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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnriaN | 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 |
---|---|
islpln2a.l | ⊢ ≤ = (le‘𝐾) |
islpln2a.j | ⊢ ∨ = (join‘𝐾) |
islpln2a.a | ⊢ 𝐴 = (Atoms‘𝐾) |
islpln2a.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
islpln2a.y | ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆) |
Ref | Expression |
---|---|
lplnriaN | ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → ¬ 𝑄 ≤ (𝑅 ∨ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islpln2a.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | islpln2a.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | islpln2a.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | islpln2a.p | . . . 4 ⊢ 𝑃 = (LPlanes‘𝐾) | |
5 | islpln2a.y | . . . 4 ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆) | |
6 | 1, 2, 3, 4, 5 | islpln2ah 36689 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑌 ∈ 𝑃 ↔ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅)))) |
7 | 1, 2, 3 | hlatcon3 36591 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))) → ¬ 𝑄 ≤ (𝑅 ∨ 𝑆)) |
8 | 7 | 3expia 1117 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅)) → ¬ 𝑄 ≤ (𝑅 ∨ 𝑆))) |
9 | 6, 8 | sylbid 242 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑌 ∈ 𝑃 → ¬ 𝑄 ≤ (𝑅 ∨ 𝑆))) |
10 | 9 | 3impia 1113 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → ¬ 𝑄 ≤ (𝑅 ∨ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 lecple 16575 joincjn 17557 Atomscatm 36403 HLchlt 36490 LPlanesclpl 36632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-lat 17659 df-clat 17721 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-llines 36638 df-lplanes 36639 |
This theorem is referenced by: lplnri2N 36694 lplnllnneN 36696 |
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