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 36492 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
2 | lhpoc.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | lhpoc.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
4 | 2, 3 | opoccl 36324 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ 𝐵) → ( ⊥ ‘𝑊) ∈ 𝐵) |
5 | 1, 4 | sylan 582 | . . 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 37144 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑊) ∈ 𝐵) → (( ⊥ ‘𝑊) ∈ 𝐻 ↔ ( ⊥ ‘( ⊥ ‘𝑊)) ∈ 𝐴)) |
9 | 5, 8 | syldan 593 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (( ⊥ ‘𝑊) ∈ 𝐻 ↔ ( ⊥ ‘( ⊥ ‘𝑊)) ∈ 𝐴)) |
10 | 2, 3 | opococ 36325 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑊)) = 𝑊) |
11 | 1, 10 | sylan 582 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑊)) = 𝑊) |
12 | 11 | eleq1d 2897 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑊)) ∈ 𝐴 ↔ 𝑊 ∈ 𝐴)) |
13 | 9, 12 | bitr2d 282 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐴 ↔ ( ⊥ ‘𝑊) ∈ 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 Basecbs 16477 occoc 16567 OPcops 36302 Atomscatm 36393 HLchlt 36480 LHypclh 37114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-p0 17643 df-p1 17644 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-hlat 36481 df-lhyp 37118 |
This theorem is referenced by: lhprelat3N 37170 |
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