cdleme23b - Mathbox for Norm Megill

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Theorem cdleme23b 40549

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 1285	. . . . . 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 39560	. . . . . 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 1287	. . . . . 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 1289	. . . . . 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 39566	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ 𝐵)
11	2, 5, 6, 10	syl3anc 1373	. . . . 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 39563	. . . . . 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 1200	. . . . . . 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 1286	. . . . . . . 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 40197	. . . . . . . 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 18361	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
20	12, 13, 17, 19	syl3anc 1373	. . . . . 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 18360	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)) ∈ 𝐵)
22	12, 11, 20, 21	syl3anc 1373	. . . . 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 39428	. . . . 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 1374	. . . 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 18369	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ (𝑋 ∧ 𝑊)))
27	12, 11, 20, 26	syl3anc 1373	. . . . . 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 18388	. . . . . . 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 1373	. . . . . 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 232	. . . . 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 7371	. . . 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 39488	. . . . . . . . 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 39488	. . . . . . . . 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 18413	. . . . . . . 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 1374	. . . . . . 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 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)) → (𝑆 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
39		simp33 1212	. . . . . . . 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 7374	. . . . . . 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 18383	. . . . . . . 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 2773	. . . . . 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 7371	. . . . 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 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) ∧ (𝑋 ∧ 𝑊)))
46	24, 31, 45	3eqtr3d 2777	. . 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 1288	. . . 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 1210	. . . 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 40242	. . . 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 1388	. . 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 2835	. 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 2838	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 class class class wbr 5096 ‘cfv 6490 (class class class)co 7356 Basecbs 17134 lecple 17182 joincjn 18232 meetcmee 18233 Latclat 18352 OLcol 39373 Atomscatm 39462 HLchlt 39549 LHypclh 40183

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

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 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-proset 18215 df-poset 18234 df-plt 18249 df-lub 18265 df-glb 18266 df-join 18267 df-meet 18268 df-p0 18344 df-p1 18345 df-lat 18353 df-clat 18420 df-oposet 39375 df-ol 39377 df-oml 39378 df-covers 39465 df-ats 39466 df-atl 39497 df-cvlat 39521 df-hlat 39550 df-lhyp 40187

This theorem is referenced by: cdleme28a 40569

W3C validator