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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 2821 | . . . . 5 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | eqid 2821 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
4 | trlle.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 1, 2, 3, 4 | lhpocnel 37169 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ (Atoms‘𝐾) ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) |
6 | 5 | adantr 483 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((oc‘𝐾)‘𝑊) ∈ (Atoms‘𝐾) ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) |
7 | eqid 2821 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
8 | eqid 2821 | . . . 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 37314 | . . 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 1458 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊)) |
13 | hllat 36514 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
14 | 13 | ad2antrr 724 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ Lat) |
15 | hlop 36513 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
16 | 15 | ad2antrr 724 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ OP) |
17 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
18 | 17, 4 | lhpbase 37149 | . . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
19 | 18 | ad2antlr 725 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝑊 ∈ (Base‘𝐾)) |
20 | 17, 2 | opoccl 36345 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) |
21 | 16, 19, 20 | syl2anc 586 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) |
22 | 17, 4, 9 | ltrncl 37276 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (𝐹‘((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾)) |
23 | 21, 22 | mpd3an3 1458 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹‘((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾)) |
24 | 17, 7 | latjcl 17661 | . . . 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 1367 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊))) ∈ (Base‘𝐾)) |
26 | 17, 1, 8 | latmle2 17687 | . . 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 1367 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((((oc‘𝐾)‘𝑊)(join‘𝐾)(𝐹‘((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) ≤ 𝑊) |
28 | 12, 27 | eqbrtrd 5088 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 lecple 16572 occoc 16573 joincjn 17554 meetcmee 17555 Latclat 17655 OPcops 36323 Atomscatm 36414 HLchlt 36501 LHypclh 37135 LTrncltrn 37252 trLctrl 37309 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 |
This theorem is referenced by: trlne 37336 cdlemc5 37346 cdlemg6c 37771 cdlemg10c 37790 cdlemg10 37792 cdlemg17dALTN 37815 cdlemg27a 37843 cdlemg31b0N 37845 cdlemg31b0a 37846 cdlemg27b 37847 cdlemg31c 37850 cdlemg35 37864 cdlemh2 37967 cdlemh 37968 cdlemk3 37984 cdlemk9 37990 cdlemk9bN 37991 cdlemk10 37994 cdlemk12 38001 cdlemk14 38005 cdlemk12u 38023 cdlemkfid1N 38072 cdlemk47 38100 dia1N 38204 dia1dim 38212 dia2dimlem1 38215 dia2dimlem10 38224 dib1dim 38316 cdlemn2a 38347 dih1dimb 38391 dihopelvalcpre 38399 dihwN 38440 dihglblem5apreN 38442 dih1dimatlem 38480 |
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