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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 31878 | . 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 2108 ⟶wf 6557 ℋchba 30938 LinOpclo 30966 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-hilex 31018 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-lnop 31860 |
| This theorem is referenced by: lnopaddi 31990 lnopsubi 31993 hoddii 32008 nmlnop0iALT 32014 nmlnopgt0i 32016 lnopmi 32019 lnophsi 32020 lnophdi 32021 lnopcoi 32022 lnopco0i 32023 lnopeq0lem1 32024 lnopeq0i 32026 lnopeqi 32027 lnopunilem1 32029 lnopunilem2 32030 lnophmlem2 32036 lnophmi 32037 nmbdoplbi 32043 nmcopexi 32046 nmcoplbi 32047 lnopconi 32053 imaelshi 32077 rnelshi 32078 cnlnadjlem2 32087 cnlnadjlem6 32091 cnlnadjlem7 32092 cnlnadjeui 32096 nmopcoi 32114 bdopcoi 32117 hmopidmchi 32170 hmopidmpji 32171 |
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