Theorem cvrat4 37979

Description: A condition implying existence of an atom with the properties shown. Lemma 3.2.20 in [PtakPulmannova] p. 68. Also Lemma 9.2(delta) in [MaedaMaeda] p. 41. (atcvat4i 31402 analog.) (Contributed by NM, 30-Nov-2011.)

Hypotheses

Ref	Expression
cvrat4.b	⊢ 𝐵 = (Base‘𝐾)
cvrat4.l	⊢ ≤ = (le‘𝐾)
cvrat4.j	⊢ ∨ = (join‘𝐾)
cvrat4.z	⊢ 0 = (0.‘𝐾)
cvrat4.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
cvrat4	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))

Distinct variable groups:   𝐴,𝑟   𝐵,𝑟   ∨ ,𝑟   𝐾,𝑟   ≤ ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑋,𝑟

Allowed substitution hint:   0 (𝑟)

Proof of Theorem cvrat4

Step	Hyp	Ref	Expression
1		hlatl 37895	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2	1	adantr 481	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ AtLat)
3		simpr1 1194	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
4		cvrat4.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐾)
5		cvrat4.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
6		cvrat4.z	. . . . . . . . . . 11 ⊢ 0 = (0.‘𝐾)
7		cvrat4.a	. . . . . . . . . . 11 ⊢ 𝐴 = (Atoms‘𝐾)
8	4, 5, 6, 7	atlex 37851	. . . . . . . . . 10 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋)
9	8	3exp 1119	. . . . . . . . 9 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝐵 → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋)))
10	2, 3, 9	sylc 65	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋))
11	10	adantr 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋))
12		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝐾 ∈ HL)
13		simplr3 1217	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ∈ 𝐴)
14		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴)
15		cvrat4.j	. . . . . . . . . . . . . . 15 ⊢ ∨ = (join‘𝐾)
16	5, 15, 7	hlatlej1 37910	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑟))
17	12, 13, 14, 16	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑟))
18		breq1 5113	. . . . . . . . . . . . 13 ⊢ (𝑃 = 𝑄 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑄 ≤ (𝑄 ∨ 𝑟)))
19	17, 18	imbitrrid 245	. . . . . . . . . . . 12 ⊢ (𝑃 = 𝑄 → (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑃 ≤ (𝑄 ∨ 𝑟)))
20	19	expd 416	. . . . . . . . . . 11 ⊢ (𝑃 = 𝑄 → ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑟 ∈ 𝐴 → 𝑃 ≤ (𝑄 ∨ 𝑟))))
21	20	impcom 408	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑟 ∈ 𝐴 → 𝑃 ≤ (𝑄 ∨ 𝑟)))
22	21	anim2d 612	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → ((𝑟 ≤ 𝑋 ∧ 𝑟 ∈ 𝐴) → (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
23	22	expcomd 417	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑟 ∈ 𝐴 → (𝑟 ≤ 𝑋 → (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
24	23	reximdvai 3158	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋 → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
25	11, 24	syld 47	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
26	25	ex 413	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 = 𝑄 → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
27	26	a1i 11	. . . 4 ⊢ (𝑃 ≤ (𝑋 ∨ 𝑄) → ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 = 𝑄 → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))))
28	27	com4l 92	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 = 𝑄 → (𝑋 ≠ 0 → (𝑃 ≤ (𝑋 ∨ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))))
29	28	imp4a 423	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 = 𝑄 → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
30		hllat 37898	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
31	30	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat)
32		simpr3 1196	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
33	4, 7	atbase 37824	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
34	32, 33	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐵)
35	4, 5, 15	latleeqj2 18355	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ≤ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
36	31, 34, 3, 35	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ≤ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
37	36	biimpa 477	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) = 𝑋)
38	37	breq2d 5122	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ 𝑃 ≤ 𝑋))
39	38	biimpa 477	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ 𝑋)
40	39	expl 458	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ 𝑋))
41		simpl 483	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ HL)
42		simpr2 1195	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
43	5, 15, 7	hlatlej2 37911	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → 𝑃 ≤ (𝑄 ∨ 𝑃))
44	41, 32, 42, 43	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ≤ (𝑄 ∨ 𝑃))
45	40, 44	jctird 527	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
46	45, 42	jctild 526	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃)))))
47	46	impl 456	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
48		breq1 5113	. . . . . . 7 ⊢ (𝑟 = 𝑃 → (𝑟 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
49		oveq2 7370	. . . . . . . 8 ⊢ (𝑟 = 𝑃 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑃))
50	49	breq2d 5122	. . . . . . 7 ⊢ (𝑟 = 𝑃 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑃)))
51	48, 50	anbi12d 631	. . . . . 6 ⊢ (𝑟 = 𝑃 → ((𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)) ↔ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
52	51	rspcev 3582	. . . . 5 ⊢ ((𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
53	47, 52	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
54	53	adantrl 714	. . 3 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ (𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
55	54	exp31 420	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ≤ 𝑋 → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
56		simpr 485	. . 3 ⊢ ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ (𝑋 ∨ 𝑄))
57		ioran 982	. . . . 5 ⊢ (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
58		df-ne 2940	. . . . . 6 ⊢ (𝑃 ≠ 𝑄 ↔ ¬ 𝑃 = 𝑄)
59	58	anbi1i 624	. . . . 5 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
60	57, 59	bitr4i 277	. . . 4 ⊢ (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) ↔ (𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
61		eqid 2731	. . . . . . . . . 10 ⊢ (meet‘𝐾) = (meet‘𝐾)
62	4, 5, 15, 61, 7	cvrat3 37978	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴))
63	62	3expd 1353	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ≠ 𝑄 → (¬ 𝑄 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴))))
64	63	imp4c 424	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴))
65	4, 7	atbase 37824	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
66	42, 65	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐵)
67	4, 15	latjcl 18342	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
68	31, 66, 34, 67	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
69	4, 5, 61	latmle1 18367	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
