![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ellkr2 | Structured version Visualization version GIF version |
Description: Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.) |
Ref | Expression |
---|---|
lkrfval2.v | β’ π = (Baseβπ) |
lkrfval2.d | β’ π· = (Scalarβπ) |
lkrfval2.o | β’ 0 = (0gβπ·) |
lkrfval2.f | β’ πΉ = (LFnlβπ) |
lkrfval2.k | β’ πΎ = (LKerβπ) |
ellkr2.w | β’ (π β π β π) |
ellkr2.g | β’ (π β πΊ β πΉ) |
ellkr2.x | β’ (π β π β π) |
Ref | Expression |
---|---|
ellkr2 | β’ (π β (π β (πΎβπΊ) β (πΊβπ) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellkr2.w | . . 3 β’ (π β π β π) | |
2 | ellkr2.g | . . 3 β’ (π β πΊ β πΉ) | |
3 | lkrfval2.v | . . . 4 β’ π = (Baseβπ) | |
4 | lkrfval2.d | . . . 4 β’ π· = (Scalarβπ) | |
5 | lkrfval2.o | . . . 4 β’ 0 = (0gβπ·) | |
6 | lkrfval2.f | . . . 4 β’ πΉ = (LFnlβπ) | |
7 | lkrfval2.k | . . . 4 β’ πΎ = (LKerβπ) | |
8 | 3, 4, 5, 6, 7 | ellkr 38561 | . . 3 β’ ((π β π β§ πΊ β πΉ) β (π β (πΎβπΊ) β (π β π β§ (πΊβπ) = 0 ))) |
9 | 1, 2, 8 | syl2anc 583 | . 2 β’ (π β (π β (πΎβπΊ) β (π β π β§ (πΊβπ) = 0 ))) |
10 | ellkr2.x | . . 3 β’ (π β π β π) | |
11 | 10 | biantrurd 532 | . 2 β’ (π β ((πΊβπ) = 0 β (π β π β§ (πΊβπ) = 0 ))) |
12 | 9, 11 | bitr4d 282 | 1 β’ (π β (π β (πΎβπΊ) β (πΊβπ) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βcfv 6548 Basecbs 17180 Scalarcsca 17236 0gc0g 17421 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 8847 df-lfl 38530 df-lkr 38558 |
This theorem is referenced by: lclkrlem2f 40985 lclkrlem2n 40993 lcfrlem3 41017 lcfrlem25 41040 hdmapellkr 41387 hdmapip0 41388 hdmapinvlem1 41391 |
Copyright terms: Public domain | W3C validator |