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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpmcvr2 | Structured version Visualization version GIF version |
Description: Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013.) |
Ref | Expression |
---|---|
lhpmcvr2.b | ⊢ 𝐵 = (Base‘𝐾) |
lhpmcvr2.l | ⊢ ≤ = (le‘𝐾) |
lhpmcvr2.j | ⊢ ∨ = (join‘𝐾) |
lhpmcvr2.m | ⊢ ∧ = (meet‘𝐾) |
lhpmcvr2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpmcvr2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpmcvr2 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpmcvr2.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | lhpmcvr2.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | lhpmcvr2.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | eqid 2773 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
5 | lhpmcvr2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | 1, 2, 3, 4, 5 | lhpmcvr 36637 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑋 ∧ 𝑊)( ⋖ ‘𝐾)𝑋) |
7 | simpll 755 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝐾 ∈ HL) | |
8 | simprl 759 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝑋 ∈ 𝐵) | |
9 | 1, 5 | lhpbase 36612 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
10 | 9 | ad2antlr 715 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝑊 ∈ 𝐵) |
11 | lhpmcvr2.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
12 | lhpmcvr2.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
13 | 1, 2, 11, 3, 4, 12 | cvrval5 36029 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑋 ∧ 𝑊)( ⋖ ‘𝐾)𝑋 ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) |
14 | 7, 8, 10, 13 | syl3anc 1352 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ((𝑋 ∧ 𝑊)( ⋖ ‘𝐾)𝑋 ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) |
15 | 6, 14 | mpbid 224 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝑝 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∃wrex 3084 class class class wbr 4926 ‘cfv 6186 (class class class)co 6975 Basecbs 16338 lecple 16427 joincjn 17425 meetcmee 17426 ⋖ ccvr 35876 Atomscatm 35877 HLchlt 35964 LHypclh 36598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-proset 17409 df-poset 17427 df-plt 17439 df-lub 17455 df-glb 17456 df-join 17457 df-meet 17458 df-p0 17520 df-p1 17521 df-lat 17527 df-clat 17589 df-oposet 35790 df-ol 35792 df-oml 35793 df-covers 35880 df-ats 35881 df-atl 35912 df-cvlat 35936 df-hlat 35965 df-lhyp 36602 |
This theorem is referenced by: lhpmcvr5N 36641 cdleme29ex 36988 cdleme29c 36990 cdlemefrs29cpre1 37012 cdlemefr29exN 37016 cdleme32d 37058 cdleme32f 37060 cdleme48gfv1 37150 cdlemg7fvbwN 37221 cdlemg7aN 37239 dihlsscpre 37848 dihvalcqpre 37849 dihord6apre 37870 dihord4 37872 dihord5b 37873 dihord5apre 37876 dihmeetlem1N 37904 dihglblem5apreN 37905 dihglbcpreN 37914 |
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