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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 30076 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 2736 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | lflf.d | . . 3 ⊢ 𝐷 = (Scalar‘𝑊) | |
4 | eqid 2736 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | lflf.k | . . 3 ⊢ 𝐾 = (Base‘𝐷) | |
6 | eqid 2736 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
7 | eqid 2736 | . . 3 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
8 | lflf.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | islfl 36760 | . 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 502 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 .rcmulr 16750 Scalarcsca 16752 ·𝑠 cvsca 16753 LFnlclfn 36757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-map 8488 df-lfl 36758 |
This theorem is referenced by: lflcl 36764 lfl1 36770 lfladdcl 36771 lfladdcom 36772 lfladdass 36773 lfladd0l 36774 lflnegl 36776 lflvscl 36777 lflvsdi1 36778 lflvsdi2 36779 lflvsass 36781 lfl0sc 36782 lfl1sc 36784 ellkr 36789 lkr0f 36794 lkrsc 36797 eqlkr2 36800 eqlkr3 36801 ldualvaddval 36831 ldualvsval 36838 |
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