
3 oveq1 6820 . . . . . 6 (𝑢 = 𝑈 → (𝑢 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑤))
43fveq1d 6354 . . . . 5 (𝑢 = 𝑈 → ((𝑢 normOpOLD 𝑤)‘𝑡) = ((𝑈 normOpOLD 𝑤)‘𝑡))
54breq1d 4814 . . . 4 (𝑢 = 𝑈 → (((𝑢 normOpOLD 𝑤)‘𝑡) < +∞ ↔ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞))
62, 5rabeqbidv 3335 . . 3 (𝑢 = 𝑈 → {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞})
7 oveq2 6821 . . . . 5 (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = (𝑈 LnOp 𝑊))
8 bloval.4 . . . . 5 𝐿 = (𝑈 LnOp 𝑊)
97, 8syl6eqr 2812 . . . 4 (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = 𝐿)
10 oveq2 6821 . . . . . . 7 (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑊))
11 bloval.3 . . . . . . 7 𝑁 = (𝑈 normOpOLD 𝑊)
1210, 11syl6eqr 2812 . . . . . 6 (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = 𝑁)
1312fveq1d 6354 . . . . 5 (𝑤 = 𝑊 → ((𝑈 normOpOLD 𝑤)‘𝑡) = (𝑁𝑡))
1413breq1d 4814 . . . 4 (𝑤 = 𝑊 → (((𝑈 normOpOLD 𝑤)‘𝑡) < +∞ ↔ (𝑁𝑡) < +∞))
159, 14rabeqbidv 3335 . . 3 (𝑤 = 𝑊 → {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
16 df-blo 27910 . . 3 BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
17 ovex 6841 . . . . 5 (𝑈 LnOp 𝑊) ∈ V
188, 17eqeltri 2835 . . . 4 𝐿 ∈ V
1918rabex 4964 . . 3 {𝑡𝐿 ∣ (𝑁𝑡) < +∞} ∈ V
206, 15, 16, 19ovmpt2 6961 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 BLnOp 𝑊) = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
211, 20syl5eq 2806 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340   class class class wbr 4804  cfv 6049  (class class class)co 6813  +∞cpnf 10263   < clt 10266  NrmCVeccnv 27748   LnOp clno 27904   normOpOLD cnmoo 27905   BLnOp cblo 27906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-blo 27910
This theorem is referenced by:  isblo  27946  hhbloi  29070
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