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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 31689 | . 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 2098 ⟶wf 6549 ℋchba 30749 LinOpclo 30777 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-hilex 30829 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8853 df-lnop 31671 |
This theorem is referenced by: lnopaddi 31801 lnopsubi 31804 hoddii 31819 nmlnop0iALT 31825 nmlnopgt0i 31827 lnopmi 31830 lnophsi 31831 lnophdi 31832 lnopcoi 31833 lnopco0i 31834 lnopeq0lem1 31835 lnopeq0i 31837 lnopeqi 31838 lnopunilem1 31840 lnopunilem2 31841 lnophmlem2 31847 lnophmi 31848 nmbdoplbi 31854 nmcopexi 31857 nmcoplbi 31858 lnopconi 31864 imaelshi 31888 rnelshi 31889 cnlnadjlem2 31898 cnlnadjlem6 31902 cnlnadjlem7 31903 cnlnadjeui 31907 nmopcoi 31925 bdopcoi 31928 hmopidmchi 31981 hmopidmpji 31982 |
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