| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpoc2N | Structured version Visualization version GIF version | ||
| Description: The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| lhpoc.b | ⊢ 𝐵 = (Base‘𝐾) |
| lhpoc.o | ⊢ ⊥ = (oc‘𝐾) |
| lhpoc.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| lhpoc.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| Ref | Expression |
|---|---|
| lhpoc2N | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐴 ↔ ( ⊥ ‘𝑊) ∈ 𝐻)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlop 39363 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
| 2 | lhpoc.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
| 3 | lhpoc.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
| 4 | 2, 3 | opoccl 39195 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ 𝐵) → ( ⊥ ‘𝑊) ∈ 𝐵) |
| 5 | 1, 4 | sylan 580 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → ( ⊥ ‘𝑊) ∈ 𝐵) |
| 6 | lhpoc.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 7 | lhpoc.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 8 | 2, 3, 6, 7 | lhpoc 40016 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑊) ∈ 𝐵) → (( ⊥ ‘𝑊) ∈ 𝐻 ↔ ( ⊥ ‘( ⊥ ‘𝑊)) ∈ 𝐴)) |
| 9 | 5, 8 | syldan 591 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (( ⊥ ‘𝑊) ∈ 𝐻 ↔ ( ⊥ ‘( ⊥ ‘𝑊)) ∈ 𝐴)) |
| 10 | 2, 3 | opococ 39196 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑊)) = 𝑊) |
| 11 | 1, 10 | sylan 580 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑊)) = 𝑊) |
| 12 | 11 | eleq1d 2826 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑊)) ∈ 𝐴 ↔ 𝑊 ∈ 𝐴)) |
| 13 | 9, 12 | bitr2d 280 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐴 ↔ ( ⊥ ‘𝑊) ∈ 𝐻)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Basecbs 17247 occoc 17305 OPcops 39173 Atomscatm 39264 HLchlt 39351 LHypclh 39986 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-p0 18470 df-p1 18471 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-hlat 39352 df-lhyp 39990 |
| This theorem is referenced by: lhprelat3N 40042 |
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