Theorem cdleme35d 40446

Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013.)

Hypotheses

Ref	Expression
cdleme35.l	⊢ ≤ = (le‘𝐾)
cdleme35.j	⊢ ∨ = (join‘𝐾)
cdleme35.m	⊢ ∧ = (meet‘𝐾)
cdleme35.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme35.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme35.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme35.f	⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme35d	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∨ 𝐹) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊))

Proof of Theorem cdleme35d

Step	Hyp	Ref	Expression
1		cdleme35.l	. . . 4 ⊢ ≤ = (le‘𝐾)
2		cdleme35.j	. . . 4 ⊢ ∨ = (join‘𝐾)
3		cdleme35.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
4		cdleme35.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5		cdleme35.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdleme35.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
7		cdleme35.f	. . . 4 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
8	1, 2, 3, 4, 5, 6, 7	cdleme35c 40445	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑄 ∨ 𝐹) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
9	8	oveq1d 7402	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∨ 𝐹) ∧ 𝑊) = ((𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) ∧ 𝑊))
10		simp11l 1285	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ HL)
11		simp13l 1289	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴)
12	10	hllatd 39357	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ Lat)
13		simp12l 1287	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ∈ 𝐴)
14		simp2rl 1243	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑅 ∈ 𝐴)
15		eqid 2729	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
16	15, 2, 4	hlatjcl 39360	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
17	10, 13, 14, 16	syl3anc 1373	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
18		simp11r 1286	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑊 ∈ 𝐻)
19	15, 5	lhpbase 39992	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
20	18, 19	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑊 ∈ (Base‘𝐾))
21	15, 3	latmcl 18399	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑅) ∧ 𝑊) ∈ (Base‘𝐾))
22	12, 17, 20, 21	syl3anc 1373	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑃 ∨ 𝑅) ∧ 𝑊) ∈ (Base‘𝐾))
23	15, 1, 3	latmle2 18424	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑅) ∧ 𝑊) ≤ 𝑊)
24	12, 17, 20, 23	syl3anc 1373	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑃 ∨ 𝑅) ∧ 𝑊) ≤ 𝑊)
25	15, 1, 2, 3, 4	atmod4i2 39861	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ ((𝑃 ∨ 𝑅) ∧ 𝑊) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ ((𝑃 ∨ 𝑅) ∧ 𝑊) ≤ 𝑊) → ((𝑄 ∧ 𝑊) ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) = ((𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) ∧ 𝑊))
26	10, 11, 22, 20, 24, 25	syl131anc 1385	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∧ 𝑊) ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) = ((𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) ∧ 𝑊))
27		simp11 1204	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28		simp13 1206	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
29		eqid 2729	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
30	1, 3, 29, 4, 5	lhpmat 40024	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∧ 𝑊) = (0.‘𝐾))
31	27, 28, 30	syl2anc 584	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑄 ∧ 𝑊) = (0.‘𝐾))
32	31	oveq1d 7402	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∧ 𝑊) ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) = ((0.‘𝐾) ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
33		hlol 39354	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
34	10, 33	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ OL)
35	15, 2, 29	olj02 39219	. . . 4 ⊢ ((𝐾 ∈ OL ∧ ((𝑃 ∨ 𝑅) ∧ 𝑊) ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) = ((𝑃 ∨ 𝑅) ∧ 𝑊))
36	34, 22, 35	syl2anc 584	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((0.‘𝐾) ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) = ((𝑃 ∨ 𝑅) ∧ 𝑊))
37	32, 36	eqtrd 2764	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∧ 𝑊) ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) = ((𝑃 ∨ 𝑅) ∧ 𝑊))
38	9, 26, 37	3eqtr2d 2770	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∨ 𝐹) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  lecple 17227  joincjn 18272  meetcmee 18273  0.cp0 18382  Latclat 18390  OLcol 39167  Atomscatm 39256  HLchlt 39343  LHypclh 39978

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-p1 18385  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-psubsp 39497  df-pmap 39498  df-padd 39790  df-lhyp 39982

This theorem is referenced by:  cdleme35e  40447