cdleme23b - Mathbox for Norm Megill

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Theorem cdleme23b 38813

Description: Part of proof of Lemma E in [Crawley] p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012.)

**Hypotheses**
Ref	Expression
cdleme23.b	⊢ 𝐵 = (Base‘𝐾)
cdleme23.l	⊢ ≤ = (le‘𝐾)
cdleme23.j	⊢ ∨ = (join‘𝐾)
cdleme23.m	⊢ ∧ = (meet‘𝐾)
cdleme23.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme23.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme23.v	⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ (𝑋 ∧ 𝑊))

**Assertion**
Ref	Expression
cdleme23b	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑉 ∈ 𝐴)

**Proof of Theorem cdleme23b**
Step	Hyp	Ref	Expression
1		cdleme23.v	. 2 ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ (𝑋 ∧ 𝑊))
2		simp11l 1284	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
3		hlol 37823	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
4	2, 3	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ OL)
5		simp12l 1286	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑆 ∈ 𝐴)
6		simp13l 1288	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑇 ∈ 𝐴)
7		cdleme23.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
8		cdleme23.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
9		cdleme23.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
10	7, 8, 9	hlatjcl 37829	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ 𝐵)
11	2, 5, 6, 10	syl3anc 1371	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑆 ∨ 𝑇) ∈ 𝐵)
12	2	hllatd 37826	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
13		simp2l 1199	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑋 ∈ 𝐵)
14		simp11r 1285	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐻)
15		cdleme23.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
16	7, 15	lhpbase 38461	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
17	14, 16	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐵)
18		cdleme23.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
19	7, 18	latmcl 18329	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
20	12, 13, 17, 19	syl3anc 1371	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
21	7, 8	latjcl 18328	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ∈ 𝐵)
22	12, 11, 20, 21	syl3anc 1371	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ∈ 𝐵)
23	7, 18	latmassOLD 37691	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ ((𝑆 ∨ 𝑇) ∈ 𝐵 ∧ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (((𝑆 ∨ 𝑇) ∧ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊))) ∧ 𝑊) = ((𝑆 ∨ 𝑇) ∧ (((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊)))
24	4, 11, 22, 17, 23	syl13anc 1372	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (((𝑆 ∨ 𝑇) ∧ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊))) ∧ 𝑊) = ((𝑆 ∨ 𝑇) ∧ (((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊)))
25		cdleme23.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
26	7, 25, 8	latlej1 18337	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)))
27	12, 11, 20, 26	syl3anc 1371	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)))
28	7, 25, 18	latleeqm1 18356	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ 𝐵 ∧ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ∈ 𝐵) → ((𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ↔ ((𝑆 ∨ 𝑇) ∧ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊))) = (𝑆 ∨ 𝑇)))
29	12, 11, 22, 28	syl3anc 1371	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ↔ ((𝑆 ∨ 𝑇) ∧ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊))) = (𝑆 ∨ 𝑇)))
30	27, 29	mpbid 231	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑆 ∨ 𝑇) ∧ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊))) = (𝑆 ∨ 𝑇))
31	30	oveq1d 7372	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (((𝑆 ∨ 𝑇) ∧ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊))) ∧ 𝑊) = ((𝑆 ∨ 𝑇) ∧ 𝑊))
32	7, 9	atbase 37751	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐵)
33	5, 32	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑆 ∈ 𝐵)
34	7, 9	atbase 37751	. . . . . . . . 9 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ 𝐵)
35	6, 34	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑇 ∈ 𝐵)
36	7, 8	latjjdir 18381	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵)) → ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) = ((𝑆 ∨ (𝑋 ∧ 𝑊)) ∨ (𝑇 ∨ (𝑋 ∧ 𝑊))))
37	12, 33, 35, 20, 36	syl13anc 1372	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) = ((𝑆 ∨ (𝑋 ∧ 𝑊)) ∨ (𝑇 ∨ (𝑋 ∧ 𝑊))))
38		simp32 1210	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
39		simp33 1211	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
40	38, 39	oveq12d 7375	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑆 ∨ (𝑋 ∧ 𝑊)) ∨ (𝑇 ∨ (𝑋 ∧ 𝑊))) = (𝑋 ∨ 𝑋))
41	7, 8	latjidm 18351	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋)
42	12, 13, 41	syl2anc 584	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑋 ∨ 𝑋) = 𝑋)
43	37, 40, 42	3eqtrd 2780	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) = 𝑋)
44	43	oveq1d 7372	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊) = (𝑋 ∧ 𝑊))
45	44	oveq2d 7373	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑆 ∨ 𝑇) ∧ (((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊)) = ((𝑆 ∨ 𝑇) ∧ (𝑋 ∧ 𝑊)))
46	24, 31, 45	3eqtr3d 2784	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑆 ∨ 𝑇) ∧ 𝑊) = ((𝑆 ∨ 𝑇) ∧ (𝑋 ∧ 𝑊)))
47		simp12r 1287	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ¬ 𝑆 ≤ 𝑊)
48		simp31 1209	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑆 ≠ 𝑇)
49	25, 8, 18, 9, 15	lhpat 38506	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) → ((𝑆 ∨ 𝑇) ∧ 𝑊) ∈ 𝐴)
50	2, 14, 5, 47, 6, 48, 49	syl222anc 1386	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑆 ∨ 𝑇) ∧ 𝑊) ∈ 𝐴)
51	46, 50	eqeltrrd 2839	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝑆 ∨ 𝑇) ∧ (𝑋 ∧ 𝑊)) ∈ 𝐴)
52	1, 51	eqeltrid 2842	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑆 ≠ 𝑇 ∧ (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑇 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑉 ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 lecple 17140 joincjn 18200 meetcmee 18201 Latclat 18320 OLcol 37636 Atomscatm 37725 HLchlt 37812 LHypclh 38447

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-lhyp 38451

This theorem is referenced by: cdleme28a 38833

W3C validator