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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lflcl | Structured version Visualization version GIF version | ||
| Description: A linear functional value is a scalar. (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 |
|---|---|
| lflcl | β’ ((π β π β§ πΊ β πΉ β§ π β π) β (πΊβπ) β πΎ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lflf.d | . . . 4 β’ π· = (Scalarβπ) | |
| 2 | lflf.k | . . . 4 β’ πΎ = (Baseβπ·) | |
| 3 | lflf.v | . . . 4 β’ π = (Baseβπ) | |
| 4 | lflf.f | . . . 4 β’ πΉ = (LFnlβπ) | |
| 5 | 1, 2, 3, 4 | lflf 38238 | . . 3 β’ ((π β π β§ πΊ β πΉ) β πΊ:πβΆπΎ) |
| 6 | 5 | 3adant3 1130 | . 2 β’ ((π β π β§ πΊ β πΉ β§ π β π) β πΊ:πβΆπΎ) |
| 7 | simp3 1136 | . 2 β’ ((π β π β§ πΊ β πΉ β§ π β π) β π β π) | |
| 8 | 6, 7 | ffvelcdmd 7088 | 1 β’ ((π β π β§ πΊ β πΉ β§ π β π) β (πΊβπ) β πΎ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1085 = wceq 1539 β wcel 2104 βΆwf 6540 βcfv 6544 Basecbs 17150 Scalarcsca 17206 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: lfl0 38240 lfladd 38241 lflsub 38242 lflmul 38243 lfl1 38245 lfladdcl 38246 lflnegcl 38250 lflvscl 38252 lkrsc 38272 eqlkr 38274 eqlkr3 38276 lkrlsp 38277 ldualvsubval 38332 dochkr1 40654 dochkr1OLDN 40655 lcfl7lem 40675 lclkrlem2m 40695 lclkrlem2o 40697 lclkrlem2p 40698 lcfrlem1 40718 lcfrlem2 40719 lcfrlem3 40720 lcfrlem29 40747 lcfrlem31 40749 lcfrlem33 40751 lcdvbasecl 40772 |
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