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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 484 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) | |
2 | dochshpsat.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑌) | |
3 | 2 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ 𝑌) |
4 | 1, 3 | eqeltrd 2829 | . . 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 2728 | . . . . 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 40577 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
13 | 8, 10, 12, 2 | lshplss 38447 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (LSubSp‘𝑈)) |
14 | eqid 2728 | . . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
15 | 14, 8 | lssss 20813 | . . . . . . 7 ⊢ (𝑋 ∈ (LSubSp‘𝑈) → 𝑋 ⊆ (Base‘𝑈)) |
16 | 13, 15 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑈)) |
17 | 5, 7, 14, 8, 6 | dochlss 40821 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈)) |
18 | 11, 16, 17 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈)) |
19 | 5, 6, 7, 8, 9, 10, 11, 18 | dochsatshpb 40919 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌)) |
20 | 19 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌)) |
21 | 4, 20 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘𝑋) ∈ 𝐴) |
22 | eqid 2728 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
23 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → 𝑈 ∈ LMod) |
24 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐴) | |
25 | 22, 9, 23, 24 | lsatn0 38465 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ≠ {(0g‘𝑈)}) |
26 | 25 | neneqd 2941 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘𝑋) = {(0g‘𝑈)}) |
27 | 11 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
28 | 5, 7, 6, 14, 22 | doch0 40825 | . . . . . . 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 2739 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈))) |
31 | eqid 2728 | . . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
32 | 5, 31, 7, 14, 6 | dochcl 40820 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
33 | 11, 16, 32 | syl2anc 583 | . . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
34 | 5, 31, 7, 22 | dih0rn 40751 | . . . . . . . 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 40840 | . . . . . 6 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0g‘𝑈)})) |
37 | 36 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0g‘𝑈)})) |
38 | 30, 37 | bitr3d 281 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘𝑋) = {(0g‘𝑈)})) |
39 | 26, 38 | mtbird 325 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈)) |
40 | 5, 6, 7, 14, 10, 11, 2 | dochshpncl 40851 | . . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑋 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈))) |
41 | 40 | necon1bbid 2976 | . . . 4 ⊢ (𝜑 → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
42 | 41 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
43 | 39, 42 | mpbid 231 | . 2 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
44 | 21, 43 | impbida 800 | 1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3945 {csn 4624 ran crn 5673 ‘cfv 6542 Basecbs 17173 0gc0g 17414 LModclmod 20736 LSubSpclss 20808 LSAtomsclsa 38440 LSHypclsh 38441 HLchlt 38816 LHypclh 39451 DVecHcdvh 40545 DIsoHcdih 40695 ocHcoch 40814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-riotaBAD 38419 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17416 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-cntz 19261 df-lsm 19584 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-drng 20619 df-lmod 20738 df-lss 20809 df-lsp 20849 df-lvec 20981 df-lsatoms 38442 df-lshyp 38443 df-oposet 38642 df-ol 38644 df-oml 38645 df-covers 38732 df-ats 38733 df-atl 38764 df-cvlat 38788 df-hlat 38817 df-llines 38965 df-lplanes 38966 df-lvols 38967 df-lines 38968 df-psubsp 38970 df-pmap 38971 df-padd 39263 df-lhyp 39455 df-laut 39456 df-ldil 39571 df-ltrn 39572 df-trl 39626 df-tgrp 40210 df-tendo 40222 df-edring 40224 df-dveca 40470 df-disoa 40496 df-dvech 40546 df-dib 40606 df-dic 40640 df-dih 40696 df-doch 40815 df-djh 40862 |
This theorem is referenced by: mapdordlem2 41104 |
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