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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 6896 | . . 3 ⊢ (𝑡 = 𝑇 → (normop‘𝑡) = (normop‘𝑇)) | |
2 | 1 | breq1d 5159 | . 2 ⊢ (𝑡 = 𝑇 → ((normop‘𝑡) < +∞ ↔ (normop‘𝑇) < +∞)) |
3 | df-bdop 31724 | . 2 ⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (normop‘𝑡) < +∞} | |
4 | 2, 3 | elrab2 3682 | 1 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 +∞cpnf 11277 < clt 11280 normopcnop 30827 LinOpclo 30829 BndLinOpcbo 30830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-bdop 31724 |
This theorem is referenced by: bdopln 31743 nmopre 31752 elbdop2 31753 0bdop 31875 |
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