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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 19871 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
6 | 5 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → 𝑊 ∈ LMod) |
7 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → (𝐾‘𝐺) ∈ 𝐻) | |
8 | 1, 2, 6, 7 | lshpne 36278 | . . 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 36390 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
15 | 5, 9, 14 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
16 | 15 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
17 | 16 | necon3bid 3031 | . . 3 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → ((𝐾‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ (𝑉 × { 0 }))) |
18 | 8, 17 | mpbid 235 | . 2 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → 𝐺 ≠ (𝑉 × { 0 })) |
19 | 3 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → 𝑊 ∈ LVec) |
20 | 9 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → 𝐺 ∈ 𝐹) |
21 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → 𝐺 ≠ (𝑉 × { 0 })) | |
22 | 1, 10, 11, 2, 12, 13 | lkrshp 36401 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × { 0 })) → (𝐾‘𝐺) ∈ 𝐻) |
23 | 19, 20, 21, 22 | syl3anc 1368 | . 2 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → (𝐾‘𝐺) ∈ 𝐻) |
24 | 18, 23 | impbida 800 | 1 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ↔ 𝐺 ≠ (𝑉 × { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {csn 4525 × cxp 5517 ‘cfv 6324 Basecbs 16475 Scalarcsca 16560 0gc0g 16705 LModclmod 19627 LVecclvec 19867 LSHypclsh 36271 LFnlclfn 36353 LKerclk 36381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lshyp 36273 df-lfl 36354 df-lkr 36382 |
This theorem is referenced by: lshpset2N 36415 lduallkr3 36458 |
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