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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 2823 | . . 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 37274 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ∃𝑝 ∈ 𝐴 (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) |
7 | simp11 1199 | . . . 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 1133 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) → 𝑝 ∈ 𝐴) | |
9 | simp3l 1197 | . . . 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 1200 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝑝 ∈ 𝐴 ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) → 𝐹 ∈ 𝑇) | |
11 | simp3r 1198 | . . . 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 37307 | . . . 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 1375 | . . 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 3288 | . 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 class class class wbr 5068 I cid 5461 ↾ cres 5559 ‘cfv 6357 Basecbs 16485 lecple 16574 Atomscatm 36401 HLchlt 36488 LHypclh 37122 LTrncltrn 37239 trLctrl 37296 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 |
This theorem is referenced by: ltrnnidn 37312 trlnidatb 37315 trlcone 37866 cdlemg46 37873 trljco 37878 cdlemh2 37954 cdlemh 37955 tendotr 37968 cdlemk3 37971 cdlemk12 37988 cdlemkole 37991 cdlemk14 37992 cdlemk15 37993 cdlemk1u 37997 cdlemk5u 37999 cdlemk12u 38010 cdlemk37 38052 cdlemk39 38054 cdlemkid1 38060 cdlemk47 38087 cdlemk51 38091 cdlemk52 38092 cdleml1N 38114 |
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