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| Mirrors > Home > HSE Home > Th. List > elbdop | Structured version Visualization version GIF version | ||
| Description: Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| elbdop | ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . . 3 ⊢ (𝑡 = 𝑇 → (normop‘𝑡) = (normop‘𝑇)) | |
| 2 | 1 | breq1d 5153 | . 2 ⊢ (𝑡 = 𝑇 → ((normop‘𝑡) < +∞ ↔ (normop‘𝑇) < +∞)) |
| 3 | df-bdop 31861 | . 2 ⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (normop‘𝑡) < +∞} | |
| 4 | 2, 3 | elrab2 3695 | 1 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 +∞cpnf 11292 < clt 11295 normopcnop 30964 LinOpclo 30966 BndLinOpcbo 30967 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-bdop 31861 |
| This theorem is referenced by: bdopln 31880 nmopre 31889 elbdop2 31890 0bdop 32012 |
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