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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpjat1 | Structured version Visualization version GIF version |
Description: The join of a co-atom (hyperplane) and an atom not under it is the lattice unity. (Contributed by NM, 18-May-2012.) |
Ref | Expression |
---|---|
lhpjat.l | ⊢ ≤ = (le‘𝐾) |
lhpjat.j | ⊢ ∨ = (join‘𝐾) |
lhpjat.u | ⊢ 1 = (1.‘𝐾) |
lhpjat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpjat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpjat1 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 766 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL) | |
2 | eqid 2733 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | lhpjat.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 2, 3 | lhpbase 38807 | . . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
5 | 4 | ad2antlr 726 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾)) |
6 | simprl 770 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴) | |
7 | lhpjat.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
8 | eqid 2733 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | 7, 8, 3 | lhp1cvr 38808 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊( ⋖ ‘𝐾) 1 ) |
10 | 9 | adantr 482 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊( ⋖ ‘𝐾) 1 ) |
11 | simprr 772 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ 𝑃 ≤ 𝑊) | |
12 | lhpjat.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
13 | lhpjat.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
14 | lhpjat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
15 | 2, 12, 13, 7, 8, 14 | 1cvrjat 38284 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ (Base‘𝐾) ∧ 𝑃 ∈ 𝐴) ∧ (𝑊( ⋖ ‘𝐾) 1 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = 1 ) |
16 | 1, 5, 6, 10, 11, 15 | syl32anc 1379 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 lecple 17200 joincjn 18260 1.cp1 18373 ⋖ ccvr 38070 Atomscatm 38071 HLchlt 38158 LHypclh 38793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-oposet 37984 df-ol 37986 df-oml 37987 df-covers 38074 df-ats 38075 df-atl 38106 df-cvlat 38130 df-hlat 38159 df-lhyp 38797 |
This theorem is referenced by: lhpjat2 38830 lhpj1 38831 trljat1 38975 trljat2 38976 cdlemc1 39000 cdlemc6 39005 cdleme20c 39120 cdleme20j 39127 trlcolem 39535 |
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