Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrcl | Structured version Visualization version GIF version | ||
| Description: A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.) |
| Ref | Expression |
|---|---|
| lkrcl.v | β’ π = (Baseβπ) |
| lkrcl.f | β’ πΉ = (LFnlβπ) |
| lkrcl.k | β’ πΎ = (LKerβπ) |
| Ref | Expression |
|---|---|
| lkrcl | β’ ((π β π β§ πΊ β πΉ β§ π β (πΎβπΊ)) β π β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lkrcl.v | . . . 4 β’ π = (Baseβπ) | |
| 2 | eqid 2732 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
| 3 | eqid 2732 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
| 4 | lkrcl.f | . . . 4 β’ πΉ = (LFnlβπ) | |
| 5 | lkrcl.k | . . . 4 β’ πΎ = (LKerβπ) | |
| 6 | 1, 2, 3, 4, 5 | ellkr 38262 | . . 3 β’ ((π β π β§ πΊ β πΉ) β (π β (πΎβπΊ) β (π β π β§ (πΊβπ) = (0gβ(Scalarβπ))))) |
| 7 | 6 | simprbda 499 | . 2 β’ (((π β π β§ πΊ β πΉ) β§ π β (πΎβπΊ)) β π β π) |
| 8 | 7 | 3impa 1110 | 1 β’ ((π β π β§ πΊ β πΉ β§ π β (πΎβπΊ)) β π β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6543 Basecbs 17148 Scalarcsca 17204 0gc0g 17389 LFnlclfn 38230 LKerclk 38258 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-map 8824 df-lfl 38231 df-lkr 38259 |
| This theorem is referenced by: lkrlss 38268 lkrin 38337 |
| Copyright terms: Public domain | W3C validator |