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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 29637 | . 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 2114 class class class wbr 5066 ‘cfv 6355 +∞cpnf 10672 < clt 10675 normopcnop 28722 LinOpclo 28724 BndLinOpcbo 28725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-bdop 29619 |
This theorem is referenced by: bdopf 29639 nmbdoplbi 29801 bdophmi 29809 lncnopbd 29814 nmopcoi 29872 bdophsi 29873 bdopcoi 29875 nmopcoadj0i 29880 unierri 29881 |
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