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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 32069 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 2734 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | lflf.d | . . 3 ⊢ 𝐷 = (Scalar‘𝑊) | |
4 | eqid 2734 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | lflf.k | . . 3 ⊢ 𝐾 = (Base‘𝐷) | |
6 | eqid 2734 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
7 | eqid 2734 | . . 3 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
8 | lflf.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | islfl 39041 | . 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 498 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 LFnlclfn 39038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-map 8866 df-lfl 39039 |
This theorem is referenced by: lflcl 39045 lfl1 39051 lfladdcl 39052 lfladdcom 39053 lfladdass 39054 lfladd0l 39055 lflnegl 39057 lflvscl 39058 lflvsdi1 39059 lflvsdi2 39060 lflvsass 39062 lfl0sc 39063 lfl1sc 39065 ellkr 39070 lkr0f 39075 lkrsc 39078 eqlkr2 39081 eqlkr3 39082 ldualvaddval 39112 ldualvsval 39119 |
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