cdleme30a - Mathbox for Norm Megill

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Theorem cdleme30a 41251

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Feb-2013.)

**Hypotheses**
Ref	Expression
cdleme30.b	⊢ 𝐵 = (Base‘𝐾)
cdleme30.l	⊢ ≤ = (le‘𝐾)
cdleme30.j	⊢ ∨ = (join‘𝐾)
cdleme30.m	⊢ ∧ = (meet‘𝐾)
cdleme30.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme30.h	⊢ 𝐻 = (LHyp‘𝐾)

**Assertion**
Ref	Expression
cdleme30a	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ (𝑌 ∧ 𝑊)) = 𝑌)

**Proof of Theorem cdleme30a**
Step	Hyp	Ref	Expression
1		simp1l 1216	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ HL)
2	1	hllatd 40237	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ Lat)
3		simp21 1225	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑠 ∈ 𝐴)
4		cdleme30.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
5		cdleme30.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 40162	. . . 4 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ 𝐵)
7	3, 6	syl 18	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑠 ∈ 𝐵)
8		simp23 1227	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ 𝐵)
9		simp1r 1217	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑊 ∈ 𝐻)
10		cdleme30.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
11	4, 10	lhpbase 40871	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
12	9, 11	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑊 ∈ 𝐵)
13		cdleme30.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
14	4, 13	latmcl 18528	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ∈ 𝐵)
15	2, 8, 12, 14	syl3anc 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑌 ∧ 𝑊) ∈ 𝐵)
16		simp22l 1311	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ∈ 𝐵)
17		cdleme30.j	. . . 4 ⊢ ∨ = (join‘𝐾)
18	4, 17	latjass 18571	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑠 ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑠 ∨ (𝑌 ∧ 𝑊)) ∨ 𝑋) = (𝑠 ∨ ((𝑌 ∧ 𝑊) ∨ 𝑋)))
19	2, 7, 15, 16, 18	syl13anc 1399	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → ((𝑠 ∨ (𝑌 ∧ 𝑊)) ∨ 𝑋) = (𝑠 ∨ ((𝑌 ∧ 𝑊) ∨ 𝑋)))
20		simp3l 1220	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
21		simp3r 1221	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ≤ 𝑌)
22		cdleme30.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
23	4, 22, 13	latmlem1 18557	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊)))
24	2, 16, 8, 12, 23	syl13anc 1399	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ≤ 𝑌 → (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊)))
25	21, 24	mpd 16	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊))
26	4, 13	latmcl 18528	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
27	2, 16, 12, 26	syl3anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
28	4, 22, 17	latjlej2 18542	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ∈ 𝐵 ∧ 𝑠 ∈ 𝐵)) → ((𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊) → (𝑠 ∨ (𝑋 ∧ 𝑊)) ≤ (𝑠 ∨ (𝑌 ∧ 𝑊))))
29	2, 27, 15, 7, 28	syl13anc 1399	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → ((𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊) → (𝑠 ∨ (𝑋 ∧ 𝑊)) ≤ (𝑠 ∨ (𝑌 ∧ 𝑊))))
30	25, 29	mpd 16	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ (𝑋 ∧ 𝑊)) ≤ (𝑠 ∨ (𝑌 ∧ 𝑊)))
31	20, 30	eqbrtrrd 5129	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ≤ (𝑠 ∨ (𝑌 ∧ 𝑊)))
32	4, 17	latjcl 18527	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑠 ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ∈ 𝐵) → (𝑠 ∨ (𝑌 ∧ 𝑊)) ∈ 𝐵)
