![]() |
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 29486 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 2777 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | lflf.d | . . 3 ⊢ 𝐷 = (Scalar‘𝑊) | |
4 | eqid 2777 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | lflf.k | . . 3 ⊢ 𝐾 = (Base‘𝐷) | |
6 | eqid 2777 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
7 | eqid 2777 | . . 3 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
8 | lflf.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | islfl 35208 | . 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 494 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 .rcmulr 16339 Scalarcsca 16341 ·𝑠 cvsca 16342 LFnlclfn 35205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-map 8142 df-lfl 35206 |
This theorem is referenced by: lflcl 35212 lfl1 35218 lfladdcl 35219 lfladdcom 35220 lfladdass 35221 lfladd0l 35222 lflnegl 35224 lflvscl 35225 lflvsdi1 35226 lflvsdi2 35227 lflvsass 35229 lfl0sc 35230 lfl1sc 35232 ellkr 35237 lkr0f 35242 lkrsc 35245 eqlkr2 35248 eqlkr3 35249 ldualvaddval 35279 ldualvsval 35286 |
Copyright terms: Public domain | W3C validator |