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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 2818 | . . 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 37034 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) |
6 | lhpocnel2.p | . . . 4 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
7 | 6 | eleq1i 2900 | . . 3 ⊢ (𝑃 ∈ 𝐴 ↔ ((oc‘𝐾)‘𝑊) ∈ 𝐴) |
8 | 6 | breq1i 5064 | . . . 4 ⊢ (𝑃 ≤ 𝑊 ↔ ((oc‘𝐾)‘𝑊) ≤ 𝑊) |
9 | 8 | notbii 321 | . . 3 ⊢ (¬ 𝑃 ≤ 𝑊 ↔ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊) |
10 | 7, 9 | anbi12i 626 | . 2 ⊢ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ↔ (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) |
11 | 5, 10 | sylibr 235 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 lecple 16560 occoc 16561 Atomscatm 36279 HLchlt 36366 LHypclh 37000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-lhyp 37004 |
This theorem is referenced by: cdlemk56w 37989 diclspsn 38210 cdlemn3 38213 cdlemn4 38214 cdlemn4a 38215 cdlemn6 38218 cdlemn8 38220 cdlemn9 38221 cdlemn11a 38223 dihordlem7b 38231 dihopelvalcpre 38264 dihmeetlem1N 38306 dihglblem5apreN 38307 dihglbcpreN 38316 dihmeetlem4preN 38322 dihmeetlem13N 38335 dih1dimatlem0 38344 dih1dimatlem 38345 dihpN 38352 dihatexv 38354 dihjatcclem3 38436 dihjatcclem4 38437 |
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