Theorem cvrat4 40250

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 32796 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 40167	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2	1	adantr 486	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ AtLat)
3		simpr1 1213	. . . . . . . . 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 40123	. . . . . . . . . 10 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋)
9	8	3exp 1137	. . . . . . . . 9 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝐵 → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋)))
10	2, 3, 9	sylc 66	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋))
11	10	adantr 486	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋))
12		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝐾 ∈ HL)
13		simplr3 1236	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ∈ 𝐴)
14		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴)
15		cvrat4.j	. . . . . . . . . . . . . . 15 ⊢ ∨ = (join‘𝐾)
16	5, 15, 7	hlatlej1 40182	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑟))
17	12, 13, 14, 16	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑟))
18		breq1 5114	. . . . . . . . . . . . 13 ⊢ (𝑃 = 𝑄 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑄 ≤ (𝑄 ∨ 𝑟)))
19	17, 18	imbitrrid 249	. . . . . . . . . . . 12 ⊢ (𝑃 = 𝑄 → (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑃 ≤ (𝑄 ∨ 𝑟)))
20	19	expd 421	. . . . . . . . . . 11 ⊢ (𝑃 = 𝑄 → ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑟 ∈ 𝐴 → 𝑃 ≤ (𝑄 ∨ 𝑟))))
21	20	impcom 413	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑟 ∈ 𝐴 → 𝑃 ≤ (𝑄 ∨ 𝑟)))
22	21	anim2d 624	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → ((𝑟 ≤ 𝑋 ∧ 𝑟 ∈ 𝐴) → (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
23	22	expcomd 422	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑟 ∈ 𝐴 → (𝑟 ≤ 𝑋 → (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
24	23	reximdvai 3178	. . . . . . 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 48	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
26	25	ex 418	. . . . 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 93	. . 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 428	. 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 40170	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
31	30	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat)
32		simpr3 1215	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
33	4, 7	atbase 40096	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
34	32, 33	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐵)
35	4, 5, 15	latleeqj2 18526	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ≤ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
36	31, 34, 3, 35	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ≤ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
37	36	biimpa 482	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) = 𝑋)
38	37	breq2d 5123	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ 𝑃 ≤ 𝑋))
39	38	biimpa 482	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ 𝑋)
40	39	expl 463	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ 𝑋))
41		simpl 488	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ HL)
42		simpr2 1214	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
43	5, 15, 7	hlatlej2 40183	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → 𝑃 ≤ (𝑄 ∨ 𝑃))
44	41, 32, 42, 43	syl3anc 1398	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ≤ (𝑄 ∨ 𝑃))
45	40, 44	jctird 536	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
46	45, 42	jctild 535	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃)))))
47	46	impl 461	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
48		breq1 5114	. . . . . . 7 ⊢ (𝑟 = 𝑃 → (𝑟 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
49		oveq2 7424	. . . . . . . 8 ⊢ (𝑟 = 𝑃 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑃))
50	49	breq2d 5123	. . . . . . 7 ⊢ (𝑟 = 𝑃 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑃)))
51	48, 50	anbi12d 644	. . . . . 6 ⊢ (𝑟 = 𝑃 → ((𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)) ↔ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
52	51	rspcev 3583	. . . . 5 ⊢ ((𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
53	47, 52	syl 18	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
54	53	adantrl 729	. . 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 425	. 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 490	. . 3 ⊢ ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ (𝑋 ∨ 𝑄))
57		ioran 999	. . . . 5 ⊢ (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
58		df-ne 2961	. . . . . 6 ⊢ (𝑃 ≠ 𝑄 ↔ ¬ 𝑃 = 𝑄)
59	58	anbi1i 636	. . . . 5 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
60	57, 59	bitr4i 281	. . . 4 ⊢ (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) ↔ (𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
61		eqid 2765	. . . . . . . . . 10 ⊢ (meet‘𝐾) = (meet‘𝐾)
62	4, 5, 15, 61, 7	cvrat3 40249	. . . . . . . . 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 1372	. . . . . . . 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 429	. . . . . . 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 40096	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
66	42, 65	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐵)
67	4, 15	latjcl 18513	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
68	31, 66, 34, 67	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
69	4, 5, 61	latmle1 18538	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
70	31, 3, 68, 69	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
71	70	adantr 486	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
72		simpll 779	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝐾 ∈ HL)
73	63	imp44 434	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴)
74		simplr2 1235	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
75	34	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑄 ∈ 𝐵)
76	73, 74, 75	3jca 1146	. . . . . . . . . . 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 521	. . . . . . . . . 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 40124	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
79	2, 32, 3, 78	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
80	4, 61	latmcom 18537	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
81	31, 34, 3, 80	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
82	81	eqeq1d 2767	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄(meet‘𝐾)𝑋) = 0 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
83	79, 82	bitrd 282	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
84	4, 61	latmcl 18514	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
85	31, 3, 68, 84	syl3anc 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
86	85, 3, 34	3jca 1146	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵))
87	31, 86	jca 521	. . . . . . . . . . . . . . . . . 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 18544	. . . . . . . . . . . . . . . . . 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 66	. . . . . . . . . . . . . . . . 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 5139	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑋(meet‘𝐾)𝑄))
91		breq2 5115	. . . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . 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 40169	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
94	93	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ OP)
95	4, 61	latmcl 18514	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ∈ 𝐵)
97	4, 5, 6	ople0 39994	. . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . 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 243	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ))
101	100	imp 412	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 )
102	101	adantrl 729	. . . . . . . . . . 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 730	. . . . . . . . . 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 18539	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
105	31, 3, 68, 104	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
106	4, 15	latjcom 18521	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
107	31, 66, 34, 106	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
108	105, 107	breqtrd 5139	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃))
109	108	adantr 486	. . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → 𝐾 ∈ Lat)
111		simpr3 1215	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → 𝑄 ∈ 𝐵)
112		simpr1 1213	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴)
113	4, 7	atbase 40096	. . . . . . . . . . . . . 14 ⊢ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
114	112, 113	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
115	4, 61	latmcom 18537	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 2767	. . . . . . . . . . 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 40198	. . . . . . . . . . . 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 1139	. . . . . . . . . . 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 243	. . . . . . . . . 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 67	. . . . . . . . 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 521	. . . . . . . 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 418	. . . . . . 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 522	. . . . . 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 5114	. . . . . . . 8 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑟 ≤ 𝑋 ↔ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋))
126		oveq2 7424	. . . . . . . . 9 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑄 ∨ 𝑟) = (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))
127	126	breq2d 5123	. . . . . . . 8 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)))))
128	125, 127	anbi12d 644	. . . . . . 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 3583	. . . . . 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 36	. . . . 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 421	. . . 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 245	. . 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 75	. 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 1050	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  lecple 17335  joincjn 18385  meetcmee 18386  0.cp0 18495  Latclat 18505  OPcops 39979  Atomscatm 40070  AtLatcal 40071  HLchlt 40157

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-proset 18368  df-poset 18387  df-plt 18402  df-lub 18418  df-glb 18419  df-join 18420  df-meet 18421  df-p0 18497  df-lat 18506  df-clat 18573  df-oposet 39983  df-ol 39985  df-oml 39986  df-covers 40073  df-ats 40074  df-atl 40105  df-cvlat 40129  df-hlat 40158

This theorem is used by:  cvrat42  40251  ps-2  40285