Theorem 4atexlemunv 36647

Description: Lemma for 4atexlem7 36656. (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 36637	. 2 ⊢ (𝜑 → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
3	1	4atexlemk 36628	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ HL)
4	1	4atexlemp 36631	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
5	1	4atexlems 36633	. . . . . . 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 35957	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑆 ≤ (𝑃 ∨ 𝑆))
10	3, 4, 5, 9	syl3anc 1351	. . . . . 6 ⊢ (𝜑 → 𝑆 ≤ (𝑃 ∨ 𝑆))
11	10	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝑆 ≤ (𝑃 ∨ 𝑆))
12		4thatlem0.v	. . . . . . . . 9 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
13	1	4atexlemkl 36638	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Lat)
14	1, 7, 8	4atexlempsb 36641	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
15		4thatlem0.h	. . . . . . . . . . 11 ⊢ 𝐻 = (LHyp‘𝐾)
16	1, 15	4atexlemwb 36640	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
17		eqid 2778	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		4thatlem0.m	. . . . . . . . . . 11 ⊢ ∧ = (meet‘𝐾)
19	17, 6, 18	latmle1 17547	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
20	13, 14, 16, 19	syl3anc 1351	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
21	12, 20	syl5eqbr 4965	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ≤ (𝑃 ∨ 𝑆))
22	1	4atexlemkc 36639	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CvLat)
23		4thatlem0.u	. . . . . . . . . 10 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
24	1, 6, 7, 18, 8, 15, 23, 12	4atexlemv 36646	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
25	17, 6, 18	latmle2 17548	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
26	13, 14, 16, 25	syl3anc 1351	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ 𝑊)
27	12, 26	syl5eqbr 4965	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
28	1	4atexlempw 36630	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
29	28	simprd 488	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑃 ≤ 𝑊)
30		nbrne2 4950	. . . . . . . . . 10 ⊢ ((𝑉 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊) → 𝑉 ≠ 𝑃)
31	27, 29, 30	syl2anc 576	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ≠ 𝑃)
32	6, 7, 8	cvlatexchb1 35915	. . . . . . . . 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 1363	. . . . . . . 8 ⊢ (𝜑 → (𝑉 ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑆)))
34	21, 33	mpbid 224	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑆))
35	34	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑆))
36		oveq2 6986	. . . . . . . 8 ⊢ (𝑈 = 𝑉 → (𝑃 ∨ 𝑈) = (𝑃 ∨ 𝑉))
37	36	eqcomd 2784	. . . . . . 7 ⊢ (𝑈 = 𝑉 → (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑈))
38	1	4atexlemq 36632	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
39	17, 7, 8	hlatjcl 35948	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
40	3, 4, 38, 39	syl3anc 1351	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
41	17, 6, 18	latmle1 17547	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
42	13, 40, 16, 41	syl3anc 1351	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
43	23, 42	syl5eqbr 4965	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ (𝑃 ∨ 𝑄))
44	1, 6, 7, 18, 8, 15, 23	4atexlemu 36645	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
45	17, 6, 18	latmle2 17548	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
46	13, 40, 16, 45	syl3anc 1351	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
47	23, 46	syl5eqbr 4965	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
48		nbrne2 4950	. . . . . . . . . 10 ⊢ ((𝑈 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊) → 𝑈 ≠ 𝑃)
49	47, 29, 48	syl2anc 576	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≠ 𝑃)
50	6, 7, 8	cvlatexchb1 35915	. . . . . . . . 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 1363	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑈) = (𝑃 ∨ 𝑄)))
52	43, 51	mpbid 224	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑈) = (𝑃 ∨ 𝑄))
53	37, 52	sylan9eqr 2836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑃 ∨ 𝑉) = (𝑃 ∨ 𝑄))
54	35, 53	eqtr3d 2816	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑃 ∨ 𝑆) = (𝑃 ∨ 𝑄))
55	11, 54	breqtrd 4956	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝑆 ≤ (𝑃 ∨ 𝑄))
56	55	ex 405	. . 3 ⊢ (𝜑 → (𝑈 = 𝑉 → 𝑆 ≤ (𝑃 ∨ 𝑄)))
57	56	necon3bd 2981	. 2 ⊢ (𝜑 → (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) → 𝑈 ≠ 𝑉))
58	2, 57	mpd 15	1 ⊢ (𝜑 → 𝑈 ≠ 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2967   class class class wbr 4930  ‘cfv 6190  (class class class)co 6978  Basecbs 16342  lecple 16431  joincjn 17415  meetcmee 17416  Latclat 17516  Atomscatm 35844  CvLatclc 35846  HLchlt 35931  LHypclh 36565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-riota 6939  df-ov 6981  df-oprab 6982  df-proset 17399  df-poset 17417  df-plt 17429  df-lub 17445  df-glb 17446  df-join 17447  df-meet 17448  df-p0 17510  df-p1 17511  df-lat 17517  df-clat 17579  df-oposet 35757  df-ol 35759  df-oml 35760  df-covers 35847  df-ats 35848  df-atl 35879  df-cvlat 35903  df-hlat 35932  df-lhyp 36569

This theorem is referenced by:  4atexlemtlw  36648  4atexlemntlpq  36649  4atexlemc  36650  4atexlemnclw  36651