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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 32060 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 2737 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 3 | lflf.d | . . 3 ⊢ 𝐷 = (Scalar‘𝑊) | |
| 4 | eqid 2737 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 5 | lflf.k | . . 3 ⊢ 𝐾 = (Base‘𝐷) | |
| 6 | eqid 2737 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
| 7 | eqid 2737 | . . 3 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
| 8 | lflf.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | islfl 39061 | . 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 1540 ∈ wcel 2108 ∀wral 3061 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 LFnlclfn 39058 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-lfl 39059 |
| This theorem is referenced by: lflcl 39065 lfl1 39071 lfladdcl 39072 lfladdcom 39073 lfladdass 39074 lfladd0l 39075 lflnegl 39077 lflvscl 39078 lflvsdi1 39079 lflvsdi2 39080 lflvsass 39082 lfl0sc 39083 lfl1sc 39085 ellkr 39090 lkr0f 39095 lkrsc 39098 eqlkr2 39101 eqlkr3 39102 ldualvaddval 39132 ldualvsval 39139 |
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