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Mirrors > Home > HSE Home > Th. List > lnfnf | Structured version Visualization version GIF version |
Description: A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnfnf | ⊢ (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellnfn 29818 | . 2 ⊢ (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ⟶wf 6336 ‘cfv 6340 (class class class)co 7171 ℂcc 10614 + caddc 10619 · cmul 10621 ℋchba 28854 +ℎ cva 28855 ·ℎ csm 28856 LinFnclf 28889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-hilex 28934 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3683 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-fv 6348 df-ov 7174 df-oprab 7175 df-mpo 7176 df-map 8440 df-lnfn 29783 |
This theorem is referenced by: nmfn0 29922 lnfnfi 29976 rnbra 30042 |
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