Nearby theorems
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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrshpor | Structured version Visualization version GIF version | ||
| Description: The kernel of a functional is either a hyperplane or the full vector space. (Contributed by NM, 7-Oct-2014.) |
| Ref | Expression |
|---|---|
| lkrshpor.v | β’ π = (Baseβπ) |
| lkrshpor.h | β’ π» = (LSHypβπ) |
| lkrshpor.f | β’ πΉ = (LFnlβπ) |
| lkrshpor.k | β’ πΎ = (LKerβπ) |
| lkrshpor.w | β’ (π β π β LVec) |
| lkrshpor.g | β’ (π β πΊ β πΉ) |
| Ref | Expression |
|---|---|
| lkrshpor | β’ (π β ((πΎβπΊ) β π» β¨ (πΎβπΊ) = π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lkrshpor.w | . . . . . 6 β’ (π β π β LVec) | |
| 2 | lveclmod 20709 | . . . . . 6 β’ (π β LVec β π β LMod) | |
| 3 | 1, 2 | syl 17 | . . . . 5 β’ (π β π β LMod) |
| 4 | lkrshpor.g | . . . . 5 β’ (π β πΊ β πΉ) | |
| 5 | eqid 2732 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 6 | eqid 2732 | . . . . . 6 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
| 7 | lkrshpor.v | . . . . . 6 β’ π = (Baseβπ) | |
| 8 | lkrshpor.f | . . . . . 6 β’ πΉ = (LFnlβπ) | |
| 9 | lkrshpor.k | . . . . . 6 β’ πΎ = (LKerβπ) | |
| 10 | 5, 6, 7, 8, 9 | lkr0f 37952 | . . . . 5 β’ ((π β LMod β§ πΊ β πΉ) β ((πΎβπΊ) = π β πΊ = (π Γ {(0gβ(Scalarβπ))}))) |
| 11 | 3, 4, 10 | syl2anc 584 | . . . 4 β’ (π β ((πΎβπΊ) = π β πΊ = (π Γ {(0gβ(Scalarβπ))}))) |
| 12 | 11 | biimpar 478 | . . 3 β’ ((π β§ πΊ = (π Γ {(0gβ(Scalarβπ))})) β (πΎβπΊ) = π) |
| 13 | 12 | olcd 872 | . 2 β’ ((π β§ πΊ = (π Γ {(0gβ(Scalarβπ))})) β ((πΎβπΊ) β π» β¨ (πΎβπΊ) = π)) |
| 14 | 1 | adantr 481 | . . . 4 β’ ((π β§ πΊ β (π Γ {(0gβ(Scalarβπ))})) β π β LVec) |
| 15 | 4 | adantr 481 | . . . 4 β’ ((π β§ πΊ β (π Γ {(0gβ(Scalarβπ))})) β πΊ β πΉ) |
| 16 | simpr 485 | . . . 4 β’ ((π β§ πΊ β (π Γ {(0gβ(Scalarβπ))})) β πΊ β (π Γ {(0gβ(Scalarβπ))})) | |
| 17 | lkrshpor.h | . . . . 5 β’ π» = (LSHypβπ) | |
| 18 | 7, 5, 6, 17, 8, 9 | lkrshp 37963 | . . . 4 β’ ((π β LVec β§ πΊ β πΉ β§ πΊ β (π Γ {(0gβ(Scalarβπ))})) β (πΎβπΊ) β π») |
| 19 | 14, 15, 16, 18 | syl3anc 1371 | . . 3 β’ ((π β§ πΊ β (π Γ {(0gβ(Scalarβπ))})) β (πΎβπΊ) β π») |
| 20 | 19 | orcd 871 | . 2 β’ ((π β§ πΊ β (π Γ {(0gβ(Scalarβπ))})) β ((πΎβπΊ) β π» β¨ (πΎβπΊ) = π)) |
| 21 | 13, 20 | pm2.61dane 3029 | 1 β’ (π β ((πΎβπΊ) β π» β¨ (πΎβπΊ) = π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 396 β¨ wo 845 = wceq 1541 β wcel 2106 β wne 2940 {csn 4627 Γ cxp 5673 βcfv 6540 Basecbs 17140 Scalarcsca 17196 0gc0g 17381 LModclmod 20463 LVecclvec 20705 LSHypclsh 37833 LFnlclfn 37915 LKerclk 37943 |
| 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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-lsm 19498 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-drng 20309 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lvec 20706 df-lshyp 37835 df-lfl 37916 df-lkr 37944 |
| This theorem is referenced by: lkrshp4 37966 lkrpssN 38021 dochlkr 40244 dochkrshp 40245 lclkrlem2e 40370 lclkrlem2h 40373 lclkrlem2s 40384 |
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