Theorem 4atexlemnclw 41318

Description: Lemma for 4atexlem7 41323. (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)
4thatlem0.c	⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))

Assertion

Ref	Expression
4atexlemnclw	⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)

Proof of Theorem 4atexlemnclw

Step	Hyp	Ref	Expression
1		4thatlem0.c	. . . 4 ⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
2		4thatlem.ph	. . . . . 6 ⊢ (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))))
3	2	4atexlemkl 41305	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
4		4thatlem0.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
5		4thatlem0.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
6	2, 4, 5	4atexlemqtb 41309	. . . . 5 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
7	2, 4, 5	4atexlempsb 41308	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
8		eqid 2763	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		4thatlem0.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
10		4thatlem0.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
11	8, 9, 10	latmle1 18664	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
12	3, 6, 7, 11	syl3anc 1399	. . . 4 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
13	1, 12	eqbrtrid 5143	. . 3 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
14		simp13r 1308	. . . . 5 ⊢ ((((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))) → ¬ 𝑄 ≤ 𝑊)
15	2, 14	sylbi 220	. . . 4 ⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊)
16	2	4atexlemkc 41306	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ CvLat)
17		4thatlem0.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
18		4thatlem0.u	. . . . . . 7 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
19		4thatlem0.v	. . . . . . 7 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
20	2, 9, 4, 10, 5, 17, 18, 19	4atexlemv 41313	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
21	2	4atexlemq 41299	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
22	2	4atexlemt 41301	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
23	2, 9, 4, 10, 5, 17, 18	4atexlemu 41312	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
24	2, 9, 4, 10, 5, 17, 18, 19	4atexlemunv 41314	. . . . . . 7 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
25	2	4atexlemutvt 41302	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
26	5, 4	cvlsupr6 40595	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ≠ 𝑉 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) → 𝑇 ≠ 𝑉)
27	26	necomd 3014	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ≠ 𝑉 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) → 𝑉 ≠ 𝑇)
28	16, 23, 20, 22, 24, 25, 27	syl132anc 1416	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ 𝑇)
29	9, 4, 5	cvlatexch2 40585	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑉 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑉 ≠ 𝑇) → (𝑉 ≤ (𝑄 ∨ 𝑇) → 𝑄 ≤ (𝑉 ∨ 𝑇)))
30	16, 20, 21, 22, 28, 29	syl131anc 1411	. . . . 5 ⊢ (𝜑 → (𝑉 ≤ (𝑄 ∨ 𝑇) → 𝑄 ≤ (𝑉 ∨ 𝑇)))
31	2, 17	4atexlemwb 41307	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
32	8, 9, 10	latmle2 18665	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
33	3, 7, 31, 32	syl3anc 1399	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
34	19, 33	eqbrtrid 5143	. . . . . . 7 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
35	2, 9, 4, 10, 5, 17, 18, 19	4atexlemtlw 41315	. . . . . . 7 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
36	8, 5	atbase 40537	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
37	20, 36	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
38	8, 5	atbase 40537	. . . . . . . . 9 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
39	22, 38	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
40	8, 9, 4	latjle12 18650	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑉 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑉 ∨ 𝑇) ≤ 𝑊))
41	3, 37, 39, 31, 40	syl13anc 1400	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑉 ∨ 𝑇) ≤ 𝑊))
42	34, 35, 41	mpbi2and 725	. . . . . 6 ⊢ (𝜑 → (𝑉 ∨ 𝑇) ≤ 𝑊)
43	8, 5	atbase 40537	. . . . . . . 8 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
44	21, 43	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
45	2	4atexlemk 41295	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
46	8, 4, 5	hlatjcl 40615	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑉 ∨ 𝑇) ∈ (Base‘𝐾))
47	45, 20, 22, 46	syl3anc 1399	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∨ 𝑇) ∈ (Base‘𝐾))
48	8, 9	lattr 18644	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑉 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ≤ (𝑉 ∨ 𝑇) ∧ (𝑉 ∨ 𝑇) ≤ 𝑊) → 𝑄 ≤ 𝑊))
49	3, 44, 47, 31, 48	syl13anc 1400	. . . . . 6 ⊢ (𝜑 → ((𝑄 ≤ (𝑉 ∨ 𝑇) ∧ (𝑉 ∨ 𝑇) ≤ 𝑊) → 𝑄 ≤ 𝑊))
50	42, 49	mpan2d 707	. . . . 5 ⊢ (𝜑 → (𝑄 ≤ (𝑉 ∨ 𝑇) → 𝑄 ≤ 𝑊))
51	30, 50	syld 48	. . . 4 ⊢ (𝜑 → (𝑉 ≤ (𝑄 ∨ 𝑇) → 𝑄 ≤ 𝑊))
52	15, 51	mtod 201	. . 3 ⊢ (𝜑 → ¬ 𝑉 ≤ (𝑄 ∨ 𝑇))
53		nbrne2 5128	. . 3 ⊢ ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ ¬ 𝑉 ≤ (𝑄 ∨ 𝑇)) → 𝐶 ≠ 𝑉)
54	13, 52, 53	syl2anc 596	. 2 ⊢ (𝜑 → 𝐶 ≠ 𝑉)
55	2	4atexlemw 41296	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
56	45, 55	jca 521	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
57	2	4atexlempw 41297	. . 3 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
58	2	4atexlems 41300	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
59	2, 9, 4, 10, 5, 17, 18, 19, 1	4atexlemc 41317	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
60	2, 9, 4, 5	4atexlempns 41310	. . 3 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
61	8, 9, 10	latmle2 18665	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
62	3, 6, 7, 61	syl3anc 1399	. . . 4 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
63	1, 62	eqbrtrid 5143	. . 3 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
64	9, 4, 10, 5, 17, 19	lhpat3 41294	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑃 ≠ 𝑆 ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))) → (¬ 𝐶 ≤ 𝑊 ↔ 𝐶 ≠ 𝑉))
65	56, 57, 58, 59, 60, 63, 64	syl222anc 1414	. 2 ⊢ (𝜑 → (¬ 𝐶 ≤ 𝑊 ↔ 𝐶 ≠ 𝑉))
66	54, 65	mpbird 260	1 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   class class class wbr 5106  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  lecple 17461  joincjn 18511  meetcmee 18512  Latclat 18631  Atomscatm 40511  CvLatclc 40513  HLchlt 40598  LHypclh 41232

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-proset 18494  df-poset 18513  df-plt 18528  df-lub 18544  df-glb 18545  df-join 18546  df-meet 18547  df-p0 18623  df-p1 18624  df-lat 18632  df-clat 18699  df-oposet 40424  df-ol 40426  df-oml 40427  df-covers 40514  df-ats 40515  df-atl 40546  df-cvlat 40570  df-hlat 40599  df-llines 40746  df-lplanes 40747  df-lhyp 41236

This theorem is used by:  4atexlemex2  41319  4atexlemcnd  41320