Theorem atnlej2 37131

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 37114	. . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	3ad2ant1 1135	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝐾 ∈ Lat)
3		simp21 1208	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴)
4		eqid 2737	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		atnlej.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 37040	. . 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 1209	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴)
9	4, 5	atbase 37040	. . 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 1210	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴)
12	4, 5	atbase 37040	. . 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 1140	. 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 17964	. 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 1385	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940   class class class wbr 5053  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  joincjn 17818  Latclat 17937  Atomscatm 37014  HLchlt 37101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-lub 17852  df-join 17854  df-lat 17938  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102

This theorem is referenced by:  lplnri2N  37305  lplnri3N  37306  lplnexllnN  37315  dalem41  37464  paddasslem2  37572  4atexlemc  37820