Nearby theorems
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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrlsp2 | Structured version Visualization version GIF version | ||
| Description: The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 12-May-2014.) |
| Ref | Expression |
|---|---|
| lkrlsp2.v | β’ π = (Baseβπ) |
| lkrlsp2.n | β’ π = (LSpanβπ) |
| lkrlsp2.p | β’ β = (LSSumβπ) |
| lkrlsp2.f | β’ πΉ = (LFnlβπ) |
| lkrlsp2.k | β’ πΎ = (LKerβπ) |
| Ref | Expression |
|---|---|
| lkrlsp2 | β’ ((π β LVec β§ (π β π β§ πΊ β πΉ) β§ Β¬ π β (πΎβπΊ)) β ((πΎβπΊ) β (πβ{π})) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp2l 1197 | . . . . . 6 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ) β§ (πΊβπ) = (0gβ(Scalarβπ))) β π β π) | |
| 2 | simp3 1136 | . . . . . 6 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ) β§ (πΊβπ) = (0gβ(Scalarβπ))) β (πΊβπ) = (0gβ(Scalarβπ))) | |
| 3 | simp1 1134 | . . . . . . 7 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ) β§ (πΊβπ) = (0gβ(Scalarβπ))) β π β LVec) | |
| 4 | simp2r 1198 | . . . . . . 7 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ) β§ (πΊβπ) = (0gβ(Scalarβπ))) β πΊ β πΉ) | |
| 5 | lkrlsp2.v | . . . . . . . 8 β’ π = (Baseβπ) | |
| 6 | eqid 2730 | . . . . . . . 8 β’ (Scalarβπ) = (Scalarβπ) | |
| 7 | eqid 2730 | . . . . . . . 8 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
| 8 | lkrlsp2.f | . . . . . . . 8 β’ πΉ = (LFnlβπ) | |
| 9 | lkrlsp2.k | . . . . . . . 8 β’ πΎ = (LKerβπ) | |
| 10 | 5, 6, 7, 8, 9 | ellkr 38262 | . . . . . . 7 β’ ((π β LVec β§ πΊ β πΉ) β (π β (πΎβπΊ) β (π β π β§ (πΊβπ) = (0gβ(Scalarβπ))))) |
| 11 | 3, 4, 10 | syl2anc 582 | . . . . . 6 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ) β§ (πΊβπ) = (0gβ(Scalarβπ))) β (π β (πΎβπΊ) β (π β π β§ (πΊβπ) = (0gβ(Scalarβπ))))) |
| 12 | 1, 2, 11 | mpbir2and 709 | . . . . 5 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ) β§ (πΊβπ) = (0gβ(Scalarβπ))) β π β (πΎβπΊ)) |
| 13 | 12 | 3expia 1119 | . . . 4 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ)) β ((πΊβπ) = (0gβ(Scalarβπ)) β π β (πΎβπΊ))) |
| 14 | 13 | necon3bd 2952 | . . 3 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ)) β (Β¬ π β (πΎβπΊ) β (πΊβπ) β (0gβ(Scalarβπ)))) |
| 15 | 14 | 3impia 1115 | . 2 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ) β§ Β¬ π β (πΎβπΊ)) β (πΊβπ) β (0gβ(Scalarβπ))) |
| 16 | lkrlsp2.n | . . 3 β’ π = (LSpanβπ) | |
| 17 | lkrlsp2.p | . . 3 β’ β = (LSSumβπ) | |
| 18 | 6, 7, 5, 16, 17, 8, 9 | lkrlsp 38275 | . 2 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ) β§ (πΊβπ) β (0gβ(Scalarβπ))) β ((πΎβπΊ) β (πβ{π})) = π) |
| 19 | 15, 18 | syld3an3 1407 | 1 β’ ((π β LVec β§ (π β π β§ πΊ β πΉ) β§ Β¬ π β (πΎβπΊ)) β ((πΎβπΊ) β (πβ{π})) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 β wne 2938 {csn 4627 βcfv 6542 (class class class)co 7411 Basecbs 17148 Scalarcsca 17204 0gc0g 17389 LSSumclsm 19543 LSpanclspn 20726 LVecclvec 20857 LFnlclfn 38230 LKerclk 38258 |
| 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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| 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 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19039 df-cntz 19222 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-drng 20502 df-lmod 20616 df-lss 20687 df-lsp 20727 df-lvec 20858 df-lfl 38231 df-lkr 38259 |
| This theorem is referenced by: lkrlsp3 38277 |
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