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Mirrors > Home > HSE Home > Th. List > bdopln | Structured version Visualization version GIF version |
Description: A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bdopln | ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elbdop 29631 | . 2 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞)) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 +∞cpnf 10666 < clt 10669 normopcnop 28716 LinOpclo 28718 BndLinOpcbo 28719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-iota 6309 df-fv 6358 df-bdop 29613 |
This theorem is referenced by: bdopf 29633 nmbdoplbi 29795 bdophmi 29803 lncnopbd 29808 nmopcoi 29866 bdophsi 29867 bdopcoi 29869 nmopcoadj0i 29874 unierri 29875 |
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