33	2, 7, 15, 32	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ (𝑌 ∧ 𝑊)) ∈ 𝐵)
34	4, 22, 17	latleeqj2 18540	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑠 ∨ (𝑌 ∧ 𝑊)) ∈ 𝐵) → (𝑋 ≤ (𝑠 ∨ (𝑌 ∧ 𝑊)) ↔ ((𝑠 ∨ (𝑌 ∧ 𝑊)) ∨ 𝑋) = (𝑠 ∨ (𝑌 ∧ 𝑊))))
35	2, 16, 33, 34	syl3anc 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ≤ (𝑠 ∨ (𝑌 ∧ 𝑊)) ↔ ((𝑠 ∨ (𝑌 ∧ 𝑊)) ∨ 𝑋) = (𝑠 ∨ (𝑌 ∧ 𝑊))))
36	31, 35	mpbid 235	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → ((𝑠 ∨ (𝑌 ∧ 𝑊)) ∨ 𝑋) = (𝑠 ∨ (𝑌 ∧ 𝑊)))
37		simp1 1154	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
38	4, 22, 17, 13, 10	lhpmod2i2 40911	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((𝑌 ∧ 𝑊) ∨ 𝑋) = (𝑌 ∧ (𝑊 ∨ 𝑋)))
39	37, 8, 16, 21, 38	syl121anc 1402	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → ((𝑌 ∧ 𝑊) ∨ 𝑋) = (𝑌 ∧ (𝑊 ∨ 𝑋)))
40	39	oveq2d 7429	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ ((𝑌 ∧ 𝑊) ∨ 𝑋)) = (𝑠 ∨ (𝑌 ∧ (𝑊 ∨ 𝑋))))
41		simp22 1226	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
42		eqid 2760	. . . . . . . 8 ⊢ (1.‘𝐾) = (1.‘𝐾)
43	4, 22, 17, 42, 10	lhpj1 40895	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑊 ∨ 𝑋) = (1.‘𝐾))
44	37, 41, 43	syl2anc 596	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑊 ∨ 𝑋) = (1.‘𝐾))
45	44	oveq2d 7429	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑌 ∧ (𝑊 ∨ 𝑋)) = (𝑌 ∧ (1.‘𝐾)))
46		hlol 40234	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
47	1, 46	syl 18	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ OL)
48	4, 13, 42	olm11 40100	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵) → (𝑌 ∧ (1.‘𝐾)) = 𝑌)
49	47, 8, 48	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑌 ∧ (1.‘𝐾)) = 𝑌)
50	45, 49	eqtrd 2795	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑌 ∧ (𝑊 ∨ 𝑋)) = 𝑌)
51	50	oveq2d 7429	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ (𝑌 ∧ (𝑊 ∨ 𝑋))) = (𝑠 ∨ 𝑌))
52	4, 22, 17	latlej1 18536	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑠 ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → 𝑠 ≤ (𝑠 ∨ (𝑋 ∧ 𝑊)))
53	2, 7, 27, 52	syl3anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑠 ≤ (𝑠 ∨ (𝑋 ∧ 𝑊)))
54	53, 20	breqtrd 5131	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑠 ≤ 𝑋)
55	4, 22, 2, 7, 16, 8, 54, 21	lattrd 18534	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑠 ≤ 𝑌)
56	4, 22, 17	latleeqj1 18539	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑠 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑠 ≤ 𝑌 ↔ (𝑠 ∨ 𝑌) = 𝑌))
57	2, 7, 8, 56	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ≤ 𝑌 ↔ (𝑠 ∨ 𝑌) = 𝑌))
58	55, 57	mpbid 235	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ 𝑌) = 𝑌)
59	40, 51, 58	3eqtrd 2799	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ ((𝑌 ∧ 𝑊) ∨ 𝑋)) = 𝑌)
60	19, 36, 59	3eqtr3d 2803	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑠 ∨ (𝑌 ∧ 𝑊)) = 𝑌)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 class class class wbr 5103 ‘cfv 6533 (class class class)co 7413 Basecbs 17301 lecple 17349 joincjn 18399 meetcmee 18400 1.cp1 18510 Latclat 18519 OLcol 40047 Atomscatm 40136 HLchlt 40223 LHypclh 40857

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7986 df-2nd 7987 df-proset 18382 df-poset 18401 df-plt 18416 df-lub 18432 df-glb 18433 df-join 18434 df-meet 18435 df-p0 18511 df-p1 18512 df-lat 18520 df-clat 18587 df-oposet 40049 df-ol 40051 df-oml 40052 df-covers 40139 df-ats 40140 df-atl 40171 df-cvlat 40195 df-hlat 40224 df-psubsp 40376 df-pmap 40377 df-padd 40669 df-lhyp 40861

This theorem is used by: cdleme32b 41315

W3C validator