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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 2826 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
3 | eqid 2826 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
4 | lhpoc.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 1, 2, 3, 4 | islhp2 36073 | . 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 35548 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊( ⋖ ‘𝐾)(1.‘𝐾) ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
9 | 5, 8 | bitrd 271 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐻 ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4874 ‘cfv 6124 Basecbs 16223 occoc 16314 1.cp1 17392 ⋖ ccvr 35338 Atomscatm 35339 HLchlt 35426 LHypclh 36060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-proset 17282 df-poset 17300 df-plt 17312 df-lub 17328 df-glb 17329 df-p0 17393 df-p1 17394 df-oposet 35252 df-ol 35254 df-oml 35255 df-covers 35342 df-ats 35343 df-hlat 35427 df-lhyp 36064 |
This theorem is referenced by: lhpoc2N 36091 lhpocnle 36092 lhpocat 36093 |
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