![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrshp3 | Structured version Visualization version GIF version |
Description: The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 17-Jul-2014.) |
Ref | Expression |
---|---|
lkrshp3.v | ⊢ 𝑉 = (Base‘𝑊) |
lkrshp3.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lkrshp3.o | ⊢ 0 = (0g‘𝐷) |
lkrshp3.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lkrshp3.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrshp3.k | ⊢ 𝐾 = (LKer‘𝑊) |
lkrshp3.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lkrshp3.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lkrshp3 | ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ↔ 𝐺 ≠ (𝑉 × { 0 }))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkrshp3.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lkrshp3.h | . . . 4 ⊢ 𝐻 = (LSHyp‘𝑊) | |
3 | lkrshp3.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 19320 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
6 | 5 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → 𝑊 ∈ LMod) |
7 | simpr 471 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → (𝐾‘𝐺) ∈ 𝐻) | |
8 | 1, 2, 6, 7 | lshpne 34792 | . . 3 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → (𝐾‘𝐺) ≠ 𝑉) |
9 | lkrshp3.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
10 | lkrshp3.d | . . . . . . 7 ⊢ 𝐷 = (Scalar‘𝑊) | |
11 | lkrshp3.o | . . . . . . 7 ⊢ 0 = (0g‘𝐷) | |
12 | lkrshp3.f | . . . . . . 7 ⊢ 𝐹 = (LFnl‘𝑊) | |
13 | lkrshp3.k | . . . . . . 7 ⊢ 𝐾 = (LKer‘𝑊) | |
14 | 10, 11, 1, 12, 13 | lkr0f 34904 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
15 | 5, 9, 14 | syl2anc 567 | . . . . 5 ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
16 | 15 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
17 | 16 | necon3bid 2987 | . . 3 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → ((𝐾‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ (𝑉 × { 0 }))) |
18 | 8, 17 | mpbid 222 | . 2 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → 𝐺 ≠ (𝑉 × { 0 })) |
19 | 3 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → 𝑊 ∈ LVec) |
20 | 9 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → 𝐺 ∈ 𝐹) |
21 | simpr 471 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → 𝐺 ≠ (𝑉 × { 0 })) | |
22 | 1, 10, 11, 2, 12, 13 | lkrshp 34915 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × { 0 })) → (𝐾‘𝐺) ∈ 𝐻) |
23 | 19, 20, 21, 22 | syl3anc 1476 | . 2 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → (𝐾‘𝐺) ∈ 𝐻) |
24 | 18, 23 | impbida 796 | 1 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ↔ 𝐺 ≠ (𝑉 × { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 {csn 4317 × cxp 5248 ‘cfv 6032 Basecbs 16065 Scalarcsca 16153 0gc0g 16309 LModclmod 19074 LVecclvec 19316 LSHypclsh 34785 LFnlclfn 34867 LKerclk 34895 |
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 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 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 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-1st 7316 df-2nd 7317 df-tpos 7505 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-er 7897 df-map 8012 df-en 8111 df-dom 8112 df-sdom 8113 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-nn 11224 df-2 11282 df-3 11283 df-ndx 16068 df-slot 16069 df-base 16071 df-sets 16072 df-ress 16073 df-plusg 16163 df-mulr 16164 df-0g 16311 df-mgm 17451 df-sgrp 17493 df-mnd 17504 df-submnd 17545 df-grp 17634 df-minusg 17635 df-sbg 17636 df-subg 17800 df-cntz 17958 df-lsm 18259 df-cmn 18403 df-abl 18404 df-mgp 18699 df-ur 18711 df-ring 18758 df-oppr 18832 df-dvdsr 18850 df-unit 18851 df-invr 18881 df-drng 18960 df-lmod 19076 df-lss 19144 df-lsp 19186 df-lvec 19317 df-lshyp 34787 df-lfl 34868 df-lkr 34896 |
This theorem is referenced by: lshpset2N 34929 lduallkr3 34972 |
Copyright terms: Public domain | W3C validator |