![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > bdopf | Structured version Visualization version GIF version |
Description: A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bdopf | ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdopln 31549 | . 2 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) | |
2 | lnopf 31547 | . 2 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⟶wf 6529 ℋchba 30607 LinOpclo 30635 BndLinOpcbo 30636 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-hilex 30687 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-map 8817 df-lnop 31529 df-bdop 31530 |
This theorem is referenced by: nmopre 31558 nmophmi 31719 adjbdln 31771 nmopadjlem 31777 nmoptrii 31782 nmopcoi 31783 bdophsi 31784 bdophdi 31785 nmoptri2i 31787 adjcoi 31788 nmopcoadji 31789 unierri 31792 |
Copyright terms: Public domain | W3C validator |