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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlnidat | Structured version Visualization version GIF version |
Description: The trace of a lattice translation other than the identity is an atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.) |
Ref | Expression |
---|---|
trlnidat.b | ⊢ 𝐵 = (Base‘𝐾) |
trlnidat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
trlnidat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trlnidat.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trlnidat.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trlnidat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → (𝑅‘𝐹) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlnidat.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | trlnidat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | trlnidat.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | trlnidat.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | ltrnnid 37154 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ∃𝑝 ∈ 𝐴 (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) |
7 | simp11 1195 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | simp2 1129 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) → 𝑝 ∈ 𝐴) | |
9 | simp3l 1193 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) → ¬ 𝑝(le‘𝐾)𝑊) | |
10 | simp12 1196 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) → 𝐹 ∈ 𝑇) | |
11 | simp3r 1194 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) → (𝐹‘𝑝) ≠ 𝑝) | |
12 | trlnidat.r | . . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
13 | 2, 3, 4, 5, 12 | trlat 37187 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝(le‘𝐾)𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑝) ≠ 𝑝)) → (𝑅‘𝐹) ∈ 𝐴) |
14 | 7, 8, 9, 10, 11, 13 | syl122anc 1371 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) → (𝑅‘𝐹) ∈ 𝐴) |
15 | 14 | rexlimdv3a 3286 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → (∃𝑝 ∈ 𝐴 (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝) → (𝑅‘𝐹) ∈ 𝐴)) |
16 | 6, 15 | mpd 15 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → (𝑅‘𝐹) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3016 ∃wrex 3139 class class class wbr 5058 I cid 5453 ↾ cres 5551 ‘cfv 6349 Basecbs 16473 lecple 16562 Atomscatm 36281 HLchlt 36368 LHypclh 37002 LTrncltrn 37119 trLctrl 37176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8398 df-proset 17528 df-poset 17546 df-plt 17558 df-lub 17574 df-glb 17575 df-join 17576 df-meet 17577 df-p0 17639 df-p1 17640 df-lat 17646 df-clat 17708 df-oposet 36194 df-ol 36196 df-oml 36197 df-covers 36284 df-ats 36285 df-atl 36316 df-cvlat 36340 df-hlat 36369 df-lhyp 37006 df-laut 37007 df-ldil 37122 df-ltrn 37123 df-trl 37177 |
This theorem is referenced by: ltrnnidn 37192 trlnidatb 37195 trlcone 37746 cdlemg46 37753 trljco 37758 cdlemh2 37834 cdlemh 37835 tendotr 37848 cdlemk3 37851 cdlemk12 37868 cdlemkole 37871 cdlemk14 37872 cdlemk15 37873 cdlemk1u 37877 cdlemk5u 37879 cdlemk12u 37890 cdlemk37 37932 cdlemk39 37934 cdlemkid1 37940 cdlemk47 37967 cdlemk51 37971 cdlemk52 37972 cdleml1N 37994 |
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