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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochlkr | Structured version Visualization version GIF version |
Description: Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.) |
Ref | Expression |
---|---|
dochlkr.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochlkr.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochlkr.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochlkr.f | ⊢ 𝐹 = (LFnl‘𝑈) |
dochlkr.y | ⊢ 𝑌 = (LSHyp‘𝑈) |
dochlkr.l | ⊢ 𝐿 = (LKer‘𝑈) |
dochlkr.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochlkr.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
dochlkr | ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochlkr.k | . . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | eqid 2800 | . . . . . . . . 9 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
3 | dochlkr.f | . . . . . . . . 9 ⊢ 𝐹 = (LFnl‘𝑈) | |
4 | dochlkr.l | . . . . . . . . 9 ⊢ 𝐿 = (LKer‘𝑈) | |
5 | dochlkr.h | . . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | dochlkr.u | . . . . . . . . . 10 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | 5, 6, 1 | dvhlmod 37130 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
8 | dochlkr.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
9 | 2, 3, 4, 7, 8 | lkrssv 35116 | . . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (Base‘𝑈)) |
10 | dochlkr.o | . . . . . . . . 9 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
11 | 5, 6, 2, 10 | dochocss 37386 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ (Base‘𝑈)) → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
12 | 1, 9, 11 | syl2anc 580 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
13 | 12 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
14 | dochlkr.y | . . . . . . 7 ⊢ 𝑌 = (LSHyp‘𝑈) | |
15 | 5, 6, 1 | dvhlvec 37129 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec) |
16 | 15 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LVec) |
17 | 7 | adantr 473 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LMod) |
18 | simpr 478 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) | |
19 | 2, 14, 17, 18 | lshpne 35002 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈)) |
20 | 19 | ex 402 | . . . . . . . . 9 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈))) |
21 | 2, 14, 3, 4, 15, 8 | lkrshpor 35127 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ 𝑌 ∨ (𝐿‘𝐺) = (Base‘𝑈))) |
22 | 21 | ord 891 | . . . . . . . . . . 11 ⊢ (𝜑 → (¬ (𝐿‘𝐺) ∈ 𝑌 → (𝐿‘𝐺) = (Base‘𝑈))) |
23 | 2fveq3 6417 | . . . . . . . . . . . . . 14 ⊢ ((𝐿‘𝐺) = (Base‘𝑈) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈)))) | |
24 | 23 | adantl 474 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈)))) |
25 | 1 | adantr 473 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
26 | 5, 6, 10, 2, 25 | dochoc1 37381 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈)) |
27 | 24, 26 | eqtrd 2834 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈)) |
28 | 27 | ex 402 | . . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿‘𝐺) = (Base‘𝑈) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈))) |
29 | 22, 28 | syld 47 | . . . . . . . . . 10 ⊢ (𝜑 → (¬ (𝐿‘𝐺) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈))) |
30 | 29 | necon1ad 2989 | . . . . . . . . 9 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈) → (𝐿‘𝐺) ∈ 𝑌)) |
31 | 20, 30 | syld 47 | . . . . . . . 8 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → (𝐿‘𝐺) ∈ 𝑌)) |
32 | 31 | imp 396 | . . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) ∈ 𝑌) |
33 | 14, 16, 32, 18 | lshpcmp 35008 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ((𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ↔ (𝐿‘𝐺) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))) |
34 | 13, 33 | mpbid 224 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
35 | 34 | eqcomd 2806 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
36 | 35, 32 | jca 508 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)) |
37 | 36 | ex 402 | . 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) |
38 | eleq1 2867 | . . 3 ⊢ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (𝐿‘𝐺) ∈ 𝑌)) | |
39 | 38 | biimpar 470 | . 2 ⊢ ((( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) |
40 | 37, 39 | impbid1 217 | 1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2972 ⊆ wss 3770 ‘cfv 6102 Basecbs 16183 LModclmod 19180 LVecclvec 19422 LSHypclsh 34995 LFnlclfn 35077 LKerclk 35105 HLchlt 35370 LHypclh 36004 DVecHcdvh 37098 ocHcoch 37367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-riotaBAD 34973 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-iin 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-tpos 7591 df-undef 7638 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-oadd 7804 df-er 7983 df-map 8098 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-n0 11580 df-z 11666 df-uz 11930 df-fz 12580 df-struct 16185 df-ndx 16186 df-slot 16187 df-base 16189 df-sets 16190 df-ress 16191 df-plusg 16279 df-mulr 16280 df-sca 16282 df-vsca 16283 df-0g 16416 df-proset 17242 df-poset 17260 df-plt 17272 df-lub 17288 df-glb 17289 df-join 17290 df-meet 17291 df-p0 17353 df-p1 17354 df-lat 17360 df-clat 17422 df-mgm 17556 df-sgrp 17598 df-mnd 17609 df-submnd 17650 df-grp 17740 df-minusg 17741 df-sbg 17742 df-subg 17903 df-cntz 18061 df-lsm 18363 df-cmn 18509 df-abl 18510 df-mgp 18805 df-ur 18817 df-ring 18864 df-oppr 18938 df-dvdsr 18956 df-unit 18957 df-invr 18987 df-dvr 18998 df-drng 19066 df-lmod 19182 df-lss 19250 df-lsp 19292 df-lvec 19423 df-lsatoms 34996 df-lshyp 34997 df-lfl 35078 df-lkr 35106 df-oposet 35196 df-ol 35198 df-oml 35199 df-covers 35286 df-ats 35287 df-atl 35318 df-cvlat 35342 df-hlat 35371 df-llines 35518 df-lplanes 35519 df-lvols 35520 df-lines 35521 df-psubsp 35523 df-pmap 35524 df-padd 35816 df-lhyp 36008 df-laut 36009 df-ldil 36124 df-ltrn 36125 df-trl 36179 df-tendo 36775 df-edring 36777 df-disoa 37049 df-dvech 37099 df-dib 37159 df-dic 37193 df-dih 37249 df-doch 37368 |
This theorem is referenced by: dochkrshp 37406 dochkrshp2 37407 mapdordlem1a 37654 mapdordlem2 37657 |
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