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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlcl | Structured version Visualization version GIF version |
Description: Closure of the trace of a lattice translation. (Contributed by NM, 22-May-2012.) |
Ref | Expression |
---|---|
trlcl.b | ⊢ 𝐵 = (Base‘𝐾) |
trlcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trlcl.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trlcl.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trlcl | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | eqid 2823 | . . . . 5 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | eqid 2823 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
4 | trlcl.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 1, 2, 3, 4 | lhpocnel 37156 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ (Atoms‘𝐾) ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊)) |
6 | 5 | adantr 483 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((oc‘𝐾)‘𝑊) ∈ (Atoms‘𝐾) ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊)) |
7 | eqid 2823 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
8 | eqid 2823 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
9 | trlcl.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
10 | trlcl.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
11 | 1, 7, 8, 3, 4, 9, 10 | trlval2 37301 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (((oc‘𝐾)‘𝑊) ∈ (Atoms‘𝐾) ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊)) → (𝑅‘𝐹) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊)) |
12 | 6, 11 | mpd3an3 1458 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊)) |
13 | hllat 36501 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
14 | 13 | ad2antrr 724 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ Lat) |
15 | hlop 36500 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
16 | 15 | ad2antrr 724 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ OP) |
17 | trlcl.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
18 | 17, 4 | lhpbase 37136 | . . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
19 | 18 | ad2antlr 725 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝑊 ∈ 𝐵) |
20 | 17, 2 | opoccl 36332 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ 𝐵) → ((oc‘𝐾)‘𝑊) ∈ 𝐵) |
21 | 16, 19, 20 | syl2anc 586 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((oc‘𝐾)‘𝑊) ∈ 𝐵) |
22 | 17, 4, 9 | ltrncl 37263 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((oc‘𝐾)‘𝑊) ∈ 𝐵) → (𝐹‘((oc‘𝐾)‘𝑊)) ∈ 𝐵) |
23 | 21, 22 | mpd3an3 1458 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹‘((oc‘𝐾)‘𝑊)) ∈ 𝐵) |
24 | 17, 7 | latjcl 17663 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑊) ∈ 𝐵 ∧ (𝐹‘((oc‘𝐾)‘𝑊)) ∈ 𝐵) → (((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊))) ∈ 𝐵) |
25 | 14, 21, 23, 24 | syl3anc 1367 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊))) ∈ 𝐵) |
26 | 17, 8 | latmcl 17664 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊))) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) ∈ 𝐵) |
27 | 14, 25, 19, 26 | syl3anc 1367 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) ∈ 𝐵) |
28 | 12, 27 | eqeltrd 2915 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 lecple 16574 occoc 16575 joincjn 17556 meetcmee 17557 Latclat 17657 OPcops 36310 Atomscatm 36401 HLchlt 36488 LHypclh 37122 LTrncltrn 37239 trLctrl 37296 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 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-ral 3145 df-rex 3146 df-reu 3147 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-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 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-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-map 8410 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-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 |
This theorem is referenced by: trljat1 37304 trljat2 37305 trlval3 37325 cdlemc3 37331 cdlemc5 37333 trlord 37707 cdlemg4c 37750 cdlemg4 37755 cdlemg6c 37758 cdlemg10c 37777 cdlemg10 37779 cdlemg12e 37785 cdlemg17dALTN 37802 cdlemg31a 37835 cdlemg31b 37836 cdlemg35 37851 cdlemg44a 37869 trljco 37878 trljco2 37879 tendoidcl 37907 tendococl 37910 tendoid 37911 tendopltp 37918 tendo0tp 37927 cdlemh1 37953 cdlemh2 37954 cdlemi1 37956 cdlemi 37958 cdlemk9 37977 cdlemk9bN 37978 cdlemkvcl 37980 cdlemk10 37981 cdlemk11 37987 cdlemk11u 38009 cdlemk37 38052 cdlemkfid1N 38059 cdlemkid1 38060 cdlemkid2 38062 cdlemk39s-id 38078 cdlemk48 38088 cdlemk50 38090 cdlemk51 38091 cdlemk52 38092 cdlemk39u 38106 tendoex 38113 dialss 38184 dia0 38190 diaglbN 38193 dia1dim 38199 dia2dimlem2 38203 dia2dimlem3 38204 dia2dimlem10 38211 cdlemm10N 38256 dib1dim 38303 diblss 38308 cdlemn2a 38334 dih1dimb 38378 dihopelvalcpre 38386 dih1 38424 dihmeetlem1N 38428 dihglblem5apreN 38429 dihglbcpreN 38438 dih1dimatlem 38467 |
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