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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 29745 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 2818 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | lflf.d | . . 3 ⊢ 𝐷 = (Scalar‘𝑊) | |
4 | eqid 2818 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | lflf.k | . . 3 ⊢ 𝐾 = (Base‘𝐷) | |
6 | eqid 2818 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
7 | eqid 2818 | . . 3 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
8 | lflf.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | islfl 36076 | . 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 499 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 LFnlclfn 36073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-lfl 36074 |
This theorem is referenced by: lflcl 36080 lfl1 36086 lfladdcl 36087 lfladdcom 36088 lfladdass 36089 lfladd0l 36090 lflnegl 36092 lflvscl 36093 lflvsdi1 36094 lflvsdi2 36095 lflvsass 36097 lfl0sc 36098 lfl1sc 36100 ellkr 36105 lkr0f 36110 lkrsc 36113 eqlkr2 36116 eqlkr3 36117 ldualvaddval 36147 ldualvsval 36154 |
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