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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 2823 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
3 | eqid 2823 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
4 | lhpoc.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 1, 2, 3, 4 | islhp2 37135 | . 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 36610 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊( ⋖ ‘𝐾)(1.‘𝐾) ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
9 | 5, 8 | bitrd 281 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐻 ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 occoc 16575 1.cp1 17650 ⋖ ccvr 36400 Atomscatm 36401 HLchlt 36488 LHypclh 37122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-p0 17651 df-p1 17652 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-hlat 36489 df-lhyp 37126 |
This theorem is referenced by: lhpoc2N 37153 lhpocnle 37154 lhpocat 37155 |
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