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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 2725 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | eqid 2725 | . . . . 5 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | eqid 2725 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
4 | trlcl.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 1, 2, 3, 4 | lhpocnel 39621 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ (Atoms‘𝐾) ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊)) |
6 | 5 | adantr 479 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((oc‘𝐾)‘𝑊) ∈ (Atoms‘𝐾) ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊)) |
7 | eqid 2725 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
8 | eqid 2725 | . . . 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 39766 | . . 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 38965 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
14 | 13 | ad2antrr 724 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ Lat) |
15 | hlop 38964 | . . . . . 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 39601 | . . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
19 | 18 | ad2antlr 725 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝑊 ∈ 𝐵) |
20 | 17, 2 | opoccl 38796 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ 𝐵) → ((oc‘𝐾)‘𝑊) ∈ 𝐵) |
21 | 16, 19, 20 | syl2anc 582 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((oc‘𝐾)‘𝑊) ∈ 𝐵) |
22 | 17, 4, 9 | ltrncl 39728 | . . . . 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 18434 | . . . 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 1368 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊))) ∈ 𝐵) |
26 | 17, 8 | latmcl 18435 | . . 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 1368 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) ∈ 𝐵) |
28 | 12, 27 | eqeltrd 2825 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 lecple 17243 occoc 17244 joincjn 18306 meetcmee 18307 Latclat 18426 OPcops 38774 Atomscatm 38865 HLchlt 38952 LHypclh 39587 LTrncltrn 39704 trLctrl 39761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-map 8847 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 |
This theorem is referenced by: trljat1 39769 trljat2 39770 trlval3 39790 cdlemc3 39796 cdlemc5 39798 trlord 40172 cdlemg4c 40215 cdlemg4 40220 cdlemg6c 40223 cdlemg10c 40242 cdlemg10 40244 cdlemg12e 40250 cdlemg17dALTN 40267 cdlemg31a 40300 cdlemg31b 40301 cdlemg35 40316 cdlemg44a 40334 trljco 40343 trljco2 40344 tendoidcl 40372 tendococl 40375 tendoid 40376 tendopltp 40383 tendo0tp 40392 cdlemh1 40418 cdlemh2 40419 cdlemi1 40421 cdlemi 40423 cdlemk9 40442 cdlemk9bN 40443 cdlemkvcl 40445 cdlemk10 40446 cdlemk11 40452 cdlemk11u 40474 cdlemk37 40517 cdlemkfid1N 40524 cdlemkid1 40525 cdlemkid2 40527 cdlemk39s-id 40543 cdlemk48 40553 cdlemk50 40555 cdlemk51 40556 cdlemk52 40557 cdlemk39u 40571 tendoex 40578 dialss 40649 dia0 40655 diaglbN 40658 dia1dim 40664 dia2dimlem2 40668 dia2dimlem3 40669 dia2dimlem10 40676 cdlemm10N 40721 dib1dim 40768 diblss 40773 cdlemn2a 40799 dih1dimb 40843 dihopelvalcpre 40851 dih1 40889 dihmeetlem1N 40893 dihglblem5apreN 40894 dihglbcpreN 40903 dih1dimatlem 40932 |
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