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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 20148 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | lkrshpor.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
5 | eqid 2737 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2737 | . . . . . 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 36850 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))}))) |
11 | 3, 4, 10 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))}))) |
12 | 11 | biimpar 481 | . . 3 ⊢ ((𝜑 ∧ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))})) → (𝐾‘𝐺) = 𝑉) |
13 | 12 | olcd 874 | . 2 ⊢ ((𝜑 ∧ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))})) → ((𝐾‘𝐺) ∈ 𝐻 ∨ (𝐾‘𝐺) = 𝑉)) |
14 | 1 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → 𝑊 ∈ LVec) |
15 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → 𝐺 ∈ 𝐹) |
16 | simpr 488 | . . . 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 36861 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → (𝐾‘𝐺) ∈ 𝐻) |
19 | 14, 15, 16, 18 | syl3anc 1373 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → (𝐾‘𝐺) ∈ 𝐻) |
20 | 19 | orcd 873 | . 2 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × {(0g‘(Scalar‘𝑊))})) → ((𝐾‘𝐺) ∈ 𝐻 ∨ (𝐾‘𝐺) = 𝑉)) |
21 | 13, 20 | pm2.61dane 3029 | 1 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ∨ (𝐾‘𝐺) = 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 {csn 4546 × cxp 5554 ‘cfv 6385 Basecbs 16765 Scalarcsca 16810 0gc0g 16949 LModclmod 19904 LVecclvec 20144 LSHypclsh 36731 LFnlclfn 36813 LKerclk 36841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-resscn 10791 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-mulcom 10798 ax-addass 10799 ax-mulass 10800 ax-distr 10801 ax-i2m1 10802 ax-1ne0 10803 ax-1rid 10804 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 ax-pre-lttri 10808 ax-pre-lttrn 10809 ax-pre-ltadd 10810 ax-pre-mulgt0 10811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-int 4865 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-riota 7175 df-ov 7221 df-oprab 7222 df-mpo 7223 df-om 7650 df-1st 7766 df-2nd 7767 df-tpos 7973 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-er 8396 df-map 8515 df-en 8632 df-dom 8633 df-sdom 8634 df-pnf 10874 df-mnf 10875 df-xr 10876 df-ltxr 10877 df-le 10878 df-sub 11069 df-neg 11070 df-nn 11836 df-2 11898 df-3 11899 df-sets 16722 df-slot 16740 df-ndx 16750 df-base 16766 df-ress 16790 df-plusg 16820 df-mulr 16821 df-0g 16951 df-mgm 18119 df-sgrp 18168 df-mnd 18179 df-submnd 18224 df-grp 18373 df-minusg 18374 df-sbg 18375 df-subg 18545 df-cntz 18716 df-lsm 19030 df-cmn 19177 df-abl 19178 df-mgp 19510 df-ur 19522 df-ring 19569 df-oppr 19646 df-dvdsr 19664 df-unit 19665 df-invr 19695 df-drng 19774 df-lmod 19906 df-lss 19974 df-lsp 20014 df-lvec 20145 df-lshyp 36733 df-lfl 36814 df-lkr 36842 |
This theorem is referenced by: lkrshp4 36864 lkrpssN 36919 dochlkr 39141 dochkrshp 39142 lclkrlem2e 39267 lclkrlem2h 39270 lclkrlem2s 39281 |
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