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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2llnma1 | Structured version Visualization version GIF version |
Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.) |
Ref | Expression |
---|---|
2llnm.l | ⊢ ≤ = (le‘𝐾) |
2llnm.j | ⊢ ∨ = (join‘𝐾) |
2llnm.m | ⊢ ∧ = (meet‘𝐾) |
2llnm.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
2llnma1 | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∨ 𝑃) ∧ (𝑄 ∨ 𝑅)) = 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ HL) | |
2 | simp21 1205 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ∈ 𝐴) | |
3 | eqid 2735 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | 2llnm.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | atbase 39271 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
6 | 2, 5 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ∈ (Base‘𝐾)) |
7 | simp22 1206 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴) | |
8 | simp23 1207 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑅 ∈ 𝐴) | |
9 | simp3 1137 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) | |
10 | 2llnm.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
11 | 10, 4 | hlatjcom 39350 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
12 | 1, 2, 7, 11 | syl3anc 1370 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
13 | 12 | breq2d 5160 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑄 ∨ 𝑃))) |
14 | 9, 13 | mtbid 324 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑅 ≤ (𝑄 ∨ 𝑃)) |
15 | 2llnm.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
16 | 2llnm.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
17 | 3, 15, 10, 16, 4 | 2llnma1b 39769 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑄 ∨ 𝑃)) → ((𝑄 ∨ 𝑃) ∧ (𝑄 ∨ 𝑅)) = 𝑄) |
18 | 1, 6, 7, 8, 14, 17 | syl131anc 1382 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∨ 𝑃) ∧ (𝑄 ∨ 𝑅)) = 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 joincjn 18369 meetcmee 18370 Atomscatm 39245 HLchlt 39332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-lat 18490 df-clat 18557 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 |
This theorem is referenced by: 2llnma3r 39771 2llnma2 39772 cdleme17c 40271 |
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