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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpocat | Structured version Visualization version GIF version |
Description: The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.) |
Ref | Expression |
---|---|
lhpocat.o | ⊢ ⊥ = (oc‘𝐾) |
lhpocat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpocat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpocat | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑊) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ 𝐻) | |
2 | eqid 2736 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | lhpocat.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 2, 3 | lhpbase 37698 | . . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
5 | lhpocat.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
6 | lhpocat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 2, 5, 6, 3 | lhpoc 37714 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑊 ∈ 𝐻 ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
8 | 4, 7 | sylan2 596 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑊 ∈ 𝐻 ↔ ( ⊥ ‘𝑊) ∈ 𝐴)) |
9 | 1, 8 | mpbid 235 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑊) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 Basecbs 16666 occoc 16757 Atomscatm 36963 HLchlt 37050 LHypclh 37684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-p0 17885 df-p1 17886 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-hlat 37051 df-lhyp 37688 |
This theorem is referenced by: lhpocnel 37718 lhpmod2i2 37738 lhpmod6i1 37739 dihat 39035 |
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