Theorem dalem48 37713

Description: Lemma for dath 37729. Analogue of dalem45 37710 for 𝑃𝑄. (Contributed by NM, 16-Aug-2012.)

Hypotheses

Ref	Expression
dalem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
dalem.l	⊢ ≤ = (le‘𝐾)
dalem.j	⊢ ∨ = (join‘𝐾)
dalem.a	⊢ 𝐴 = (Atoms‘𝐾)
dalem.ps	⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
dalem44.m	⊢ ∧ = (meet‘𝐾)
dalem44.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem44.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem44.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem44.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem44.h	⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem44.i	⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))

Assertion

Ref	Expression
dalem48	⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ (𝑃 ∨ 𝑄))

Proof of Theorem dalem48

Step	Hyp	Ref	Expression
1		dalem.ph	. . . 4 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
2	1	dalemkelat 37617	. . 3 ⊢ (𝜑 → 𝐾 ∈ Lat)
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ Lat)
4		dalem.ps	. . . 4 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
5		dalem.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	dalemcceb 37682	. . 3 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
7	6	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
8		dalem.j	. . . 4 ⊢ ∨ = (join‘𝐾)
9	1, 8, 5	dalempjqeb 37638	. . 3 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
10	9	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
11	1, 5	dalemreb 37634	. . 3 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
12	11	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ (Base‘𝐾))
13	4	dalem-ccly 37678	. . . 4 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)
14		dalem44.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
15	14	breq2i 5086	. . . 4 ⊢ (𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
16	13, 15	sylnib 327	. . 3 ⊢ (𝜓 → ¬ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
17	16	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
18		eqid 2739	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		dalem.l	. . 3 ⊢ ≤ = (le‘𝐾)
20	18, 19, 8	latnlej2l 18159	. 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) ∧ ¬ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) → ¬ 𝑐 ≤ (𝑃 ∨ 𝑄))
21	3, 7, 10, 12, 17, 20	syl131anc 1381	1 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ (𝑃 ∨ 𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109   ≠ wne 2944   class class class wbr 5078  ‘cfv 6430  (class class class)co 7268  Basecbs 16893  lecple 16950  joincjn 18010  meetcmee 18011  Latclat 18130  Atomscatm 37256  HLchlt 37343  LPlanesclpl 37485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-poset 18012  df-lub 18045  df-glb 18046  df-join 18047  df-meet 18048  df-lat 18131  df-ats 37260  df-atl 37291  df-cvlat 37315  df-hlat 37344

This theorem is referenced by:  dalem49  37714  dalem51  37716  dalem52  37717