pmodlem2 - Mathbox for Norm Megill

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Theorem pmodlem2 39866

Description: Lemma for pmod1i 39867. (Contributed by NM, 9-Mar-2012.)

**Hypotheses**
Ref	Expression
pmodlem.l	⊢ ≤ = (le‘𝐾)
pmodlem.j	⊢ ∨ = (join‘𝐾)
pmodlem.a	⊢ 𝐴 = (Atoms‘𝐾)
pmodlem.s	⊢ 𝑆 = (PSubSp‘𝐾)
pmodlem.p	⊢ + = (+_𝑃‘𝐾)

**Assertion**
Ref	Expression
pmodlem2	⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))

Proof of Theorem pmodlem2 Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → 𝑋 = ∅)
2	1	oveq1d 7420	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → (𝑋 + 𝑌) = (∅ + 𝑌))
3		simpl1 1192	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → 𝐾 ∈ HL)
4		simpl22 1253	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → 𝑌 ⊆ 𝐴)
5		pmodlem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6		pmodlem.p	. . . . . . 7 ⊢ + = (+_𝑃‘𝐾)
7	5, 6	padd02 39831	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → (∅ + 𝑌) = 𝑌)
8	3, 4, 7	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → (∅ + 𝑌) = 𝑌)
9	2, 8	eqtrd 2770	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → (𝑋 + 𝑌) = 𝑌)
10	9	ineq1d 4194	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) = (𝑌 ∩ 𝑍))
11		ssinss1 4221	. . . . 5 ⊢ (𝑌 ⊆ 𝐴 → (𝑌 ∩ 𝑍) ⊆ 𝐴)
12	4, 11	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → (𝑌 ∩ 𝑍) ⊆ 𝐴)
13		simpl21 1252	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → 𝑋 ⊆ 𝐴)
14	5, 6	sspadd2 39835	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑌 ∩ 𝑍) ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑌 ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
15	3, 12, 13, 14	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → (𝑌 ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
16	10, 15	eqsstrd 3993	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
17		oveq2 7413	. . . . 5 ⊢ (𝑌 = ∅ → (𝑋 + 𝑌) = (𝑋 + ∅))
18		simp1 1136	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) → 𝐾 ∈ HL)
19		simp21 1207	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ⊆ 𝐴)
20	5, 6	padd01 39830	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑋 + ∅) = 𝑋)
21	18, 19, 20	syl2anc 584	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) → (𝑋 + ∅) = 𝑋)
22	17, 21	sylan9eqr 2792	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → (𝑋 + 𝑌) = 𝑋)
23	22	ineq1d 4194	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) = (𝑋 ∩ 𝑍))
24		inss1 4212	. . . 4 ⊢ (𝑋 ∩ 𝑍) ⊆ 𝑋
25		simpl1 1192	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → 𝐾 ∈ HL)
26		simpl21 1252	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → 𝑋 ⊆ 𝐴)
27		simpl22 1253	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → 𝑌 ⊆ 𝐴)
28	27, 11	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → (𝑌 ∩ 𝑍) ⊆ 𝐴)
29	5, 6	sspadd1 39834	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ (𝑌 ∩ 𝑍) ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
30	25, 26, 28, 29	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → 𝑋 ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
31	24, 30	sstrid 3970	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → (𝑋 ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
32	23, 31	eqsstrd 3993	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
33		elin 3942	. . . 4 ⊢ (𝑝 ∈ ((𝑋 + 𝑌) ∩ 𝑍) ↔ (𝑝 ∈ (𝑋 + 𝑌) ∧ 𝑝 ∈ 𝑍))
34		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ ((𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅) ∧ 𝑝 ∈ 𝑍)) → 𝐾 ∈ HL)
35	34	hllatd 39382	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ ((𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅) ∧ 𝑝 ∈ 𝑍)) → 𝐾 ∈ Lat)
36		simpl21 1252	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ ((𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅) ∧ 𝑝 ∈ 𝑍)) → 𝑋 ⊆ 𝐴)
37		simpl22 1253	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ ((𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅) ∧ 𝑝 ∈ 𝑍)) → 𝑌 ⊆ 𝐴)
38		simprl 770	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ ((𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅) ∧ 𝑝 ∈ 𝑍)) → (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅))
39		pmodlem.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
40		pmodlem.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
41	39, 40, 5, 6	elpaddn0 39819	. . . . . . . . 9 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑝 ∈ (𝑋 + 𝑌) ↔ (𝑝 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟))))
