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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 unity, 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 773 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ 𝑃 ≤ 𝑊) | |
2 | hlatl 39341 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
3 | 2 | ad2antrr 726 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ AtLat) |
4 | simprl 771 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴) | |
5 | eqid 2734 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | lhpmat.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | 5, 6 | lhpbase 39980 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
8 | 7 | ad2antlr 727 | . . 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 39298 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑊 ∈ (Base‘𝐾)) → (¬ 𝑃 ≤ 𝑊 ↔ (𝑃 ∧ 𝑊) = 0 )) |
14 | 3, 4, 8, 13 | syl3anc 1370 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (¬ 𝑃 ≤ 𝑊 ↔ (𝑃 ∧ 𝑊) = 0 )) |
15 | 1, 14 | mpbid 232 | 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 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 lecple 17304 meetcmee 18369 0.cp0 18480 Atomscatm 39244 AtLatcal 39245 HLchlt 39331 LHypclh 39966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-lat 18489 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-lhyp 39970 |
This theorem is referenced by: lhpmatb 40013 lhp2at0 40014 lhpelim 40019 lhple 40024 idltrn 40132 ltrnmw 40133 trl0 40152 cdleme0e 40199 cdleme2 40210 cdleme7c 40227 cdleme22d 40325 cdlemefrs29pre00 40377 cdlemefrs29bpre0 40378 cdlemefrs29cpre1 40380 cdleme32fva 40419 cdleme35d 40434 cdleme42ke 40467 cdlemeg46frv 40507 cdleme50trn3 40535 cdlemg2fv2 40582 cdlemg8a 40609 cdlemg10bALTN 40618 cdlemh2 40798 cdlemk9 40821 cdlemk9bN 40822 dia2dimlem1 41046 dihvalcqat 41221 dihjatc1 41293 |
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