Theorem 4atexlemtlw 40362

Description: Lemma for 4atexlem7 40370. (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 2735	. 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 40352	. 2 ⊢ (𝜑 → 𝐾 ∈ Lat)
5	3	4atexlemt 40348	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
6		4thatlem0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7	1, 6	atbase 39584	. . 3 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
8	5, 7	syl 17	. 2 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
9	3	4atexlemk 40342	. . 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 40359	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
15		4thatlem0.v	. . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
16	3, 2, 10, 11, 6, 12, 13, 15	4atexlemv 40360	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
17	1, 10, 6	hlatjcl 39662	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
18	9, 14, 16, 17	syl3anc 1374	. 2 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
19	3, 12	4atexlemwb 40354	. 2 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
20	3	4atexlemkc 40353	. . 3 ⊢ (𝜑 → 𝐾 ∈ CvLat)
21	3, 2, 10, 11, 6, 12, 13, 15	4atexlemunv 40361	. . 3 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
22	3	4atexlemutvt 40349	. . 3 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
23	6, 2, 10	cvlsupr4 39640	. . 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 1391	. 2 ⊢ (𝜑 → 𝑇 ≤ (𝑈 ∨ 𝑉))
25	3	4atexlemp 40345	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
26	3	4atexlemq 40346	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
27	1, 10, 6	hlatjcl 39662	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
28	9, 25, 26, 27	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
29	1, 2, 11	latmle2 18390	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
30	4, 28, 19, 29	syl3anc 1374	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
31	13, 30	eqbrtrid 5132	. . 3 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
32	3, 10, 6	4atexlempsb 40355	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
33	1, 2, 11	latmle2 18390	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
34	4, 32, 19, 33	syl3anc 1374	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
35	15, 34	eqbrtrid 5132	. . 3 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
36	1, 6	atbase 39584	. . . . 5 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
37	14, 36	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
38	1, 6	atbase 39584	. . . . 5 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
39	16, 38	syl 17	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
40	1, 2, 10	latjle12 18375	. . . 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 1375	. . 3 ⊢ (𝜑 → ((𝑈 ≤ 𝑊 ∧ 𝑉 ≤ 𝑊) ↔ (𝑈 ∨ 𝑉) ≤ 𝑊))
42	31, 35, 41	mpbi2and 713	. 2 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ 𝑊)
43	1, 2, 4, 8, 18, 19, 24, 42	lattrd 18371	1 ⊢ (𝜑 → 𝑇 ≤ 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2931   class class class wbr 5097  ‘cfv 6491  (class class class)co 7358  Basecbs 17138  lecple 17186  joincjn 18236  meetcmee 18237  Latclat 18356  Atomscatm 39558  CvLatclc 39560  HLchlt 39645  LHypclh 40279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-proset 18219  df-poset 18238  df-plt 18253  df-lub 18269  df-glb 18270  df-join 18271  df-meet 18272  df-p0 18348  df-p1 18349  df-lat 18357  df-clat 18424  df-oposet 39471  df-ol 39473  df-oml 39474  df-covers 39561  df-ats 39562  df-atl 39593  df-cvlat 39617  df-hlat 39646  df-lhyp 40283

This theorem is referenced by:  4atexlemntlpq  40363  4atexlemnclw  40365  4atexlemcnd  40367