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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 2736 | . . 3 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
| 3 | eqid 2736 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
| 4 | lhpoc.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 5 | 1, 2, 3, 4 | islhp2 40257 | . 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 39732 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊( ⋖ ‘𝐾)(1.‘𝐾) ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
| 9 | 5, 8 | bitrd 279 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐻 ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 class class class wbr 5098 ‘cfv 6492 Basecbs 17136 occoc 17185 1.cp1 18345 ⋖ ccvr 39522 Atomscatm 39523 HLchlt 39610 LHypclh 40244 |
| 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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-p0 18346 df-p1 18347 df-oposet 39436 df-ol 39438 df-oml 39439 df-covers 39526 df-ats 39527 df-hlat 39611 df-lhyp 40248 |
| This theorem is referenced by: lhpoc2N 40275 lhpocnle 40276 lhpocat 40277 |
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