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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrssv | Structured version Visualization version GIF version | ||
| Description: The kernel of a linear functional is a set of vectors. (Contributed by NM, 1-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| lkrssv.v | ⊢ 𝑉 = (Base‘𝑊) | 
| lkrssv.f | ⊢ 𝐹 = (LFnl‘𝑊) | 
| lkrssv.k | ⊢ 𝐾 = (LKer‘𝑊) | 
| lkrssv.w | ⊢ (𝜑 → 𝑊 ∈ LMod) | 
| lkrssv.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) | 
| Ref | Expression | 
|---|---|
| lkrssv | ⊢ (𝜑 → (𝐾‘𝐺) ⊆ 𝑉) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lkrssv.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
| 2 | lkrssv.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
| 3 | lkrssv.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 4 | lkrssv.k | . . . 4 ⊢ 𝐾 = (LKer‘𝑊) | |
| 5 | eqid 2736 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
| 6 | 3, 4, 5 | lkrlss 39097 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ∈ (LSubSp‘𝑊)) | 
| 7 | 1, 2, 6 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐾‘𝐺) ∈ (LSubSp‘𝑊)) | 
| 8 | lkrssv.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
| 9 | 8, 5 | lssss 20935 | . 2 ⊢ ((𝐾‘𝐺) ∈ (LSubSp‘𝑊) → (𝐾‘𝐺) ⊆ 𝑉) | 
| 10 | 7, 9 | syl 17 | 1 ⊢ (𝜑 → (𝐾‘𝐺) ⊆ 𝑉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ‘cfv 6560 Basecbs 17248 LModclmod 20859 LSubSpclss 20930 LFnlclfn 39059 LKerclk 39087 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-sbg 18957 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-lmod 20861 df-lss 20931 df-lfl 39060 df-lkr 39088 | 
| This theorem is referenced by: lkrscss 39100 lkrlsp3 39106 lshpkr 39119 lfl1dim 39123 lfl1dim2N 39124 lkrpssN 39165 dochlkr 41388 dochkrsat 41458 dochkrsat2 41459 dochsnkrlem1 41472 dochsnkr 41475 dochfln0 41480 dochkr1 41481 dochkr1OLDN 41482 lcfl4N 41498 lcfl5 41499 lcfl6lem 41501 lcfl6 41503 lcfl9a 41508 lclkrlem2s 41528 lclkrlem2v 41531 lclkrslem1 41540 lclkrslem2 41541 lcfrvalsnN 41544 lcfrlem4 41548 lcfrlem5 41549 lcfrlem6 41550 lcfrlem16 41561 lcfrlem26 41571 lcfrlem36 41581 lcfr 41588 mapdsn 41644 mapdrvallem2 41648 mapd0 41668 hdmaplkr 41916 | 
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