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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochkrsm | Structured version Visualization version GIF version |
Description: The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 39815 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.) |
Ref | Expression |
---|---|
dochkrsm.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochkrsm.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dochkrsm.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochkrsm.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochkrsm.p | ⊢ ⊕ = (LSSum‘𝑈) |
dochkrsm.f | ⊢ 𝐹 = (LFnl‘𝑈) |
dochkrsm.l | ⊢ 𝐿 = (LKer‘𝑈) |
dochkrsm.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochkrsm.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
dochkrsm.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
dochkrsm | ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘(𝐿‘𝐺))) ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochkrsm.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dochkrsm.i | . . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
3 | dochkrsm.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | dochkrsm.p | . . 3 ⊢ ⊕ = (LSSum‘𝑈) | |
5 | eqid 2737 | . . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
6 | dochkrsm.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
8 | dochkrsm.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)) → 𝑋 ∈ ran 𝐼) |
10 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)) | |
11 | 1, 2, 3, 4, 5, 7, 9, 10 | dihsmatrn 39837 | . 2 ⊢ ((𝜑 ∧ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)) → (𝑋 ⊕ ( ⊥ ‘(𝐿‘𝐺))) ∈ ran 𝐼) |
12 | oveq2 7359 | . . . 4 ⊢ (( ⊥ ‘(𝐿‘𝐺)) = {(0g‘𝑈)} → (𝑋 ⊕ ( ⊥ ‘(𝐿‘𝐺))) = (𝑋 ⊕ {(0g‘𝑈)})) | |
13 | 1, 3, 6 | dvhlmod 39511 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
14 | eqid 2737 | . . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
15 | 1, 3, 2, 14 | dihrnlss 39678 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ∈ (LSubSp‘𝑈)) |
16 | 6, 8, 15 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (LSubSp‘𝑈)) |
17 | 14 | lsssubg 20371 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ (LSubSp‘𝑈)) → 𝑋 ∈ (SubGrp‘𝑈)) |
18 | 13, 16, 17 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘𝑈)) |
19 | eqid 2737 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
20 | 19, 4 | lsm01 19412 | . . . . 5 ⊢ (𝑋 ∈ (SubGrp‘𝑈) → (𝑋 ⊕ {(0g‘𝑈)}) = 𝑋) |
21 | 18, 20 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑋 ⊕ {(0g‘𝑈)}) = 𝑋) |
22 | 12, 21 | sylan9eqr 2799 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘(𝐿‘𝐺)) = {(0g‘𝑈)}) → (𝑋 ⊕ ( ⊥ ‘(𝐿‘𝐺))) = 𝑋) |
23 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘(𝐿‘𝐺)) = {(0g‘𝑈)}) → 𝑋 ∈ ran 𝐼) |
24 | 22, 23 | eqeltrd 2838 | . 2 ⊢ ((𝜑 ∧ ( ⊥ ‘(𝐿‘𝐺)) = {(0g‘𝑈)}) → (𝑋 ⊕ ( ⊥ ‘(𝐿‘𝐺))) ∈ ran 𝐼) |
25 | dochkrsm.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
26 | dochkrsm.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
27 | dochkrsm.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
28 | dochkrsm.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
29 | 1, 25, 3, 19, 5, 26, 27, 6, 28 | dochsat0 39858 | . 2 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈) ∨ ( ⊥ ‘(𝐿‘𝐺)) = {(0g‘𝑈)})) |
30 | 11, 24, 29 | mpjaodan 957 | 1 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘(𝐿‘𝐺))) ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4584 ran crn 5632 ‘cfv 6493 (class class class)co 7351 0gc0g 17281 SubGrpcsubg 18881 LSSumclsm 19375 LModclmod 20275 LSubSpclss 20345 LSAtomsclsa 37374 LFnlclfn 37457 LKerclk 37485 HLchlt 37750 LHypclh 38385 DVecHcdvh 39479 DIsoHcdih 39629 ocHcoch 39748 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-riotaBAD 37353 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-undef 8196 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-0g 17283 df-proset 18144 df-poset 18162 df-plt 18179 df-lub 18195 df-glb 18196 df-join 18197 df-meet 18198 df-p0 18274 df-p1 18275 df-lat 18281 df-clat 18348 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-subg 18884 df-cntz 19056 df-lsm 19377 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-invr 20054 df-dvr 20065 df-drng 20140 df-lmod 20277 df-lss 20346 df-lsp 20386 df-lvec 20517 df-lsatoms 37376 df-lshyp 37377 df-lfl 37458 df-lkr 37486 df-oposet 37576 df-ol 37578 df-oml 37579 df-covers 37666 df-ats 37667 df-atl 37698 df-cvlat 37722 df-hlat 37751 df-llines 37899 df-lplanes 37900 df-lvols 37901 df-lines 37902 df-psubsp 37904 df-pmap 37905 df-padd 38197 df-lhyp 38389 df-laut 38390 df-ldil 38505 df-ltrn 38506 df-trl 38560 df-tgrp 39144 df-tendo 39156 df-edring 39158 df-dveca 39404 df-disoa 39430 df-dvech 39480 df-dib 39540 df-dic 39574 df-dih 39630 df-doch 39749 df-djh 39796 |
This theorem is referenced by: lclkrslem2 39939 |
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