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Mathbox for Norm Megill |
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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 31559 analog.) (Contributed by NM, 15-Apr-2014.) |
Ref | Expression |
---|---|
lflf.d | β’ π· = (Scalarβπ) |
lflf.k | β’ πΎ = (Baseβπ·) |
lflf.v | β’ π = (Baseβπ) |
lflf.f | β’ πΉ = (LFnlβπ) |
Ref | Expression |
---|---|
lflf | β’ ((π β π β§ πΊ β πΉ) β πΊ:πβΆπΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lflf.v | . . 3 β’ π = (Baseβπ) | |
2 | eqid 2730 | . . 3 β’ (+gβπ) = (+gβπ) | |
3 | lflf.d | . . 3 β’ π· = (Scalarβπ) | |
4 | eqid 2730 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
5 | lflf.k | . . 3 β’ πΎ = (Baseβπ·) | |
6 | eqid 2730 | . . 3 β’ (+gβπ·) = (+gβπ·) | |
7 | eqid 2730 | . . 3 β’ (.rβπ·) = (.rβπ·) | |
8 | lflf.f | . . 3 β’ πΉ = (LFnlβπ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | islfl 38235 | . 2 β’ (π β π β (πΊ β πΉ β (πΊ:πβΆπΎ β§ βπ β πΎ βπ₯ β π βπ¦ β π (πΊβ((π( Β·π βπ)π₯)(+gβπ)π¦)) = ((π(.rβπ·)(πΊβπ₯))(+gβπ·)(πΊβπ¦))))) |
10 | 9 | simprbda 497 | 1 β’ ((π β π β§ πΊ β πΉ) β πΊ:πβΆπΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 βwral 3059 βΆwf 6540 βcfv 6544 (class class class)co 7413 Basecbs 17150 +gcplusg 17203 .rcmulr 17204 Scalarcsca 17206 Β·π cvsca 17207 LFnlclfn 38232 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7416 df-oprab 7417 df-mpo 7418 df-map 8826 df-lfl 38233 |
This theorem is referenced by: lflcl 38239 lfl1 38245 lfladdcl 38246 lfladdcom 38247 lfladdass 38248 lfladd0l 38249 lflnegl 38251 lflvscl 38252 lflvsdi1 38253 lflvsdi2 38254 lflvsass 38256 lfl0sc 38257 lfl1sc 38259 ellkr 38264 lkr0f 38269 lkrsc 38272 eqlkr2 38275 eqlkr3 38276 ldualvaddval 38306 ldualvsval 38313 |
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