Theorem 4atexlemtlw 37775

Description: Lemma for 4atexlem7 37783. (Contributed by NM, 24-Nov-2012.)

Hypotheses

Ref	Expression
4thatlem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
4thatlem0.l	⊢ ≤ = (le‘𝐾)
4thatlem0.j	⊢ ∨ = (join‘𝐾)
4thatlem0.m	⊢ ∧ = (meet‘𝐾)
4thatlem0.a	⊢ 𝐴 = (Atoms‘𝐾)
4thatlem0.h	⊢ 𝐻 = (LHyp‘𝐾)
4thatlem0.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
4thatlem0.v	⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)

Assertion

Ref	Expression
4atexlemtlw	⊢ (𝜑 → 𝑇 ≤ 𝑊)

Proof of Theorem 4atexlemtlw

Step	Hyp	Ref	Expression
1		eqid 2734	. 2 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		4thatlem0.l	. 2 ⊢ ≤ = (le‘𝐾)
3		4thatlem.ph	. . 3 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
4	3	4atexlemkl 37765	. 2 ⊢ (𝜑 → 𝐾 ∈ Lat)
5	3	4atexlemt 37761	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
6		4thatlem0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7	1, 6	atbase 36997	. . 3 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
8	5, 7	syl 17	. 2 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
9	3	4atexlemk 37755	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
10		4thatlem0.j	. . . 4 ⊢ ∨ = (join‘𝐾)
11		4thatlem0.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
12		4thatlem0.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
13		4thatlem0.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
14	3, 2, 10, 11, 6, 12, 13	4atexlemu 37772	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
15		4thatlem0.v	. . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
16	3, 2, 10, 11, 6, 12, 13, 15	4atexlemv 37773	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
17	1, 10, 6	hlatjcl 37075	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
18	9, 14, 16, 17	syl3anc 1373	. 2 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
19	3, 12	4atexlemwb 37767	. 2 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
20	3	4atexlemkc 37766	. . 3 ⊢ (𝜑 → 𝐾 ∈ CvLat)
21	3, 2, 10, 11, 6, 12, 13, 15	4atexlemunv 37774	. . 3 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
22	3	4atexlemutvt 37762	. . 3 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
23	6, 2, 10	cvlsupr4 37053	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ≠ 𝑉 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) → 𝑇 ≤ (𝑈 ∨ 𝑉))
24	20, 14, 16, 5, 21, 22, 23	syl132anc 1390	. 2 ⊢ (𝜑 → 𝑇 ≤ (𝑈 ∨ 𝑉))
25	3	4atexlemp 37758	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
26	3	4atexlemq 37759	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
27	1, 10, 6	hlatjcl 37075	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
28	9, 25, 26, 27	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
29	1, 2, 11	latmle2 17943	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
30	4, 28, 19, 29	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
31	13, 30	eqbrtrid 5078	. . 3 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
32	3, 10, 6	4atexlempsb 37768	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
33	1, 2, 11	latmle2 17943	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
34	4, 32, 19, 33	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
35	15, 34	eqbrtrid 5078	. . 3 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
36	1, 6	atbase 36997	. . . . 5 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
37	14, 36	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
38	1, 6	atbase 36997	. . . . 5 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
39	16, 38	syl 17	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
40	1, 2, 10	latjle12 17928	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑈 ≤ 𝑊 ∧ 𝑉 ≤ 𝑊) ↔ (𝑈 ∨ 𝑉) ≤ 𝑊))
41	4, 37, 39, 19, 40	syl13anc 1374	. . 3 ⊢ (𝜑 → ((𝑈 ≤ 𝑊 ∧ 𝑉 ≤ 𝑊) ↔ (𝑈 ∨ 𝑉) ≤ 𝑊))
42	31, 35, 41	mpbi2and 712	. 2 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ 𝑊)
43	1, 2, 4, 8, 18, 19, 24, 42	lattrd 17924	1 ⊢ (𝜑 → 𝑇 ≤ 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2935   class class class wbr 5043  ‘cfv 6369  (class class class)co 7202  Basecbs 16684  lecple 16774  joincjn 17790  meetcmee 17791  Latclat 17909  Atomscatm 36971  CvLatclc 36973  HLchlt 37058  LHypclh 37692

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 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512

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 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-proset 17774  df-poset 17792  df-plt 17808  df-lub 17824  df-glb 17825  df-join 17826  df-meet 17827  df-p0 17903  df-p1 17904  df-lat 17910  df-clat 17977  df-oposet 36884  df-ol 36886  df-oml 36887  df-covers 36974  df-ats 36975  df-atl 37006  df-cvlat 37030  df-hlat 37059  df-lhyp 37696

This theorem is referenced by:  4atexlemntlpq  37776  4atexlemnclw  37778  4atexlemcnd  37780