70	31, 3, 68, 69	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
71	70	adantr 481	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
72		simpll 765	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝐾 ∈ HL)
73	63	imp44 429	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴)
74		simplr2 1216	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
75	34	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑄 ∈ 𝐵)
76	73, 74, 75	3jca 1128	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵))
77	72, 76	jca 512	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)))
78	4, 5, 61, 6, 7	atnle 37852	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
79	2, 32, 3, 78	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
80	4, 61	latmcom 18366	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
81	31, 34, 3, 80	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
82	81	eqeq1d 2733	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄(meet‘𝐾)𝑋) = 0 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
83	79, 82	bitrd 278	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
84	4, 61	latmcl 18343	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
85	31, 3, 68, 84	syl3anc 1371	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
86	85, 3, 34	3jca 1128	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵))
87	31, 86	jca 512	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵)))
88	4, 5, 61	latmlem2 18373	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵)) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑄(meet‘𝐾)𝑋)))
89	87, 70, 88	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑄(meet‘𝐾)𝑋))
90	89, 81	breqtrd 5136	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑋(meet‘𝐾)𝑄))
91		breq2 5114	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋(meet‘𝐾)𝑄) = 0 → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑋(meet‘𝐾)𝑄) ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ 0 ))
92	90, 91	syl5ibcom 244	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋(meet‘𝐾)𝑄) = 0 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ 0 ))
93		hlop 37897	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
94	93	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ OP)
95	4, 61	latmcl 18343	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ∈ 𝐵)
96	31, 34, 85, 95	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ∈ 𝐵)
97	4, 5, 6	ople0 37722	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ OP ∧ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ∈ 𝐵) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ 0 ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ))
98	94, 96, 97	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ 0 ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ))
99	92, 98	sylibd 238	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋(meet‘𝐾)𝑄) = 0 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ))
100	83, 99	sylbid 239	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ))
101	100	imp 407	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 )
102	101	adantrl 714	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 )
103	102	adantrr 715	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 )
104	4, 5, 61	latmle2 18368	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
105	31, 3, 68, 104	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
106	4, 15	latjcom 18350	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
107	31, 66, 34, 106	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
108	105, 107	breqtrd 5136	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃))
109	108	adantr 481	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃))
110	30	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → 𝐾 ∈ Lat)
111		simpr3 1196	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → 𝑄 ∈ 𝐵)
112		simpr1 1194	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴)
113	4, 7	atbase 37824	. . . . . . . . . . . . . 14 ⊢ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
114	112, 113	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
115	4, 61	latmcom 18366	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))(meet‘𝐾)𝑄))
116	110, 111, 114, 115	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))(meet‘𝐾)𝑄))
117	116	eqeq1d 2733	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ↔ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))(meet‘𝐾)𝑄) = 0 ))
118	4, 5, 15, 61, 6, 7	hlexch3 37927	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))(meet‘𝐾)𝑄) = 0 ) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃) → 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)))))
119	118	3expia 1121	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))(meet‘𝐾)𝑄) = 0 → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃) → 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))))
120	117, 119	sylbid 239	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃) → 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))))
121	77, 103, 109, 120	syl3c 66	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))
122	71, 121	jca 512	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)))))
123	122	ex 413	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))))
124	64, 123	jcad 513	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)))))))
125		breq1 5113	. . . . . . . 8 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑟 ≤ 𝑋 ↔ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋))
126		oveq2 7370	. . . . . . . . 9 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑄 ∨ 𝑟) = (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))
127	126	breq2d 5122	. . . . . . . 8 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)))))
128	125, 127	anbi12d 631	. . . . . . 7 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → ((𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)) ↔ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))))
129	128	rspcev 3582	. . . . . 6 ⊢ (((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
130	124, 129	syl6 35	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
131	130	expd 416	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
132	60, 131	biimtrid 241	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
133	56, 132	syl7 74	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
134	29, 55, 133	ecase3d 1032	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∃wrex 3069   class class class wbr 5110  ‘cfv 6501  (class class class)co 7362  Basecbs 17094  lecple 17154  joincjn 18214  meetcmee 18215  0.cp0 18326  Latclat 18334  OPcops 37707  Atomscatm 37798  AtLatcal 37799  HLchlt 37885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-proset 18198  df-poset 18216  df-plt 18233  df-lub 18249  df-glb 18250  df-join 18251  df-meet 18252  df-p0 18328  df-lat 18335  df-clat 18402  df-oposet 37711  df-ol 37713  df-oml 37714  df-covers 37801  df-ats 37802  df-atl 37833  df-cvlat 37857  df-hlat 37886

This theorem is referenced by:  cvrat42  37980  ps-2  38014