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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 29641 | . 2 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) | |
2 | lnopf 29639 | . 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 2113 ⟶wf 6354 ℋchba 28699 LinOpclo 28727 BndLinOpcbo 28728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-hilex 28779 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-map 8411 df-lnop 29621 df-bdop 29622 |
This theorem is referenced by: nmopre 29650 nmophmi 29811 adjbdln 29863 nmopadjlem 29869 nmoptrii 29874 nmopcoi 29875 bdophsi 29876 bdophdi 29877 nmoptri2i 29879 adjcoi 29880 nmopcoadji 29881 unierri 29884 |
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