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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochkrsat2 | Structured version Visualization version GIF version |
Description: The orthocomplement of a kernel is an atom iff the double orthocomplement is not the vector space. (Contributed by NM, 1-Jan-2015.) |
Ref | Expression |
---|---|
dochkrsat2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochkrsat2.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochkrsat2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochkrsat2.v | ⊢ 𝑉 = (Base‘𝑈) |
dochkrsat2.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
dochkrsat2.f | ⊢ 𝐹 = (LFnl‘𝑈) |
dochkrsat2.l | ⊢ 𝐿 = (LKer‘𝑈) |
dochkrsat2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochkrsat2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
dochkrsat2 | ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochkrsat2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dochkrsat2.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | dochkrsat2.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | dochkrsat2.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | eqid 2771 | . . 3 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
6 | dochkrsat2.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | dochkrsat2.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑈) | |
8 | dochkrsat2.l | . . . 4 ⊢ 𝐿 = (LKer‘𝑈) | |
9 | 1, 3, 6 | dvhlmod 36913 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
10 | dochkrsat2.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | 4, 7, 8, 9, 10 | lkrssv 34898 | . . 3 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
12 | 1, 2, 3, 4, 5, 6, 11 | dochn0nv 37178 | . 2 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ≠ {(0g‘𝑈)} ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)) |
13 | dochkrsat2.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
14 | 1, 2, 3, 13, 7, 8, 5, 6, 10 | dochkrsat 37258 | . 2 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ≠ {(0g‘𝑈)} ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴)) |
15 | 12, 14 | bitr3d 270 | 1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 {csn 4316 ‘cfv 6029 Basecbs 16057 0gc0g 16301 LSAtomsclsa 34776 LFnlclfn 34859 LKerclk 34887 HLchlt 35152 LHypclh 35785 DVecHcdvh 36881 ocHcoch 37150 |
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 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-riotaBAD 34754 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 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-iin 4657 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 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-tpos 7502 df-undef 7549 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-map 8009 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-nn 11221 df-2 11279 df-3 11280 df-4 11281 df-5 11282 df-6 11283 df-n0 11493 df-z 11578 df-uz 11887 df-fz 12527 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16155 df-mulr 16156 df-sca 16158 df-vsca 16159 df-0g 16303 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-p1 17241 df-lat 17247 df-clat 17309 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-subg 17792 df-cntz 17950 df-lsm 18251 df-cmn 18395 df-abl 18396 df-mgp 18691 df-ur 18703 df-ring 18750 df-oppr 18824 df-dvdsr 18842 df-unit 18843 df-invr 18873 df-dvr 18884 df-drng 18952 df-lmod 19068 df-lss 19136 df-lsp 19178 df-lvec 19309 df-lsatoms 34778 df-lshyp 34779 df-lfl 34860 df-lkr 34888 df-oposet 34978 df-ol 34980 df-oml 34981 df-covers 35068 df-ats 35069 df-atl 35100 df-cvlat 35124 df-hlat 35153 df-llines 35299 df-lplanes 35300 df-lvols 35301 df-lines 35302 df-psubsp 35304 df-pmap 35305 df-padd 35597 df-lhyp 35789 df-laut 35790 df-ldil 35905 df-ltrn 35906 df-trl 35961 df-tgrp 36545 df-tendo 36557 df-edring 36559 df-dveca 36805 df-disoa 36832 df-dvech 36882 df-dib 36942 df-dic 36976 df-dih 37032 df-doch 37151 df-djh 37198 |
This theorem is referenced by: dochsnkrlem2 37273 dochkr1 37281 dochkr1OLDN 37282 lcfl3 37297 lcfl8b 37307 |
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