Theorem 4atexlemnclw 40088

Description: Lemma for 4atexlem7 40093. (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 40075	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
4		4thatlem0.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
5		4thatlem0.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
6	2, 4, 5	4atexlemqtb 40079	. . . . 5 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
7	2, 4, 5	4atexlempsb 40078	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
8		eqid 2730	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		4thatlem0.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
10		4thatlem0.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
11	8, 9, 10	latmle1 18362	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
12	3, 6, 7, 11	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
13	1, 12	eqbrtrid 5124	. . 3 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
14		simp13r 1290	. . . . 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 217	. . . 4 ⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊)
16	2	4atexlemkc 40076	. . . . . 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 40083	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
21	2	4atexlemq 40069	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
22	2	4atexlemt 40071	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
23	2, 9, 4, 10, 5, 17, 18	4atexlemu 40082	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
24	2, 9, 4, 10, 5, 17, 18, 19	4atexlemunv 40084	. . . . . . 7 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
25	2	4atexlemutvt 40072	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
26	5, 4	cvlsupr6 39365	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ≠ 𝑉 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) → 𝑇 ≠ 𝑉)
27	26	necomd 2981	. . . . . . 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 1390	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ 𝑇)
29	9, 4, 5	cvlatexch2 39355	. . . . . 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 1385	. . . . 5 ⊢ (𝜑 → (𝑉 ≤ (𝑄 ∨ 𝑇) → 𝑄 ≤ (𝑉 ∨ 𝑇)))
31	2, 17	4atexlemwb 40077	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
32	8, 9, 10	latmle2 18363	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
33	3, 7, 31, 32	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
34	19, 33	eqbrtrid 5124	. . . . . . 7 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
35	2, 9, 4, 10, 5, 17, 18, 19	4atexlemtlw 40085	. . . . . . 7 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
36	8, 5	atbase 39307	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
37	20, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
38	8, 5	atbase 39307	. . . . . . . . 9 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
39	22, 38	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
40	8, 9, 4	latjle12 18348	. . . . . . . 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 1374	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑉 ∨ 𝑇) ≤ 𝑊))
42	34, 35, 41	mpbi2and 712	. . . . . 6 ⊢ (𝜑 → (𝑉 ∨ 𝑇) ≤ 𝑊)
43	8, 5	atbase 39307	. . . . . . . 8 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
44	21, 43	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
45	2	4atexlemk 40065	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
46	8, 4, 5	hlatjcl 39385	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑉 ∨ 𝑇) ∈ (Base‘𝐾))
47	45, 20, 22, 46	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∨ 𝑇) ∈ (Base‘𝐾))
48	8, 9	lattr 18342	. . . . . . 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 1374	. . . . . 6 ⊢ (𝜑 → ((𝑄 ≤ (𝑉 ∨ 𝑇) ∧ (𝑉 ∨ 𝑇) ≤ 𝑊) → 𝑄 ≤ 𝑊))
50	42, 49	mpan2d 694	. . . . 5 ⊢ (𝜑 → (𝑄 ≤ (𝑉 ∨ 𝑇) → 𝑄 ≤ 𝑊))
51	30, 50	syld 47	. . . 4 ⊢ (𝜑 → (𝑉 ≤ (𝑄 ∨ 𝑇) → 𝑄 ≤ 𝑊))
52	15, 51	mtod 198	. . 3 ⊢ (𝜑 → ¬ 𝑉 ≤ (𝑄 ∨ 𝑇))
53		nbrne2 5109	. . 3 ⊢ ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ ¬ 𝑉 ≤ (𝑄 ∨ 𝑇)) → 𝐶 ≠ 𝑉)
54	13, 52, 53	syl2anc 584	. 2 ⊢ (𝜑 → 𝐶 ≠ 𝑉)
55	2	4atexlemw 40066	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
56	45, 55	jca 511	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
57	2	4atexlempw 40067	. . 3 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
58	2	4atexlems 40070	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
59	2, 9, 4, 10, 5, 17, 18, 19, 1	4atexlemc 40087	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
60	2, 9, 4, 5	4atexlempns 40080	. . 3 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
61	8, 9, 10	latmle2 18363	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
62	3, 6, 7, 61	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
63	1, 62	eqbrtrid 5124	. . 3 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
64	9, 4, 10, 5, 17, 19	lhpat3 40064	. . 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 1388	. 2 ⊢ (𝜑 → (¬ 𝐶 ≤ 𝑊 ↔ 𝐶 ≠ 𝑉))
66	54, 65	mpbird 257	1 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110   ≠ wne 2926   class class class wbr 5089  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  lecple 17160  joincjn 18209  meetcmee 18210  Latclat 18329  Atomscatm 39281  CvLatclc 39283  HLchlt 39368  LHypclh 40002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-proset 18192  df-poset 18211  df-plt 18226  df-lub 18242  df-glb 18243  df-join 18244  df-meet 18245  df-p0 18321  df-p1 18322  df-lat 18330  df-clat 18397  df-oposet 39194  df-ol 39196  df-oml 39197  df-covers 39284  df-ats 39285  df-atl 39316  df-cvlat 39340  df-hlat 39369  df-llines 39516  df-lplanes 39517  df-lhyp 40006

This theorem is referenced by:  4atexlemex2  40089  4atexlemcnd  40090