cdleme30a - Mathbox for Norm Megill

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Theorem cdleme30a 37990

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 1195	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ HL)
2	1	hllatd 36976	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ Lat)
3		simp21 1204	. . . 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 36901	. . . 4 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ 𝐵)
7	3, 6	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑠 ∈ 𝐵)
8		simp23 1206	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ 𝐵)
9		simp1r 1196	. . . . 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 37610	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
12	9, 11	syl 17	. . . 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 17743	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ∈ 𝐵)
15	2, 8, 12, 14	syl3anc 1369	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑌 ∧ 𝑊) ∈ 𝐵)
16		simp22l 1290	. . 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 17786	. . 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 1370	. 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 1199	. . . 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 1200	. . . . . 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 17772	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊)))
24	2, 16, 8, 12, 23	syl13anc 1370	. . . . . 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 15	. . . . 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 17743	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
27	2, 16, 12, 26	syl3anc 1369	. . . . . 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 17757	. . . . . 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 1370	. . . . 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 15	. . . 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 5061	. . 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 17742	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑠 ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ∈ 𝐵) → (𝑠 ∨ (𝑌 ∧ 𝑊)) ∈ 𝐵)
33	2, 7, 15, 32	syl3anc 1369	. . . 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 17755	. . . 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 1369	. . 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 1134	. . . . 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 37650	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((𝑌 ∧ 𝑊) ∨ 𝑋) = (𝑌 ∧ (𝑊 ∨ 𝑋)))
39	37, 8, 16, 21, 38	syl121anc 1373	. . . 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 7173	. . 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 1205	. . . . . . 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 2759	. . . . . . . 8 ⊢ (1.‘𝐾) = (1.‘𝐾)
43	4, 22, 17, 42, 10	lhpj1 37634	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑊 ∨ 𝑋) = (1.‘𝐾))
44	37, 41, 43	syl2anc 587	. . . . . 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 7173	. . . . 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 36973	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
47	1, 46	syl 17	. . . . . 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 36839	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵) → (𝑌 ∧ (1.‘𝐾)) = 𝑌)
49	47, 8, 48	syl2anc 587	. . . . 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 2794	. . . 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 7173	. . 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 17751	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑠 ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → 𝑠 ≤ (𝑠 ∨ (𝑋 ∧ 𝑊)))
53	2, 7, 27, 52	syl3anc 1369	. . . . . 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 5063	. . . . 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 17749	. . . 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 17754	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑠 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑠 ≤ 𝑌 ↔ (𝑠 ∨ 𝑌) = 𝑌))
57	2, 7, 8, 56	syl3anc 1369	. . . 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 2798	. 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 2802	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 class class class wbr 5037 ‘cfv 6341 (class class class)co 7157 Basecbs 16556 lecple 16645 joincjn 17635 meetcmee 17636 1.cp1 17729 Latclat 17736 OLcol 36786 Atomscatm 36875 HLchlt 36962 LHypclh 37596

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5161 ax-sep 5174 ax-nul 5181 ax-pow 5239 ax-pr 5303 ax-un 7466

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-ral 3076 df-rex 3077 df-reu 3078 df-rab 3080 df-v 3412 df-sbc 3700 df-csb 3809 df-dif 3864 df-un 3866 df-in 3868 df-ss 3878 df-nul 4229 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4803 df-iun 4889 df-iin 4890 df-br 5038 df-opab 5100 df-mpt 5118 df-id 5435 df-xp 5535 df-rel 5536 df-cnv 5537 df-co 5538 df-dm 5539 df-rn 5540 df-res 5541 df-ima 5542 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7115 df-ov 7160 df-oprab 7161 df-mpo 7162 df-1st 7700 df-2nd 7701 df-proset 17619 df-poset 17637 df-plt 17649 df-lub 17665 df-glb 17666 df-join 17667 df-meet 17668 df-p0 17730 df-p1 17731 df-lat 17737 df-clat 17799 df-oposet 36788 df-ol 36790 df-oml 36791 df-covers 36878 df-ats 36879 df-atl 36910 df-cvlat 36934 df-hlat 36963 df-psubsp 37115 df-pmap 37116 df-padd 37408 df-lhyp 37600

This theorem is referenced by: cdleme32b 38054

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