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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 | ⊢ 𝑉 = (Base‘𝑊) |
lkrshpor.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lkrshpor.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrshpor.k | ⊢ 𝐾 = (LKer‘𝑊) |
lkrshpor.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lkrshpor.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lkrshpor | ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ∨ (𝐾‘𝐺) = 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkrshpor.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | lveclmod 19319 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | lkrshpor.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
5 | eqid 2771 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2771 | . . . . . 6 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
7 | lkrshpor.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
8 | lkrshpor.f | . . . . . 6 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | lkrshpor.k | . . . . . 6 ⊢ 𝐾 = (LKer‘𝑊) | |
10 | 5, 6, 7, 8, 9 | lkr0f 34903 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))}))) |
11 | 3, 4, 10 | syl2anc 573 | . . . 4 ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))}))) |
12 | 11 | biimpar 463 | . . 3 ⊢ ((𝜑 ∧ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))})) → (𝐾‘𝐺) = 𝑉) |
13 | 12 | olcd 863 | . 2 ⊢ ((𝜑 ∧ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))})) → ((𝐾‘𝐺) ∈ 𝐻 ∨ (𝐾‘𝐺) = 𝑉)) |
14 | 1 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → 𝑊 ∈ LVec) |
15 | 4 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → 𝐺 ∈ 𝐹) |
16 | simpr 471 | . . . 4 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) | |
17 | lkrshpor.h | . . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊) | |
18 | 7, 5, 6, 17, 8, 9 | lkrshp 34914 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → (𝐾‘𝐺) ∈ 𝐻) |
19 | 14, 15, 16, 18 | syl3anc 1476 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → (𝐾‘𝐺) ∈ 𝐻) |
20 | 19 | orcd 862 | . 2 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → ((𝐾‘𝐺) ∈ 𝐻 ∨ (𝐾‘𝐺) = 𝑉)) |
21 | 13, 20 | pm2.61dane 3030 | 1 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ∨ (𝐾‘𝐺) = 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∨ wo 836 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 {csn 4316 × cxp 5247 ‘cfv 6031 Basecbs 16064 Scalarcsca 16152 0gc0g 16308 LModclmod 19073 LVecclvec 19315 LSHypclsh 34784 LFnlclfn 34866 LKerclk 34894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-tpos 7504 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-grp 17633 df-minusg 17634 df-sbg 17635 df-subg 17799 df-cntz 17957 df-lsm 18258 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-invr 18880 df-drng 18959 df-lmod 19075 df-lss 19143 df-lsp 19185 df-lvec 19316 df-lshyp 34786 df-lfl 34867 df-lkr 34895 |
This theorem is referenced by: lkrshp4 34917 lkrpssN 34972 dochlkr 37195 dochkrshp 37196 lclkrlem2e 37321 lclkrlem2h 37324 lclkrlem2s 37335 |
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