Theorem atnlej2 38715

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.)

Hypotheses

Ref	Expression
atnlej.l	⊢ ≤ = (le‘𝐾)
atnlej.j	⊢ ∨ = (join‘𝐾)
atnlej.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
atnlej2	⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑅)

Proof of Theorem atnlej2

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)

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