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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpocnel | 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, 25-May-2012.) |
Ref | Expression |
---|---|
lhpocnel.l | ⊢ ≤ = (le‘𝐾) |
lhpocnel.o | ⊢ ⊥ = (oc‘𝐾) |
lhpocnel.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpocnel.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpocnel | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊) ≤ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpocnel.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
2 | lhpocnel.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | lhpocnel.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 1, 2, 3 | lhpocat 36627 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑊) ∈ 𝐴) |
5 | lhpocnel.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
6 | 5, 1, 3 | lhpocnle 36626 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ¬ ( ⊥ ‘𝑊) ≤ 𝑊) |
7 | 4, 6 | jca 504 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊) ≤ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 class class class wbr 4925 ‘cfv 6185 lecple 16426 occoc 16427 Atomscatm 35873 HLchlt 35960 LHypclh 36594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-proset 17408 df-poset 17426 df-plt 17438 df-lub 17454 df-glb 17455 df-meet 17457 df-p0 17519 df-p1 17520 df-lat 17526 df-oposet 35786 df-ol 35788 df-oml 35789 df-covers 35876 df-ats 35877 df-atl 35908 df-cvlat 35932 df-hlat 35961 df-lhyp 36598 |
This theorem is referenced by: lhpocnel2 36629 trlcl 36774 trlle 36794 cdlemk19w 37582 dia2dimlem8 37681 dicssdvh 37796 dicvaddcl 37800 dicvscacl 37801 dicn0 37802 dih1 37896 dihatlat 37944 |
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