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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrf0 | Structured version Visualization version GIF version | ||
| Description: The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.) |
| Ref | Expression |
|---|---|
| lkrf0.d | β’ π· = (Scalarβπ) |
| lkrf0.o | β’ 0 = (0gβπ·) |
| lkrf0.f | β’ πΉ = (LFnlβπ) |
| lkrf0.k | β’ πΎ = (LKerβπ) |
| Ref | Expression |
|---|---|
| lkrf0 | β’ ((π β π β§ πΊ β πΉ β§ π β (πΎβπΊ)) β (πΊβπ) = 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2728 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
| 2 | lkrf0.d | . . . 4 β’ π· = (Scalarβπ) | |
| 3 | lkrf0.o | . . . 4 β’ 0 = (0gβπ·) | |
| 4 | lkrf0.f | . . . 4 β’ πΉ = (LFnlβπ) | |
| 5 | lkrf0.k | . . . 4 β’ πΎ = (LKerβπ) | |
| 6 | 1, 2, 3, 4, 5 | ellkr 38561 | . . 3 β’ ((π β π β§ πΊ β πΉ) β (π β (πΎβπΊ) β (π β (Baseβπ) β§ (πΊβπ) = 0 ))) |
| 7 | 6 | simplbda 499 | . 2 β’ (((π β π β§ πΊ β πΉ) β§ π β (πΎβπΊ)) β (πΊβπ) = 0 ) |
| 8 | 7 | 3impa 1108 | 1 β’ ((π β π β§ πΊ β πΉ β§ π β (πΎβπΊ)) β (πΊβπ) = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 βcfv 6548 Basecbs 17179 Scalarcsca 17235 0gc0g 17420 LFnlclfn 38529 LKerclk 38557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8846 df-lfl 38530 df-lkr 38558 |
| This theorem is referenced by: lkrlss 38567 lkrshp 38577 lkrin 38636 lcfrlem12N 41027 |
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