Theorem 4atexlemtlw 40055

Description: Lemma for 4atexlem7 40063. (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 2729	. 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 40045	. 2 ⊢ (𝜑 → 𝐾 ∈ Lat)
5	3	4atexlemt 40041	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
6		4thatlem0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7	1, 6	atbase 39276	. . 3 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
8	5, 7	syl 17	. 2 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
9	3	4atexlemk 40035	. . 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 40052	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
15		4thatlem0.v	. . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
16	3, 2, 10, 11, 6, 12, 13, 15	4atexlemv 40053	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
17	1, 10, 6	hlatjcl 39354	. . 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 40047	. 2 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
20	3	4atexlemkc 40046	. . 3 ⊢ (𝜑 → 𝐾 ∈ CvLat)
21	3, 2, 10, 11, 6, 12, 13, 15	4atexlemunv 40054	. . 3 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
22	3	4atexlemutvt 40042	. . 3 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
23	6, 2, 10	cvlsupr4 39332	. . 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 40038	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
26	3	4atexlemq 40039	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
27	1, 10, 6	hlatjcl 39354	. . . . . 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 18407	. . . . 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 5137	. . 3 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
32	3, 10, 6	4atexlempsb 40048	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
33	1, 2, 11	latmle2 18407	. . . . 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 5137	. . 3 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
36	1, 6	atbase 39276	. . . . 5 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
37	14, 36	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
38	1, 6	atbase 39276	. . . . 5 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
39	16, 38	syl 17	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
40	1, 2, 10	latjle12 18392	. . . 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 18388	1 ⊢ (𝜑 → 𝑇 ≤ 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  Basecbs 17156  lecple 17204  joincjn 18253  meetcmee 18254  Latclat 18373  Atomscatm 39250  CvLatclc 39252  HLchlt 39337  LHypclh 39972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-proset 18236  df-poset 18255  df-plt 18270  df-lub 18286  df-glb 18287  df-join 18288  df-meet 18289  df-p0 18365  df-p1 18366  df-lat 18374  df-clat 18441  df-oposet 39163  df-ol 39165  df-oml 39166  df-covers 39253  df-ats 39254  df-atl 39285  df-cvlat 39309  df-hlat 39338  df-lhyp 39976

This theorem is referenced by:  4atexlemntlpq  40056  4atexlemnclw  40058  4atexlemcnd  40060