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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpoc | Structured version Visualization version GIF version |
Description: The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.) |
Ref | Expression |
---|---|
lhpoc.b | ⊢ 𝐵 = (Base‘𝐾) |
lhpoc.o | ⊢ ⊥ = (oc‘𝐾) |
lhpoc.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpoc.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpoc | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐻 ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpoc.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2731 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
3 | eqid 2731 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
4 | lhpoc.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 1, 2, 3, 4 | islhp2 38673 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐻 ↔ 𝑊( ⋖ ‘𝐾)(1.‘𝐾))) |
6 | lhpoc.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
7 | lhpoc.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 1, 2, 6, 3, 7 | 1cvrco 38148 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊( ⋖ ‘𝐾)(1.‘𝐾) ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
9 | 5, 8 | bitrd 278 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐻 ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5141 ‘cfv 6532 Basecbs 17126 occoc 17187 1.cp1 18359 ⋖ ccvr 37937 Atomscatm 37938 HLchlt 38025 LHypclh 38660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-proset 18230 df-poset 18248 df-plt 18265 df-lub 18281 df-glb 18282 df-p0 18360 df-p1 18361 df-oposet 37851 df-ol 37853 df-oml 37854 df-covers 37941 df-ats 37942 df-hlat 38026 df-lhyp 38664 |
This theorem is referenced by: lhpoc2N 38691 lhpocnle 38692 lhpocat 38693 |
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