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Mirrors > Home > MPE Home > Th. List > Mathboxes > islpln2ah | Structured version Visualization version GIF version |
Description: The predicate "is a lattice plane" for join of atoms. Version of islpln2a 37185 expressed with an abbreviation hypothesis. (Contributed by NM, 30-Jul-2012.) |
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 |
---|---|
islpln2ah | ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑌 ∈ 𝑃 ↔ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islpln2a.y | . . 3 ⊢ 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑆) | |
2 | 1 | eleq1i 2823 | . 2 ⊢ (𝑌 ∈ 𝑃 ↔ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∈ 𝑃) |
3 | islpln2a.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | islpln2a.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
5 | islpln2a.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | islpln2a.p | . . 3 ⊢ 𝑃 = (LPlanes‘𝐾) | |
7 | 3, 4, 5, 6 | islpln2a 37185 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (((𝑄 ∨ 𝑅) ∨ 𝑆) ∈ 𝑃 ↔ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅)))) |
8 | 2, 7 | syl5bb 286 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑌 ∈ 𝑃 ↔ (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 class class class wbr 5030 ‘cfv 6339 (class class class)co 7170 lecple 16675 joincjn 17670 Atomscatm 36900 HLchlt 36987 LPlanesclpl 37129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-glb 17701 df-join 17702 df-meet 17703 df-p0 17765 df-lat 17772 df-clat 17834 df-oposet 36813 df-ol 36815 df-oml 36816 df-covers 36903 df-ats 36904 df-atl 36935 df-cvlat 36959 df-hlat 36988 df-llines 37135 df-lplanes 37136 |
This theorem is referenced by: lplnriaN 37187 lplnribN 37188 lplnric 37189 lplnri1 37190 |
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