Theorem dalem48 35527

Description: Lemma for dath 35543. Analogue of dalem45 35524 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 35431	. . 3 ⊢ (𝜑 → 𝐾 ∈ Lat)
3	2	adantr 466	. 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 35496	. . 3 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
7	6	adantl 467	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
8		dalem.j	. . . 4 ⊢ ∨ = (join‘𝐾)
9	1, 8, 5	dalempjqeb 35452	. . 3 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
10	9	adantr 466	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
11	1, 5	dalemreb 35448	. . 3 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
12	11	adantr 466	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ (Base‘𝐾))
13	4	dalem-ccly 35492	. . . 4 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)
14		dalem44.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
15	14	breq2i 4795	. . . 4 ⊢ (𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
16	13, 15	sylnib 317	. . 3 ⊢ (𝜓 → ¬ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
17	16	adantl 467	. 2 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
18		eqid 2771	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		dalem.l	. . 3 ⊢ ≤ = (le‘𝐾)
20	18, 19, 8	latnlej2l 17280	. 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 1489	1 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ (𝑃 ∨ 𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943   class class class wbr 4787  ‘cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Latclat 17253  Atomscatm 35070  HLchlt 35157  LPlanesclpl 35299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-poset 17154  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-lat 17254  df-ats 35074  df-atl 35105  df-cvlat 35129  df-hlat 35158

This theorem is referenced by:  dalem49  35528  dalem51  35530  dalem52  35531