pmodlem2 - Mathbox for Norm Megill

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Theorem pmodlem2 39826

Description: Lemma for pmod1i 39827. (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 7368	. . . . 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 39791	. . . . . 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 2764	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → (𝑋 + 𝑌) = 𝑌)
10	9	ineq1d 4172	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) = (𝑌 ∩ 𝑍))
11		ssinss1 4199	. . . . 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 39795	. . . 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 3972	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
17		oveq2 7361	. . . . 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 39790	. . . . . 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 2786	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → (𝑋 + 𝑌) = 𝑋)
23	22	ineq1d 4172	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) = (𝑋 ∩ 𝑍))
24		inss1 4190	. . . 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 39794	. . . . 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 3949	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → (𝑋 ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
32	23, 31	eqsstrd 3972	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
33		elin 3921	. . . 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 39342	. . . . . . . . 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 39779	. . . . . . . . 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 39825	. . . . . . . . . . . . . 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 3185	. . . . . . . . . 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 3943	. 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 3013	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 2109 ≠ wne 2925 ∃wrex 3053 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 lecple 17186 joincjn 18235 Latclat 18355 Atomscatm 39241 HLchlt 39328 PSubSpcpsubsp 39475 +_𝑃cpadd 39774

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7931 df-2nd 7932 df-proset 18218 df-poset 18237 df-plt 18252 df-lub 18268 df-glb 18269 df-join 18270 df-meet 18271 df-p0 18347 df-lat 18356 df-covers 39244 df-ats 39245 df-atl 39276 df-cvlat 39300 df-hlat 39329 df-psubsp 39482 df-padd 39775

This theorem is referenced by: pmod1i 39827

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