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Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrshp4 | Structured version Visualization version GIF version |
Description: A kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.) |
Ref | Expression |
---|---|
lkrshp4.v | ⊢ 𝑉 = (Base‘𝑊) |
lkrshp4.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lkrshp4.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrshp4.k | ⊢ 𝐾 = (LKer‘𝑊) |
lkrshp4.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lkrshp4.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lkrshp4 | ⊢ (𝜑 → ((𝐾‘𝐺) ≠ 𝑉 ↔ (𝐾‘𝐺) ∈ 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkrshp4.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lkrshp4.h | . . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊) | |
3 | lkrshp4.f | . . . . 5 ⊢ 𝐹 = (LFnl‘𝑊) | |
4 | lkrshp4.k | . . . . 5 ⊢ 𝐾 = (LKer‘𝑊) | |
5 | lkrshp4.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lkrshp4.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | lkrshpor 37129 | . . . 4 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ∨ (𝐾‘𝐺) = 𝑉)) |
8 | 7 | orcomd 868 | . . 3 ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ∨ (𝐾‘𝐺) ∈ 𝐻)) |
9 | neor 3036 | . . 3 ⊢ (((𝐾‘𝐺) = 𝑉 ∨ (𝐾‘𝐺) ∈ 𝐻) ↔ ((𝐾‘𝐺) ≠ 𝑉 → (𝐾‘𝐺) ∈ 𝐻)) | |
10 | 8, 9 | sylib 217 | . 2 ⊢ (𝜑 → ((𝐾‘𝐺) ≠ 𝑉 → (𝐾‘𝐺) ∈ 𝐻)) |
11 | lveclmod 20378 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
12 | 5, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → 𝑊 ∈ LMod) |
14 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → (𝐾‘𝐺) ∈ 𝐻) | |
15 | 1, 2, 13, 14 | lshpne 37004 | . . 3 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → (𝐾‘𝐺) ≠ 𝑉) |
16 | 15 | ex 413 | . 2 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 → (𝐾‘𝐺) ≠ 𝑉)) |
17 | 10, 16 | impbid 211 | 1 ⊢ (𝜑 → ((𝐾‘𝐺) ≠ 𝑉 ↔ (𝐾‘𝐺) ∈ 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6426 Basecbs 16922 LModclmod 20133 LVecclvec 20374 LSHypclsh 36997 LFnlclfn 37079 LKerclk 37107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-tpos 8029 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-0g 17162 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-grp 18590 df-minusg 18591 df-sbg 18592 df-subg 18762 df-cntz 18933 df-lsm 19251 df-cmn 19398 df-abl 19399 df-mgp 19731 df-ur 19748 df-ring 19795 df-oppr 19872 df-dvdsr 19893 df-unit 19894 df-invr 19924 df-drng 20003 df-lmod 20135 df-lss 20204 df-lsp 20244 df-lvec 20375 df-lshyp 36999 df-lfl 37080 df-lkr 37108 |
This theorem is referenced by: lkrpssN 37185 dochkrshp3 39410 lcfl9a 39527 |
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