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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 6191 | . . 3 ⊢ (𝑡 = 𝑇 → (normop‘𝑡) = (normop‘𝑇)) | |
2 | 1 | breq1d 4663 | . 2 ⊢ (𝑡 = 𝑇 → ((normop‘𝑡) < +∞ ↔ (normop‘𝑇) < +∞)) |
3 | df-bdop 28701 | . 2 ⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (normop‘𝑡) < +∞} | |
4 | 2, 3 | elrab2 3366 | 1 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 +∞cpnf 10071 < clt 10074 normopcnop 27802 LinOpclo 27804 BndLinOpcbo 27805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-bdop 28701 |
This theorem is referenced by: bdopln 28720 nmopre 28729 elbdop2 28730 0bdop 28852 |
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