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Definition df-blo 27729
 Description: Define the class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
Assertion
Ref Expression
df-blo BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
Distinct variable group:   𝑢,𝑡,𝑤

Detailed syntax breakdown of Definition df-blo
StepHypRef Expression
1 cblo 27725 . 2 class BLnOp
2 vu . . 3 setvar 𝑢
3 vw . . 3 setvar 𝑤
4 cnv 27567 . . 3 class NrmCVec
5 vt . . . . . . 7 setvar 𝑡
65cv 1522 . . . . . 6 class 𝑡
72cv 1522 . . . . . . 7 class 𝑢
83cv 1522 . . . . . . 7 class 𝑤
9 cnmoo 27724 . . . . . . 7 class normOpOLD
107, 8, 9co 6690 . . . . . 6 class (𝑢 normOpOLD 𝑤)
116, 10cfv 5926 . . . . 5 class ((𝑢 normOpOLD 𝑤)‘𝑡)
12 cpnf 10109 . . . . 5 class +∞
13 clt 10112 . . . . 5 class <
1411, 12, 13wbr 4685 . . . 4 wff ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞
15 clno 27723 . . . . 5 class LnOp
167, 8, 15co 6690 . . . 4 class (𝑢 LnOp 𝑤)
1714, 5, 16crab 2945 . . 3 class {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞}
182, 3, 4, 4, 17cmpt2 6692 . 2 class (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
191, 18wceq 1523 1 wff BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
 Colors of variables: wff setvar class This definition is referenced by:  bloval  27764
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