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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 31294 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 2733 | . . 3 β’ (+gβπ) = (+gβπ) | |
3 | lflf.d | . . 3 β’ π· = (Scalarβπ) | |
4 | eqid 2733 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
5 | lflf.k | . . 3 β’ πΎ = (Baseβπ·) | |
6 | eqid 2733 | . . 3 β’ (+gβπ·) = (+gβπ·) | |
7 | eqid 2733 | . . 3 β’ (.rβπ·) = (.rβπ·) | |
8 | lflf.f | . . 3 β’ πΉ = (LFnlβπ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | islfl 37930 | . 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 500 | 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 397 = wceq 1542 β wcel 2107 βwral 3062 βΆwf 6540 βcfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 .rcmulr 17198 Scalarcsca 17200 Β·π cvsca 17201 LFnlclfn 37927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 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 7412 df-oprab 7413 df-mpo 7414 df-map 8822 df-lfl 37928 |
This theorem is referenced by: lflcl 37934 lfl1 37940 lfladdcl 37941 lfladdcom 37942 lfladdass 37943 lfladd0l 37944 lflnegl 37946 lflvscl 37947 lflvsdi1 37948 lflvsdi2 37949 lflvsass 37951 lfl0sc 37952 lfl1sc 37954 ellkr 37959 lkr0f 37964 lkrsc 37967 eqlkr2 37970 eqlkr3 37971 ldualvaddval 38001 ldualvsval 38008 |
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