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Mirrors > Home > HSE Home > Th. List > lnfnfi | Structured version Visualization version GIF version |
Description: A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnfnl.1 | ⊢ 𝑇 ∈ LinFn |
Ref | Expression |
---|---|
lnfnfi | ⊢ 𝑇: ℋ⟶ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnfnl.1 | . 2 ⊢ 𝑇 ∈ LinFn | |
2 | lnfnf 29319 | . 2 ⊢ (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝑇: ℋ⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ⟶wf 6133 ℂcc 10272 ℋchba 28352 LinFnclf 28387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-hilex 28432 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-map 8144 df-lnfn 29283 |
This theorem is referenced by: lnfn0i 29477 lnfnaddi 29478 lnfnmuli 29479 lnfnsubi 29481 nmbdfnlbi 29484 nmcfnexi 29486 nmcfnlbi 29487 lnfnconi 29490 nlelshi 29495 nlelchi 29496 riesz3i 29497 riesz4i 29498 |
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