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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpkr | Structured version Visualization version GIF version |
Description: The kernel of functional πΊ is the hyperplane defining it. (Contributed by NM, 17-Jul-2014.) |
Ref | Expression |
---|---|
lshpkr.v | β’ π = (Baseβπ) |
lshpkr.a | β’ + = (+gβπ) |
lshpkr.n | β’ π = (LSpanβπ) |
lshpkr.p | β’ β = (LSSumβπ) |
lshpkr.h | β’ π» = (LSHypβπ) |
lshpkr.w | β’ (π β π β LVec) |
lshpkr.u | β’ (π β π β π») |
lshpkr.z | β’ (π β π β π) |
lshpkr.e | β’ (π β (π β (πβ{π})) = π) |
lshpkr.d | β’ π· = (Scalarβπ) |
lshpkr.k | β’ πΎ = (Baseβπ·) |
lshpkr.t | β’ Β· = ( Β·π βπ) |
lshpkr.g | β’ πΊ = (π₯ β π β¦ (β©π β πΎ βπ¦ β π π₯ = (π¦ + (π Β· π)))) |
lshpkr.l | β’ πΏ = (LKerβπ) |
Ref | Expression |
---|---|
lshpkr | β’ (π β (πΏβπΊ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpkr.v | . . . . 5 β’ π = (Baseβπ) | |
2 | eqid 2730 | . . . . 5 β’ (LFnlβπ) = (LFnlβπ) | |
3 | lshpkr.l | . . . . 5 β’ πΏ = (LKerβπ) | |
4 | lshpkr.w | . . . . . 6 β’ (π β π β LVec) | |
5 | lveclmod 20863 | . . . . . 6 β’ (π β LVec β π β LMod) | |
6 | 4, 5 | syl 17 | . . . . 5 β’ (π β π β LMod) |
7 | lshpkr.a | . . . . . 6 β’ + = (+gβπ) | |
8 | lshpkr.n | . . . . . 6 β’ π = (LSpanβπ) | |
9 | lshpkr.p | . . . . . 6 β’ β = (LSSumβπ) | |
10 | lshpkr.h | . . . . . 6 β’ π» = (LSHypβπ) | |
11 | lshpkr.u | . . . . . 6 β’ (π β π β π») | |
12 | lshpkr.z | . . . . . 6 β’ (π β π β π) | |
13 | lshpkr.e | . . . . . 6 β’ (π β (π β (πβ{π})) = π) | |
14 | lshpkr.d | . . . . . 6 β’ π· = (Scalarβπ) | |
15 | lshpkr.k | . . . . . 6 β’ πΎ = (Baseβπ·) | |
16 | lshpkr.t | . . . . . 6 β’ Β· = ( Β·π βπ) | |
17 | lshpkr.g | . . . . . 6 β’ πΊ = (π₯ β π β¦ (β©π β πΎ βπ¦ β π π₯ = (π¦ + (π Β· π)))) | |
18 | 1, 7, 8, 9, 10, 4, 11, 12, 13, 14, 15, 16, 17, 2 | lshpkrcl 38291 | . . . . 5 β’ (π β πΊ β (LFnlβπ)) |
19 | 1, 2, 3, 6, 18 | lkrssv 38271 | . . . 4 β’ (π β (πΏβπΊ) β π) |
20 | 19 | sseld 3982 | . . 3 β’ (π β (π£ β (πΏβπΊ) β π£ β π)) |
21 | eqid 2730 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
22 | 21, 10, 6, 11 | lshplss 38156 | . . . . 5 β’ (π β π β (LSubSpβπ)) |
23 | 1, 21 | lssel 20694 | . . . . 5 β’ ((π β (LSubSpβπ) β§ π£ β π) β π£ β π) |
24 | 22, 23 | sylan 578 | . . . 4 β’ ((π β§ π£ β π) β π£ β π) |
25 | 24 | ex 411 | . . 3 β’ (π β (π£ β π β π£ β π)) |
26 | eqid 2730 | . . . . . . . 8 β’ (0gβπ·) = (0gβπ·) | |
27 | 1, 14, 26, 2, 3 | ellkr 38264 | . . . . . . 7 β’ ((π β LVec β§ πΊ β (LFnlβπ)) β (π£ β (πΏβπΊ) β (π£ β π β§ (πΊβπ£) = (0gβπ·)))) |
28 | 4, 18, 27 | syl2anc 582 | . . . . . 6 β’ (π β (π£ β (πΏβπΊ) β (π£ β π β§ (πΊβπ£) = (0gβπ·)))) |
29 | 28 | baibd 538 | . . . . 5 β’ ((π β§ π£ β π) β (π£ β (πΏβπΊ) β (πΊβπ£) = (0gβπ·))) |
30 | 4 | adantr 479 | . . . . . 6 β’ ((π β§ π£ β π) β π β LVec) |
31 | 11 | adantr 479 | . . . . . 6 β’ ((π β§ π£ β π) β π β π») |
32 | 12 | adantr 479 | . . . . . 6 β’ ((π β§ π£ β π) β π β π) |
33 | simpr 483 | . . . . . 6 β’ ((π β§ π£ β π) β π£ β π) | |
34 | 13 | adantr 479 | . . . . . 6 β’ ((π β§ π£ β π) β (π β (πβ{π})) = π) |
35 | 1, 7, 8, 9, 10, 30, 31, 32, 33, 34, 14, 15, 16, 26, 17 | lshpkrlem1 38285 | . . . . 5 β’ ((π β§ π£ β π) β (π£ β π β (πΊβπ£) = (0gβπ·))) |
36 | 29, 35 | bitr4d 281 | . . . 4 β’ ((π β§ π£ β π) β (π£ β (πΏβπΊ) β π£ β π)) |
37 | 36 | ex 411 | . . 3 β’ (π β (π£ β π β (π£ β (πΏβπΊ) β π£ β π))) |
38 | 20, 25, 37 | pm5.21ndd 378 | . 2 β’ (π β (π£ β (πΏβπΊ) β π£ β π)) |
39 | 38 | eqrdv 2728 | 1 β’ (π β (πΏβπΊ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1539 β wcel 2104 βwrex 3068 {csn 4629 β¦ cmpt 5232 βcfv 6544 β©crio 7368 (class class class)co 7413 Basecbs 17150 +gcplusg 17203 Scalarcsca 17206 Β·π cvsca 17207 0gc0g 17391 LSSumclsm 19545 LModclmod 20616 LSubSpclss 20688 LSpanclspn 20728 LVecclvec 20859 LSHypclsh 38150 LFnlclfn 38232 LKerclk 38260 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-2 12281 df-3 12282 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-0g 17393 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18708 df-grp 18860 df-minusg 18861 df-sbg 18862 df-subg 19041 df-cntz 19224 df-lsm 19547 df-cmn 19693 df-abl 19694 df-mgp 20031 df-rng 20049 df-ur 20078 df-ring 20131 df-oppr 20227 df-dvdsr 20250 df-unit 20251 df-invr 20281 df-drng 20504 df-lmod 20618 df-lss 20689 df-lsp 20729 df-lvec 20860 df-lshyp 38152 df-lfl 38233 df-lkr 38261 |
This theorem is referenced by: lshpkrex 38293 dochsnkr2 40649 |
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