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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochsnkr | Structured version Visualization version GIF version |
Description: A (closed) kernel expressed in terms of a nonzero vector in its orthocomplement. TODO: consolidate lemmas unless they're needed for something else (in which case break out as theorems). (Contributed by NM, 2-Jan-2015.) |
Ref | Expression |
---|---|
dochsnkr.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochsnkr.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochsnkr.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochsnkr.v | ⊢ 𝑉 = (Base‘𝑈) |
dochsnkr.z | ⊢ 0 = (0g‘𝑈) |
dochsnkr.f | ⊢ 𝐹 = (LFnl‘𝑈) |
dochsnkr.l | ⊢ 𝐿 = (LKer‘𝑈) |
dochsnkr.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochsnkr.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
dochsnkr.x | ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) |
Ref | Expression |
---|---|
dochsnkr | ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochsnkr.z | . . . 4 ⊢ 0 = (0g‘𝑈) | |
2 | eqid 2733 | . . . 4 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
3 | eqid 2733 | . . . 4 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
4 | dochsnkr.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dochsnkr.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | dochsnkr.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 4, 5, 6 | dvhlvec 39918 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec) |
8 | dochsnkr.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
9 | dochsnkr.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
10 | dochsnkr.f | . . . . 5 ⊢ 𝐹 = (LFnl‘𝑈) | |
11 | dochsnkr.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑈) | |
12 | dochsnkr.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
13 | dochsnkr.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) | |
14 | 4, 8, 5, 9, 1, 10, 11, 6, 12, 13, 3 | dochsnkrlem2 40279 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)) |
15 | 13 | eldifad 3959 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺))) |
16 | eldifsni 4792 | . . . . 5 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) → 𝑋 ≠ 0 ) | |
17 | 13, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
18 | 1, 2, 3, 7, 14, 15, 17 | lsatel 37813 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) = ((LSpan‘𝑈)‘{𝑋})) |
19 | 18 | fveq2d 6892 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) |
20 | 4, 8, 5, 9, 1, 10, 11, 6, 12, 13 | dochsnkrlem3 40280 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
21 | 4, 5, 6 | dvhlmod 39919 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
22 | 9, 10, 11, 21, 12 | lkrssv 37904 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
23 | 4, 5, 9, 8 | dochssv 40164 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
24 | 6, 22, 23 | syl2anc 585 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
25 | 24 | ssdifssd 4141 | . . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) ⊆ 𝑉) |
26 | 25, 13 | sseldd 3982 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
27 | 26 | snssd 4811 | . . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
28 | 4, 5, 8, 9, 2, 6, 27 | dochocsp 40188 | . 2 ⊢ (𝜑 → ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})) = ( ⊥ ‘{𝑋})) |
29 | 19, 20, 28 | 3eqtr3d 2781 | 1 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 ‘cfv 6540 Basecbs 17140 0gc0g 17381 LSpanclspn 20570 LSAtomsclsa 37782 LFnlclfn 37865 LKerclk 37893 HLchlt 38158 LHypclh 38793 DVecHcdvh 39887 ocHcoch 40156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37761 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-lsm 19497 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-dvr 20204 df-drng 20306 df-lmod 20461 df-lss 20531 df-lsp 20571 df-lvec 20702 df-lsatoms 37784 df-lshyp 37785 df-lfl 37866 df-lkr 37894 df-oposet 37984 df-ol 37986 df-oml 37987 df-covers 38074 df-ats 38075 df-atl 38106 df-cvlat 38130 df-hlat 38159 df-llines 38307 df-lplanes 38308 df-lvols 38309 df-lines 38310 df-psubsp 38312 df-pmap 38313 df-padd 38605 df-lhyp 38797 df-laut 38798 df-ldil 38913 df-ltrn 38914 df-trl 38968 df-tgrp 39552 df-tendo 39564 df-edring 39566 df-dveca 39812 df-disoa 39838 df-dvech 39888 df-dib 39948 df-dic 39982 df-dih 40038 df-doch 40157 df-djh 40204 |
This theorem is referenced by: dochfln0 40286 lcfl6lem 40307 |
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