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Mirrors > Home > HSE Home > Th. List > lnopfi | Structured version Visualization version GIF version |
Description: A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopl.1 | ⊢ 𝑇 ∈ LinOp |
Ref | Expression |
---|---|
lnopfi | ⊢ 𝑇: ℋ⟶ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnopl.1 | . 2 ⊢ 𝑇 ∈ LinOp | |
2 | lnopf 31887 | . 2 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑇: ℋ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ⟶wf 6558 ℋchba 30947 LinOpclo 30975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-hilex 31027 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-map 8866 df-lnop 31869 |
This theorem is referenced by: lnopaddi 31999 lnopsubi 32002 hoddii 32017 nmlnop0iALT 32023 nmlnopgt0i 32025 lnopmi 32028 lnophsi 32029 lnophdi 32030 lnopcoi 32031 lnopco0i 32032 lnopeq0lem1 32033 lnopeq0i 32035 lnopeqi 32036 lnopunilem1 32038 lnopunilem2 32039 lnophmlem2 32045 lnophmi 32046 nmbdoplbi 32052 nmcopexi 32055 nmcoplbi 32056 lnopconi 32062 imaelshi 32086 rnelshi 32087 cnlnadjlem2 32096 cnlnadjlem6 32100 cnlnadjlem7 32101 cnlnadjeui 32105 nmopcoi 32123 bdopcoi 32126 hmopidmchi 32179 hmopidmpji 32180 |
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