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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 2797 | . . . . . . . . 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 37798 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
8 | dochlkr.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
9 | 2, 3, 4, 7, 8 | lkrssv 35784 | . . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (Base‘𝑈)) |
10 | dochlkr.o | . . . . . . . . 9 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
11 | 5, 6, 2, 10 | dochocss 38054 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ (Base‘𝑈)) → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
12 | 1, 9, 11 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
13 | 12 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
14 | dochlkr.y | . . . . . . 7 ⊢ 𝑌 = (LSHyp‘𝑈) | |
15 | 5, 6, 1 | dvhlvec 37797 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec) |
16 | 15 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LVec) |
17 | 7 | adantr 481 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LMod) |
18 | simpr 485 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) | |
19 | 2, 14, 17, 18 | lshpne 35670 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈)) |
20 | 19 | ex 413 | . . . . . . . . 9 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈))) |
21 | 2, 14, 3, 4, 15, 8 | lkrshpor 35795 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ 𝑌 ∨ (𝐿‘𝐺) = (Base‘𝑈))) |
22 | 21 | ord 859 | . . . . . . . . . . 11 ⊢ (𝜑 → (¬ (𝐿‘𝐺) ∈ 𝑌 → (𝐿‘𝐺) = (Base‘𝑈))) |
23 | 2fveq3 6550 | . . . . . . . . . . . . . 14 ⊢ ((𝐿‘𝐺) = (Base‘𝑈) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈)))) | |
24 | 23 | adantl 482 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈)))) |
25 | 1 | adantr 481 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
26 | 5, 6, 10, 2, 25 | dochoc1 38049 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈)) |
27 | 24, 26 | eqtrd 2833 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈)) |
28 | 27 | ex 413 | . . . . . . . . . . 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 3003 | . . . . . . . . 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 407 | . . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) ∈ 𝑌) |
33 | 14, 16, 32, 18 | lshpcmp 35676 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ((𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ↔ (𝐿‘𝐺) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))) |
34 | 13, 33 | mpbid 233 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
35 | 34 | eqcomd 2803 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
36 | 35, 32 | jca 512 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)) |
37 | 36 | ex 413 | . 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) |
38 | eleq1 2872 | . . 3 ⊢ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (𝐿‘𝐺) ∈ 𝑌)) | |
39 | 38 | biimpar 478 | . 2 ⊢ ((( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) |
40 | 37, 39 | impbid1 226 | 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 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 ⊆ wss 3865 ‘cfv 6232 Basecbs 16316 LModclmod 19328 LVecclvec 19568 LSHypclsh 35663 LFnlclfn 35745 LKerclk 35773 HLchlt 36038 LHypclh 36672 DVecHcdvh 37766 ocHcoch 38035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-riotaBAD 35641 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-tpos 7750 df-undef 7797 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-sca 16414 df-vsca 16415 df-0g 16548 df-proset 17371 df-poset 17389 df-plt 17401 df-lub 17417 df-glb 17418 df-join 17419 df-meet 17420 df-p0 17482 df-p1 17483 df-lat 17489 df-clat 17551 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-grp 17868 df-minusg 17869 df-sbg 17870 df-subg 18034 df-cntz 18192 df-lsm 18495 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-ring 18993 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-dvr 19127 df-drng 19198 df-lmod 19330 df-lss 19398 df-lsp 19438 df-lvec 19569 df-lsatoms 35664 df-lshyp 35665 df-lfl 35746 df-lkr 35774 df-oposet 35864 df-ol 35866 df-oml 35867 df-covers 35954 df-ats 35955 df-atl 35986 df-cvlat 36010 df-hlat 36039 df-llines 36186 df-lplanes 36187 df-lvols 36188 df-lines 36189 df-psubsp 36191 df-pmap 36192 df-padd 36484 df-lhyp 36676 df-laut 36677 df-ldil 36792 df-ltrn 36793 df-trl 36847 df-tendo 37443 df-edring 37445 df-disoa 37717 df-dvech 37767 df-dib 37827 df-dic 37861 df-dih 37917 df-doch 38036 |
This theorem is referenced by: dochkrshp 38074 dochkrshp2 38075 mapdordlem1a 38322 mapdordlem2 38325 |
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