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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpmat | Structured version Visualization version GIF version |
Description: An element covered by the lattice unit, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.) |
Ref | Expression |
---|---|
lhpmat.l | ⊢ ≤ = (le‘𝐾) |
lhpmat.m | ⊢ ∧ = (meet‘𝐾) |
lhpmat.z | ⊢ 0 = (0.‘𝐾) |
lhpmat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpmat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpmat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∧ 𝑊) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 813 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ 𝑃 ≤ 𝑊) | |
2 | hlatl 35150 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
3 | 2 | ad2antrr 764 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ AtLat) |
4 | simprl 811 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴) | |
5 | eqid 2760 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | lhpmat.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | 5, 6 | lhpbase 35787 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
8 | 7 | ad2antlr 765 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾)) |
9 | lhpmat.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
10 | lhpmat.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
11 | lhpmat.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
12 | lhpmat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
13 | 5, 9, 10, 11, 12 | atnle 35107 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑊 ∈ (Base‘𝐾)) → (¬ 𝑃 ≤ 𝑊 ↔ (𝑃 ∧ 𝑊) = 0 )) |
14 | 3, 4, 8, 13 | syl3anc 1477 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (¬ 𝑃 ≤ 𝑊 ↔ (𝑃 ∧ 𝑊) = 0 )) |
15 | 1, 14 | mpbid 222 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∧ 𝑊) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 lecple 16150 meetcmee 17146 0.cp0 17238 Atomscatm 35053 AtLatcal 35054 HLchlt 35140 LHypclh 35773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-lat 17247 df-covers 35056 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 df-lhyp 35777 |
This theorem is referenced by: lhpmatb 35820 lhp2at0 35821 lhpelim 35826 lhple 35831 idltrn 35939 ltrnmw 35940 trl0 35960 cdleme0e 36007 cdleme2 36018 cdleme7c 36035 cdleme22d 36133 cdlemefrs29pre00 36185 cdlemefrs29bpre0 36186 cdlemefrs29cpre1 36188 cdleme32fva 36227 cdleme35d 36242 cdleme42ke 36275 cdlemeg46frv 36315 cdleme50trn3 36343 cdlemg2fv2 36390 cdlemg8a 36417 cdlemg10bALTN 36426 cdlemh2 36606 cdlemk9 36629 cdlemk9bN 36630 dia2dimlem1 36855 dihvalcqat 37030 dihjatc1 37102 |
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