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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 2825 | . . 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 36088 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) |
6 | lhpocnel2.p | . . . 4 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
7 | 6 | eleq1i 2897 | . . 3 ⊢ (𝑃 ∈ 𝐴 ↔ ((oc‘𝐾)‘𝑊) ∈ 𝐴) |
8 | 6 | breq1i 4882 | . . . 4 ⊢ (𝑃 ≤ 𝑊 ↔ ((oc‘𝐾)‘𝑊) ≤ 𝑊) |
9 | 8 | notbii 312 | . . 3 ⊢ (¬ 𝑃 ≤ 𝑊 ↔ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊) |
10 | 7, 9 | anbi12i 620 | . 2 ⊢ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ↔ (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊)) |
11 | 5, 10 | sylibr 226 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ‘cfv 6127 lecple 16319 occoc 16320 Atomscatm 35333 HLchlt 35420 LHypclh 36054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-oposet 35246 df-ol 35248 df-oml 35249 df-covers 35336 df-ats 35337 df-atl 35368 df-cvlat 35392 df-hlat 35421 df-lhyp 36058 |
This theorem is referenced by: cdlemk56w 37043 diclspsn 37264 cdlemn3 37267 cdlemn4 37268 cdlemn4a 37269 cdlemn6 37272 cdlemn8 37274 cdlemn9 37275 cdlemn11a 37277 dihordlem7b 37285 dihopelvalcpre 37318 dihmeetlem1N 37360 dihglblem5apreN 37361 dihglbcpreN 37370 dihmeetlem4preN 37376 dihmeetlem13N 37389 dih1dimatlem0 37398 dih1dimatlem 37399 dihpN 37406 dihatexv 37408 dihjatcclem3 37490 dihjatcclem4 37491 |
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