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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochkrsat | Structured version Visualization version GIF version |
Description: The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.) |
Ref | Expression |
---|---|
dochkrsat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochkrsat.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochkrsat.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochkrsat.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
dochkrsat.f | ⊢ 𝐹 = (LFnl‘𝑈) |
dochkrsat.l | ⊢ 𝐿 = (LKer‘𝑈) |
dochkrsat.z | ⊢ 0 = (0g‘𝑈) |
dochkrsat.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochkrsat.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
dochkrsat | ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ≠ { 0 } ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochkrsat.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dochkrsat.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | dochkrsat.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | eqid 2726 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
5 | eqid 2726 | . . 3 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈) | |
6 | dochkrsat.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
7 | dochkrsat.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
8 | dochkrsat.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | dochkrsat.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | dochkrshp 41085 | . 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ (LSHyp‘𝑈))) |
11 | dochkrsat.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
12 | 1, 3, 8 | dvhlmod 40809 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
13 | 4, 6, 7, 12, 9 | lkrssv 38794 | . . 3 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (Base‘𝑈)) |
14 | 1, 2, 3, 4, 11, 8, 13 | dochn0nv 41074 | . 2 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ≠ { 0 } ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈))) |
15 | eqid 2726 | . . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
16 | dochkrsat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
17 | 1, 3, 4, 15, 2 | dochlss 41053 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈)) |
18 | 8, 13, 17 | syl2anc 582 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈)) |
19 | 1, 2, 3, 15, 16, 5, 8, 18 | dochsatshpb 41151 | . 2 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ (LSHyp‘𝑈))) |
20 | 10, 14, 19 | 3bitr4d 310 | 1 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ≠ { 0 } ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ⊆ wss 3947 {csn 4633 ‘cfv 6554 Basecbs 17213 0gc0g 17454 LSubSpclss 20908 LSAtomsclsa 38672 LSHypclsh 38673 LFnlclfn 38755 LKerclk 38783 HLchlt 39048 LHypclh 39683 DVecHcdvh 40777 ocHcoch 41046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-riotaBAD 38651 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-undef 8288 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-0g 17456 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-cntz 19311 df-lsm 19634 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-drng 20709 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lvec 21081 df-lsatoms 38674 df-lshyp 38675 df-lfl 38756 df-lkr 38784 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 df-lvols 39199 df-lines 39200 df-psubsp 39202 df-pmap 39203 df-padd 39495 df-lhyp 39687 df-laut 39688 df-ldil 39803 df-ltrn 39804 df-trl 39858 df-tgrp 40442 df-tendo 40454 df-edring 40456 df-dveca 40702 df-disoa 40728 df-dvech 40778 df-dib 40838 df-dic 40872 df-dih 40928 df-doch 41047 df-djh 41094 |
This theorem is referenced by: dochkrsat2 41155 dochsat0 41156 |
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