Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochshpsat | Structured version Visualization version GIF version |
Description: A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.) |
Ref | Expression |
---|---|
dochshpsat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochshpsat.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochshpsat.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochshpsat.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
dochshpsat.y | ⊢ 𝑌 = (LSHyp‘𝑈) |
dochshpsat.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochshpsat.x | ⊢ (𝜑 → 𝑋 ∈ 𝑌) |
Ref | Expression |
---|---|
dochshpsat | ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) | |
2 | dochshpsat.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑌) | |
3 | 2 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ 𝑌) |
4 | 1, 3 | eqeltrd 2915 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌) |
5 | dochshpsat.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | dochshpsat.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
7 | dochshpsat.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | eqid 2823 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
9 | dochshpsat.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
10 | dochshpsat.y | . . . . 5 ⊢ 𝑌 = (LSHyp‘𝑈) | |
11 | dochshpsat.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | 5, 7, 11 | dvhlmod 38248 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
13 | 8, 10, 12, 2 | lshplss 36119 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (LSubSp‘𝑈)) |
14 | eqid 2823 | . . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
15 | 14, 8 | lssss 19710 | . . . . . . 7 ⊢ (𝑋 ∈ (LSubSp‘𝑈) → 𝑋 ⊆ (Base‘𝑈)) |
16 | 13, 15 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑈)) |
17 | 5, 7, 14, 8, 6 | dochlss 38492 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈)) |
18 | 11, 16, 17 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈)) |
19 | 5, 6, 7, 8, 9, 10, 11, 18 | dochsatshpb 38590 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌)) |
20 | 19 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌)) |
21 | 4, 20 | mpbird 259 | . 2 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘𝑋) ∈ 𝐴) |
22 | eqid 2823 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
23 | 12 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → 𝑈 ∈ LMod) |
24 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐴) | |
25 | 22, 9, 23, 24 | lsatn0 36137 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ≠ {(0g‘𝑈)}) |
26 | 25 | neneqd 3023 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘𝑋) = {(0g‘𝑈)}) |
27 | 11 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
28 | 5, 7, 6, 14, 22 | doch0 38496 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈)) |
29 | 27, 28 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈)) |
30 | 29 | eqeq2d 2834 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈))) |
31 | eqid 2823 | . . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
32 | 5, 31, 7, 14, 6 | dochcl 38491 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
33 | 11, 16, 32 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
34 | 5, 31, 7, 22 | dih0rn 38422 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {(0g‘𝑈)} ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
35 | 11, 34 | syl 17 | . . . . . . 7 ⊢ (𝜑 → {(0g‘𝑈)} ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
36 | 5, 31, 6, 11, 33, 35 | doch11 38511 | . . . . . 6 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0g‘𝑈)})) |
37 | 36 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0g‘𝑈)})) |
38 | 30, 37 | bitr3d 283 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘𝑋) = {(0g‘𝑈)})) |
39 | 26, 38 | mtbird 327 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈)) |
40 | 5, 6, 7, 14, 10, 11, 2 | dochshpncl 38522 | . . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑋 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈))) |
41 | 40 | necon1bbid 3057 | . . . 4 ⊢ (𝜑 → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
42 | 41 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
43 | 39, 42 | mpbid 234 | . 2 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
44 | 21, 43 | impbida 799 | 1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 {csn 4569 ran crn 5558 ‘cfv 6357 Basecbs 16485 0gc0g 16715 LModclmod 19636 LSubSpclss 19705 LSAtomsclsa 36112 LSHypclsh 36113 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 DIsoHcdih 38366 ocHcoch 38485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533 |
This theorem is referenced by: mapdordlem2 38775 |
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