![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lflf | Structured version Visualization version GIF version |
Description: A linear functional is a function from vectors to scalars. (lnfnfi 31923 analog.) (Contributed by NM, 15-Apr-2014.) |
Ref | Expression |
---|---|
lflf.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lflf.k | ⊢ 𝐾 = (Base‘𝐷) |
lflf.v | ⊢ 𝑉 = (Base‘𝑊) |
lflf.f | ⊢ 𝐹 = (LFnl‘𝑊) |
Ref | Expression |
---|---|
lflf | ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lflf.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | eqid 2725 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | lflf.d | . . 3 ⊢ 𝐷 = (Scalar‘𝑊) | |
4 | eqid 2725 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | lflf.k | . . 3 ⊢ 𝐾 = (Base‘𝐷) | |
6 | eqid 2725 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
7 | eqid 2725 | . . 3 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
8 | lflf.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | islfl 38662 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝐺 ∈ 𝐹 ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘𝐷)(𝐺‘𝑥))(+g‘𝐷)(𝐺‘𝑦))))) |
10 | 9 | simprbda 497 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 +gcplusg 17236 .rcmulr 17237 Scalarcsca 17239 ·𝑠 cvsca 17240 LFnlclfn 38659 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-map 8847 df-lfl 38660 |
This theorem is referenced by: lflcl 38666 lfl1 38672 lfladdcl 38673 lfladdcom 38674 lfladdass 38675 lfladd0l 38676 lflnegl 38678 lflvscl 38679 lflvsdi1 38680 lflvsdi2 38681 lflvsass 38683 lfl0sc 38684 lfl1sc 38686 ellkr 38691 lkr0f 38696 lkrsc 38699 eqlkr2 38702 eqlkr3 38703 ldualvaddval 38733 ldualvsval 38740 |
Copyright terms: Public domain | W3C validator |