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 20379 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
6 | 5 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → 𝑊 ∈ LMod) |
7 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → (𝐾‘𝐺) ∈ 𝐻) | |
8 | 1, 2, 6, 7 | lshpne 37005 | . . 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 37117 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
15 | 5, 9, 14 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
16 | 15 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
17 | 16 | necon3bid 2990 | . . 3 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → ((𝐾‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ (𝑉 × { 0 }))) |
18 | 8, 17 | mpbid 231 | . 2 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ∈ 𝐻) → 𝐺 ≠ (𝑉 × { 0 })) |
19 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → 𝑊 ∈ LVec) |
20 | 9 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → 𝐺 ∈ 𝐹) |
21 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → 𝐺 ≠ (𝑉 × { 0 })) | |
22 | 1, 10, 11, 2, 12, 13 | lkrshp 37128 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × { 0 })) → (𝐾‘𝐺) ∈ 𝐻) |
23 | 19, 20, 21, 22 | syl3anc 1370 | . 2 ⊢ ((𝜑 ∧ 𝐺 ≠ (𝑉 × { 0 })) → (𝐾‘𝐺) ∈ 𝐻) |
24 | 18, 23 | impbida 798 | 1 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ 𝐻 ↔ 𝐺 ≠ (𝑉 × { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 {csn 4567 × cxp 5588 ‘cfv 6432 Basecbs 16923 Scalarcsca 16976 0gc0g 17161 LModclmod 20134 LVecclvec 20375 LSHypclsh 36998 LFnlclfn 37080 LKerclk 37108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-tpos 8034 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-er 8490 df-map 8609 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-3 12048 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-0g 17163 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-grp 18591 df-minusg 18592 df-sbg 18593 df-subg 18763 df-cntz 18934 df-lsm 19252 df-cmn 19399 df-abl 19400 df-mgp 19732 df-ur 19749 df-ring 19796 df-oppr 19873 df-dvdsr 19894 df-unit 19895 df-invr 19925 df-drng 20004 df-lmod 20136 df-lss 20205 df-lsp 20245 df-lvec 20376 df-lshyp 37000 df-lfl 37081 df-lkr 37109 |
This theorem is referenced by: lshpset2N 37142 lduallkr3 37185 |
Copyright terms: Public domain | W3C validator |