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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 2735 | . . . . 5 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | eqid 2735 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
4 | trlle.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 1, 2, 3, 4 | lhpocnel 40001 | . . . 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 2735 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
8 | eqid 2735 | . . . 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 40146 | . . 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 1461 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊)) |
13 | hllat 39345 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
14 | 13 | ad2antrr 726 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ Lat) |
15 | hlop 39344 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
16 | 15 | ad2antrr 726 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ OP) |
17 | eqid 2735 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
18 | 17, 4 | lhpbase 39981 | . . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
19 | 18 | ad2antlr 727 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝑊 ∈ (Base‘𝐾)) |
20 | 17, 2 | opoccl 39176 | . . . . 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 40108 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (𝐹‘((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾)) |
23 | 21, 22 | mpd3an3 1461 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹‘((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾)) |
24 | 17, 7 | latjcl 18497 | . . . 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 1370 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊))) ∈ (Base‘𝐾)) |
26 | 17, 1, 8 | latmle2 18523 | . . 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 1370 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) ≤ 𝑊) |
28 | 12, 27 | eqbrtrd 5170 | 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 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 occoc 17306 joincjn 18369 meetcmee 18370 Latclat 18489 OPcops 39154 Atomscatm 39245 HLchlt 39332 LHypclh 39967 LTrncltrn 40084 trLctrl 40141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 |
This theorem is referenced by: trlne 40168 cdlemc5 40178 cdlemg6c 40603 cdlemg10c 40622 cdlemg10 40624 cdlemg17dALTN 40647 cdlemg27a 40675 cdlemg31b0N 40677 cdlemg31b0a 40678 cdlemg27b 40679 cdlemg31c 40682 cdlemg35 40696 cdlemh2 40799 cdlemh 40800 cdlemk3 40816 cdlemk9 40822 cdlemk9bN 40823 cdlemk10 40826 cdlemk12 40833 cdlemk14 40837 cdlemk12u 40855 cdlemkfid1N 40904 cdlemk47 40932 dia1N 41036 dia1dim 41044 dia2dimlem1 41047 dia2dimlem10 41056 dib1dim 41148 cdlemn2a 41179 dih1dimb 41223 dihopelvalcpre 41231 dihwN 41272 dihglblem5apreN 41274 dih1dimatlem 41312 |
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