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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnri1 | Structured version Visualization version GIF version |
Description: Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) |
Ref | Expression |
---|---|
lplnri1.j | ⊢ ∨ = (join‘𝐾) |
lplnri1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lplnri1.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
lplnri1.y | ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆) |
Ref | Expression |
---|---|
lplnri1 | ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → 𝑄 ≠ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | lplnri1.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | lplnri1.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | lplnri1.p | . . . 4 ⊢ 𝑃 = (LPlanes‘𝐾) | |
5 | lplnri1.y | . . . 4 ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆) | |
6 | 1, 2, 3, 4, 5 | islpln2ah 37979 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑌 ∈ 𝑃 ↔ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆(le‘𝐾)(𝑄 ∨ 𝑅)))) |
7 | 6 | biimp3a 1469 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → (𝑄 ≠ 𝑅 ∧ ¬ 𝑆(le‘𝐾)(𝑄 ∨ 𝑅))) |
8 | 7 | simpld 495 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → 𝑄 ≠ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 class class class wbr 5103 ‘cfv 6493 (class class class)co 7353 lecple 17132 joincjn 18192 Atomscatm 37692 HLchlt 37779 LPlanesclpl 37922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-proset 18176 df-poset 18194 df-plt 18211 df-lub 18227 df-glb 18228 df-join 18229 df-meet 18230 df-p0 18306 df-lat 18313 df-clat 18380 df-oposet 37605 df-ol 37607 df-oml 37608 df-covers 37695 df-ats 37696 df-atl 37727 df-cvlat 37751 df-hlat 37780 df-llines 37928 df-lplanes 37929 |
This theorem is referenced by: dalem1 38089 dalemdea 38092 |
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