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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnri2N | 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.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lplnri1.j | ⊢ ∨ = (join‘𝐾) |
lplnri1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lplnri1.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
lplnri1.y | ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆) |
Ref | Expression |
---|---|
lplnri2N | ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → 𝑄 ≠ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | lplnri1.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | lplnri1.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | lplnri1.p | . . 3 ⊢ 𝑃 = (LPlanes‘𝐾) | |
5 | lplnri1.y | . . 3 ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆) | |
6 | 1, 2, 3, 4, 5 | lplnriaN 38290 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → ¬ 𝑄(le‘𝐾)(𝑅 ∨ 𝑆)) |
7 | 1, 2, 3 | atnlej2 38120 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)(𝑅 ∨ 𝑆)) → 𝑄 ≠ 𝑆) |
8 | 6, 7 | syld3an3 1409 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → 𝑄 ≠ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5142 ‘cfv 6533 (class class class)co 7394 lecple 17188 joincjn 18248 Atomscatm 38002 HLchlt 38089 LPlanesclpl 38232 |
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 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-proset 18232 df-poset 18250 df-plt 18267 df-lub 18283 df-glb 18284 df-join 18285 df-meet 18286 df-p0 18362 df-lat 18369 df-clat 18436 df-oposet 37915 df-ol 37917 df-oml 37918 df-covers 38005 df-ats 38006 df-atl 38037 df-cvlat 38061 df-hlat 38090 df-llines 38238 df-lplanes 38239 |
This theorem is referenced by: (None) |
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