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Mirrors > Home > MPE Home > Th. List > Mathboxes > trl0 | Structured version Visualization version GIF version |
Description: If an atom not under the fiducial co-atom 𝑊 equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.) |
Ref | Expression |
---|---|
trl0.l | ⊢ ≤ = (le‘𝐾) |
trl0.z | ⊢ 0 = (0.‘𝐾) |
trl0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
trl0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trl0.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trl0.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trl0 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑅‘𝐹) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1054 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simp3l 1082 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → 𝐹 ∈ 𝑇) | |
3 | simp2 1055 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
4 | trl0.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
5 | eqid 2610 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
6 | eqid 2610 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
7 | trl0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | trl0.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | trl0.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
10 | trl0.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
11 | 4, 5, 6, 7, 8, 9, 10 | trlval2 34468 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃(join‘𝐾)(𝐹‘𝑃))(meet‘𝐾)𝑊)) |
12 | 1, 2, 3, 11 | syl3anc 1318 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑅‘𝐹) = ((𝑃(join‘𝐾)(𝐹‘𝑃))(meet‘𝐾)𝑊)) |
13 | simp3r 1083 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝐹‘𝑃) = 𝑃) | |
14 | 13 | oveq2d 6565 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑃(join‘𝐾)(𝐹‘𝑃)) = (𝑃(join‘𝐾)𝑃)) |
15 | simp1l 1078 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → 𝐾 ∈ HL) | |
16 | simp2l 1080 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → 𝑃 ∈ 𝐴) | |
17 | 5, 7 | hlatjidm 33673 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃) |
18 | 15, 16, 17 | syl2anc 691 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑃(join‘𝐾)𝑃) = 𝑃) |
19 | 14, 18 | eqtrd 2644 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑃(join‘𝐾)(𝐹‘𝑃)) = 𝑃) |
20 | 19 | oveq1d 6564 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → ((𝑃(join‘𝐾)(𝐹‘𝑃))(meet‘𝐾)𝑊) = (𝑃(meet‘𝐾)𝑊)) |
21 | trl0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
22 | 4, 6, 21, 7, 8 | lhpmat 34334 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃(meet‘𝐾)𝑊) = 0 ) |
23 | 1, 3, 22 | syl2anc 691 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑃(meet‘𝐾)𝑊) = 0 ) |
24 | 12, 20, 23 | 3eqtrd 2648 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑅‘𝐹) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 lecple 15775 joincjn 16767 meetcmee 16768 0.cp0 16860 Atomscatm 33568 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 trLctrl 34463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-lat 16869 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 |
This theorem is referenced by: trlator0 34476 ltrnnidn 34479 trlid0 34481 trlnidatb 34482 trlnle 34491 trlval3 34492 trlval4 34493 cdlemc6 34501 cdlemg31d 35006 |
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