Theorem 3dimlem3a 40215

Description: Lemma for 3dim3 40224. (Contributed by NM, 27-Jul-2012.)

Hypotheses

Ref	Expression
3dim0.j	⊢ ∨ = (join‘𝐾)
3dim0.l	⊢ ≤ = (le‘𝐾)
3dim0.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
3dimlem3a	⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ¬ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))

Proof of Theorem 3dimlem3a

Step	Hyp	Ref	Expression
1		simp31 1228	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))
2		simp11 1222	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝐾 ∈ HL)
3	2	hllatd 40119	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝐾 ∈ Lat)
4		simp13 1224	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑄 ∈ 𝐴)
5		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		3dim0.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
7	5, 6	atbase 40044	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
8	4, 7	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑄 ∈ (Base‘𝐾))
9		simp2l 1218	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑅 ∈ 𝐴)
10	5, 6	atbase 40044	. . . . . 6 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
11	9, 10	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑅 ∈ (Base‘𝐾))
12		simp12 1223	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑃 ∈ 𝐴)
13	5, 6	atbase 40044	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
14	12, 13	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑃 ∈ (Base‘𝐾))
15		3dim0.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
16	5, 15	latjrot 18545	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
17	3, 8, 11, 14, 16	syl13anc 1399	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
18		simp33 1230	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))
19		simp2r 1219	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑆 ∈ 𝐴)
20	5, 15, 6	hlatjcl 40122	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
21	2, 4, 9, 20	syl3anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
22		simp32 1229	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ¬ 𝑃 ≤ (𝑄 ∨ 𝑅))
23		3dim0.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
24	5, 23, 15, 6	hlexchb1 40139	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → (𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ↔ ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑄 ∨ 𝑅) ∨ 𝑆)))
25	2, 12, 19, 21, 22, 24	syl131anc 1410	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ↔ ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑄 ∨ 𝑅) ∨ 𝑆)))
26	18, 25	mpbid 235	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑄 ∨ 𝑅) ∨ 𝑆))
27	17, 26	eqtr3d 2800	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑄 ∨ 𝑅) ∨ 𝑆))
28	27	breq2d 5122	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆)))
29	1, 28	mtbird 328	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ¬ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  lecple 17318  joincjn 18368  Latclat 18488  Atomscatm 40018  HLchlt 40105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-proset 18351  df-poset 18370  df-lub 18401  df-glb 18402  df-join 18403  df-meet 18404  df-lat 18489  df-ats 40022  df-atl 40053  df-cvlat 40077  df-hlat 40106

This theorem is referenced by:  3dimlem3  40216  3dim3  40224