Theorem 4atexlemunv 40067

Description: Lemma for 4atexlem7 40076. (Contributed by NM, 21-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
4atexlemunv	⊢ (𝜑 → 𝑈 ≠ 𝑉)

Proof of Theorem 4atexlemunv

Step	Hyp	Ref	Expression
1		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))))
2	1	4atexlemnslpq 40057	. 2 ⊢ (𝜑 → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
3	1	4atexlemk 40048	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ HL)
4	1	4atexlemp 40051	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
5	1	4atexlems 40053	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
6		4thatlem0.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
7		4thatlem0.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
8		4thatlem0.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
9	6, 7, 8	hlatlej2 39376	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑆 ≤ (𝑃 ∨ 𝑆))
10	3, 4, 5, 9	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → 𝑆 ≤ (𝑃 ∨ 𝑆))
11	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝑆 ≤ (𝑃 ∨ 𝑆))
12		4thatlem0.v	. . . . . . . . 9 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
13	1	4atexlemkl 40058	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Lat)
14	1, 7, 8	4atexlempsb 40061	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
15		4thatlem0.h	. . . . . . . . . . 11 ⊢ 𝐻 = (LHyp‘𝐾)
16	1, 15	4atexlemwb 40060	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
17		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		4thatlem0.m	. . . . . . . . . . 11 ⊢ ∧ = (meet‘𝐾)
19	17, 6, 18	latmle1 18430	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
20	13, 14, 16, 19	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
21	12, 20	eqbrtrid 5145	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ≤ (𝑃 ∨ 𝑆))
22	1	4atexlemkc 40059	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CvLat)
23		4thatlem0.u	. . . . . . . . . 10 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
24	1, 6, 7, 18, 8, 15, 23, 12	4atexlemv 40066	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
25	17, 6, 18	latmle2 18431	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
26	13, 14, 16, 25	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
27	12, 26	eqbrtrid 5145	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
28	1	4atexlempw 40050	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
29	28	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑃 ≤ 𝑊)
30		nbrne2 5130	. . . . . . . . . 10 ⊢ ((𝑉 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊) → 𝑉 ≠ 𝑃)
31	27, 29, 30	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ≠ 𝑃)
32	6, 7, 8	cvlatexchb1 39334	. . . . . . . . 9 ⊢ ((𝐾 ∈ CvLat ∧ (𝑉 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑉 ≠ 𝑃) → (𝑉 ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑆)))
33	22, 24, 5, 4, 31, 32	syl131anc 1385	. . . . . . . 8 ⊢ (𝜑 → (𝑉 ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑆)))
34	21, 33	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑆))
35	34	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑆))
36		oveq2 7398	. . . . . . . 8 ⊢ (𝑈 = 𝑉 → (𝑃 ∨ 𝑈) = (𝑃 ∨ 𝑉))
37	36	eqcomd 2736	. . . . . . 7 ⊢ (𝑈 = 𝑉 → (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑈))
38	1	4atexlemq 40052	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
39	17, 7, 8	hlatjcl 39367	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
40	3, 4, 38, 39	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
41	17, 6, 18	latmle1 18430	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
42	13, 40, 16, 41	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
43	23, 42	eqbrtrid 5145	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ (𝑃 ∨ 𝑄))
44	1, 6, 7, 18, 8, 15, 23	4atexlemu 40065	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
45	17, 6, 18	latmle2 18431	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
46	13, 40, 16, 45	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
47	23, 46	eqbrtrid 5145	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
48		nbrne2 5130	. . . . . . . . . 10 ⊢ ((𝑈 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊) → 𝑈 ≠ 𝑃)
49	47, 29, 48	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≠ 𝑃)
50	6, 7, 8	cvlatexchb1 39334	. . . . . . . . 9 ⊢ ((𝐾 ∈ CvLat ∧ (𝑈 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑈 ≠ 𝑃) → (𝑈 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑈) = (𝑃 ∨ 𝑄)))
51	22, 44, 38, 4, 49, 50	syl131anc 1385	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑈) = (𝑃 ∨ 𝑄)))
52	43, 51	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑈) = (𝑃 ∨ 𝑄))
53	37, 52	sylan9eqr 2787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑄))
54	35, 53	eqtr3d 2767	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑃 ∨ 𝑆) = (𝑃 ∨ 𝑄))
55	11, 54	breqtrd 5136	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝑆 ≤ (𝑃 ∨ 𝑄))
56	55	ex 412	. . 3 ⊢ (𝜑 → (𝑈 = 𝑉 → 𝑆 ≤ (𝑃 ∨ 𝑄)))
57	56	necon3bd 2940	. 2 ⊢ (𝜑 → (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) → 𝑈 ≠ 𝑉))
58	2, 57	mpd 15	1 ⊢ (𝜑 → 𝑈 ≠ 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  meetcmee 18280  Latclat 18397  Atomscatm 39263  CvLatclc 39265  HLchlt 39350  LHypclh 39985

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 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 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-proset 18262  df-poset 18281  df-plt 18296  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-p1 18392  df-lat 18398  df-clat 18465  df-oposet 39176  df-ol 39178  df-oml 39179  df-covers 39266  df-ats 39267  df-atl 39298  df-cvlat 39322  df-hlat 39351  df-lhyp 39989

This theorem is referenced by:  4atexlemtlw  40068  4atexlemntlpq  40069  4atexlemc  40070  4atexlemnclw  40071