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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atnlej2 | Structured version Visualization version GIF version |
Description: If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.) |
Ref | Expression |
---|---|
atnlej.l | ⊢ ≤ = (le‘𝐾) |
atnlej.j | ⊢ ∨ = (join‘𝐾) |
atnlej.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atnlej2 | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 38697 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | 1 | 3ad2ant1 1132 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝐾 ∈ Lat) |
3 | simp21 1205 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴) | |
4 | eqid 2731 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | atnlej.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 4, 5 | atbase 38623 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
7 | 3, 6 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃 ∈ (Base‘𝐾)) |
8 | simp22 1206 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴) | |
9 | 4, 5 | atbase 38623 | . . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
10 | 8, 9 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑄 ∈ (Base‘𝐾)) |
11 | simp23 1207 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴) | |
12 | 4, 5 | atbase 38623 | . . 3 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
13 | 11, 12 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑅 ∈ (Base‘𝐾)) |
14 | simp3 1137 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) | |
15 | atnlej.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
16 | atnlej.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
17 | 4, 15, 16 | latnlej1r 18421 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑅) |
18 | 2, 7, 10, 13, 14, 17 | syl131anc 1382 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 lecple 17211 joincjn 18274 Latclat 18394 Atomscatm 38597 HLchlt 38684 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-lub 18309 df-join 18311 df-lat 18395 df-ats 38601 df-atl 38632 df-cvlat 38656 df-hlat 38685 |
This theorem is referenced by: lplnri2N 38889 lplnri3N 38890 lplnexllnN 38899 dalem41 39048 paddasslem2 39156 4atexlemc 39404 |
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