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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnribN | 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 |
---|---|
lplnribN | ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → ¬ 𝑅 ≤ (𝑄 ∨ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islpln2a.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
2 | islpln2a.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
3 | islpln2a.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 1, 2, 3 | 3noncolr1N 36588 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))) → (𝑆 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑄))) |
5 | 4 | simprd 498 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))) → ¬ 𝑅 ≤ (𝑆 ∨ 𝑄)) |
6 | 5 | 3expia 1117 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅)) → ¬ 𝑅 ≤ (𝑆 ∨ 𝑄))) |
7 | islpln2a.p | . . . 4 ⊢ 𝑃 = (LPlanes‘𝐾) | |
8 | islpln2a.y | . . . 4 ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆) | |
9 | 1, 2, 3, 7, 8 | islpln2ah 36687 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑌 ∈ 𝑃 ↔ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅)))) |
10 | 2, 3 | hlatjcom 36506 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑄 ∨ 𝑆) = (𝑆 ∨ 𝑄)) |
11 | 10 | 3adant3r2 1179 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑄 ∨ 𝑆) = (𝑆 ∨ 𝑄)) |
12 | 11 | breq2d 5080 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑅 ≤ (𝑄 ∨ 𝑆) ↔ 𝑅 ≤ (𝑆 ∨ 𝑄))) |
13 | 12 | notbid 320 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (¬ 𝑅 ≤ (𝑄 ∨ 𝑆) ↔ ¬ 𝑅 ≤ (𝑆 ∨ 𝑄))) |
14 | 6, 9, 13 | 3imtr4d 296 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑌 ∈ 𝑃 → ¬ 𝑅 ≤ (𝑄 ∨ 𝑆))) |
15 | 14 | 3impia 1113 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑌 ∈ 𝑃) → ¬ 𝑅 ≤ (𝑄 ∨ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 lecple 16574 joincjn 17556 Atomscatm 36401 HLchlt 36488 LPlanesclpl 36630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 |
This theorem is referenced by: lplnri3N 36693 |
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