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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 31419 | . 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 2105 ⟶wf 6539 ℂcc 11114 ℋchba 30454 LinFnclf 30489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-hilex 30534 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8828 df-lnfn 31383 |
This theorem is referenced by: lnfn0i 31577 lnfnaddi 31578 lnfnmuli 31579 lnfnsubi 31581 nmbdfnlbi 31584 nmcfnexi 31586 nmcfnlbi 31587 lnfnconi 31590 nlelshi 31595 nlelchi 31596 riesz3i 31597 riesz4i 31598 |
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