pmodlem2 - Mathbox for Norm Megill

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Theorem pmodlem2 39966

Description: Lemma for pmod1i 39967. (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 7367	. . . . 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 39931	. . . . . 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 2768	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → (𝑋 + 𝑌) = 𝑌)
10	9	ineq1d 4168	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) = (𝑌 ∩ 𝑍))
11		ssinss1 4195	. . . . 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 39935	. . . 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 3965	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑋 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
17		oveq2 7360	. . . . 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 39930	. . . . . 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 2790	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → (𝑋 + 𝑌) = 𝑋)
23	22	ineq1d 4168	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) = (𝑋 ∩ 𝑍))
24		inss1 4186	. . . 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 39934	. . . . 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 3942	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → (𝑋 ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
32	23, 31	eqsstrd 3965	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) ∧ 𝑌 = ∅) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍)))
33		elin 3914	. . . 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 39483	. . . . . . . . 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 39919	. . . . . . . . 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 39965	. . . . . . . . . . . . . 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 3189	. . . . . . . . . 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 3936	. 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 3017	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 1541 ∈ wcel 2113 ≠ wne 2929 ∃wrex 3057 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 class class class wbr 5093 ‘cfv 6486 (class class class)co 7352 lecple 17170 joincjn 18219 Latclat 18339 Atomscatm 39382 HLchlt 39469 PSubSpcpsubsp 39615 +_𝑃cpadd 39914

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 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

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 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 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 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-proset 18202 df-poset 18221 df-plt 18236 df-lub 18252 df-glb 18253 df-join 18254 df-meet 18255 df-p0 18331 df-lat 18340 df-covers 39385 df-ats 39386 df-atl 39417 df-cvlat 39441 df-hlat 39470 df-psubsp 39622 df-padd 39915

This theorem is referenced by: pmod1i 39967

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