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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpocnel2 | Structured version Visualization version GIF version |
Description: The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014.) |
Ref | Expression |
---|---|
lhpocnel2.l | ⊢ ≤ = (le‘𝐾) |
lhpocnel2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpocnel2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lhpocnel2.p | ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
lhpocnel2 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpocnel2.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | eqid 2734 | . . 3 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | lhpocnel2.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | lhpocnel2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 1, 2, 3, 4 | lhpocnel 40000 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) |
6 | lhpocnel2.p | . . . 4 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
7 | 6 | eleq1i 2829 | . . 3 ⊢ (𝑃 ∈ 𝐴 ↔ ((oc‘𝐾)‘𝑊) ∈ 𝐴) |
8 | 6 | breq1i 5154 | . . . 4 ⊢ (𝑃 ≤ 𝑊 ↔ ((oc‘𝐾)‘𝑊) ≤ 𝑊) |
9 | 8 | notbii 320 | . . 3 ⊢ (¬ 𝑃 ≤ 𝑊 ↔ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊) |
10 | 7, 9 | anbi12i 628 | . 2 ⊢ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ↔ (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) |
11 | 5, 10 | sylibr 234 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 lecple 17304 occoc 17305 Atomscatm 39244 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-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-lhyp 39970 |
This theorem is referenced by: cdlemk56w 40955 diclspsn 41176 cdlemn3 41179 cdlemn4 41180 cdlemn4a 41181 cdlemn6 41184 cdlemn8 41186 cdlemn9 41187 cdlemn11a 41189 dihordlem7b 41197 dihopelvalcpre 41230 dihmeetlem1N 41272 dihglblem5apreN 41273 dihglbcpreN 41282 dihmeetlem4preN 41288 dihmeetlem13N 41301 dih1dimatlem0 41310 dih1dimatlem 41311 dihpN 41318 dihatexv 41320 dihjatcclem3 41402 dihjatcclem4 41403 |
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