| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > trlle | Structured version Visualization version GIF version | ||
| Description: The trace of a lattice translation is less than the fiducial co-atom 𝑊. (Contributed by NM, 25-May-2012.) |
| Ref | Expression |
|---|---|
| trlle.l | ⊢ ≤ = (le‘𝐾) |
| trlle.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| trlle.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
| trlle.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
| Ref | Expression |
|---|---|
| trlle | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | trlle.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
| 2 | eqid 2737 | . . . . 5 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
| 4 | trlle.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 5 | 1, 2, 3, 4 | lhpocnel 40020 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ (Atoms‘𝐾) ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) |
| 6 | 5 | adantr 480 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((oc‘𝐾)‘𝑊) ∈ (Atoms‘𝐾) ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) |
| 7 | eqid 2737 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
| 8 | eqid 2737 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
| 9 | trlle.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
| 10 | trlle.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
| 11 | 1, 7, 8, 3, 4, 9, 10 | trlval2 40165 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (((oc‘𝐾)‘𝑊) ∈ (Atoms‘𝐾) ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) → (𝑅‘𝐹) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊)) |
| 12 | 6, 11 | mpd3an3 1464 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊)) |
| 13 | hllat 39364 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
| 14 | 13 | ad2antrr 726 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ Lat) |
| 15 | hlop 39363 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
| 16 | 15 | ad2antrr 726 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ OP) |
| 17 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 18 | 17, 4 | lhpbase 40000 | . . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
| 19 | 18 | ad2antlr 727 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝑊 ∈ (Base‘𝐾)) |
| 20 | 17, 2 | opoccl 39195 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) |
| 21 | 16, 19, 20 | syl2anc 584 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) |
| 22 | 17, 4, 9 | ltrncl 40127 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (𝐹‘((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾)) |
| 23 | 21, 22 | mpd3an3 1464 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹‘((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾)) |
| 24 | 17, 7 | latjcl 18484 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾) ∧ (𝐹‘((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊))) ∈ (Base‘𝐾)) |
| 25 | 14, 21, 23, 24 | syl3anc 1373 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊))) ∈ (Base‘𝐾)) |
| 26 | 17, 1, 8 | latmle2 18510 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊))) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) ≤ 𝑊) |
| 27 | 14, 25, 19, 26 | syl3anc 1373 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) ≤ 𝑊) |
| 28 | 12, 27 | eqbrtrd 5165 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 occoc 17305 joincjn 18357 meetcmee 18358 Latclat 18476 OPcops 39173 Atomscatm 39264 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 trLctrl 40160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 |
| This theorem is referenced by: trlne 40187 cdlemc5 40197 cdlemg6c 40622 cdlemg10c 40641 cdlemg10 40643 cdlemg17dALTN 40666 cdlemg27a 40694 cdlemg31b0N 40696 cdlemg31b0a 40697 cdlemg27b 40698 cdlemg31c 40701 cdlemg35 40715 cdlemh2 40818 cdlemh 40819 cdlemk3 40835 cdlemk9 40841 cdlemk9bN 40842 cdlemk10 40845 cdlemk12 40852 cdlemk14 40856 cdlemk12u 40874 cdlemkfid1N 40923 cdlemk47 40951 dia1N 41055 dia1dim 41063 dia2dimlem1 41066 dia2dimlem10 41075 dib1dim 41167 cdlemn2a 41198 dih1dimb 41242 dihopelvalcpre 41250 dihwN 41291 dihglblem5apreN 41293 dih1dimatlem 41331 |
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