42	35, 36, 37, 38, 41	syl31anc 1375	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ ((𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅) ∧ 𝑝 ∈ 𝑍)) → (𝑝 ∈ (𝑋 + 𝑌) ↔ (𝑝 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟))))
43		simpl1 1192	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑝 ∈ 𝑍 ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝐾 ∈ HL)
44		simpl21 1252	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑝 ∈ 𝑍 ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑋 ⊆ 𝐴)
45		simpl22 1253	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑝 ∈ 𝑍 ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑌 ⊆ 𝐴)
46		simpl23 1254	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑝 ∈ 𝑍 ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑍 ∈ 𝑆)
47		simpl3 1194	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑝 ∈ 𝑍 ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑋 ⊆ 𝑍)
48		simpr1 1195	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑝 ∈ 𝑍 ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑝 ∈ 𝑍)
49		simpr2l 1233	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑝 ∈ 𝑍 ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑞 ∈ 𝑋)
50		simpr2r 1234	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑝 ∈ 𝑍 ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑟 ∈ 𝑌)
51		simpr3 1197	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑝 ∈ 𝑍 ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑝 ≤ (𝑞 ∨ 𝑟))
52		pmodlem.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (PSubSp‘𝐾)
53	39, 40, 5, 52, 6	pmodlem1 39865	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ∈ 𝑆 ∧ 𝑋 ⊆ 𝑍 ∧ 𝑝 ∈ 𝑍) ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌 ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍)))
54	43, 44, 45, 46, 47, 48, 49, 50, 51, 53	syl333anc 1404	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑝 ∈ 𝑍 ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍)))
55	54	3exp2 1355	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) → (𝑝 ∈ 𝑍 → ((𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) → (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍))))))
56	55	imp 406	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑝 ∈ 𝑍) → ((𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌) → (𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍)))))
57	56	rexlimdvv 3197	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑝 ∈ 𝑍) → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍))))
58	57	adantld 490	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑝 ∈ 𝑍) → ((𝑝 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍))))
59	58	adantrl 716	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ ((𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅) ∧ 𝑝 ∈ 𝑍)) → ((𝑝 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍))))
60	42, 59	sylbid 240	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ ((𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅) ∧ 𝑝 ∈ 𝑍)) → (𝑝 ∈ (𝑋 + 𝑌) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍))))
61	60	exp32 420	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) → ((𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅) → (𝑝 ∈ 𝑍 → (𝑝 ∈ (𝑋 + 𝑌) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍))))))
62	61	com34 91	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) → ((𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅) → (𝑝 ∈ (𝑋 + 𝑌) → (𝑝 ∈ 𝑍 → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍))))))
63	62	imp4b 421	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑝 ∈ (𝑋 + 𝑌) ∧ 𝑝 ∈ 𝑍) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍))))
64	33, 63	biimtrid 242	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑝 ∈ ((𝑋 + 𝑌) ∩ 𝑍) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍))))
65	64	ssrdv 3964	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
66	16, 32, 65	pm2.61da2ne 3020	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∃wrex 3060 ∩ cin 3925 ⊆ wss 3926 ∅c0 4308 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 lecple 17278 joincjn 18323 Latclat 18441 Atomscatm 39281 HLchlt 39368 PSubSpcpsubsp 39515 +_𝑃cpadd 39814

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-proset 18306 df-poset 18325 df-plt 18340 df-lub 18356 df-glb 18357 df-join 18358 df-meet 18359 df-p0 18435 df-lat 18442 df-covers 39284 df-ats 39285 df-atl 39316 df-cvlat 39340 df-hlat 39369 df-psubsp 39522 df-padd 39815

This theorem is referenced by: pmod1i 39867

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