Theorem cvrat4 39903

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 32483 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 39820	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2	1	adantr 480	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ AtLat)
3		simpr1 1196	. . . . . . . . 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 39776	. . . . . . . . . 10 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋)
9	8	3exp 1120	. . . . . . . . 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 480	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋))
12		simpll 767	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝐾 ∈ HL)
13		simplr3 1219	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ∈ 𝐴)
14		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴)
15		cvrat4.j	. . . . . . . . . . . . . . 15 ⊢ ∨ = (join‘𝐾)
16	5, 15, 7	hlatlej1 39835	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑟))
17	12, 13, 14, 16	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑟))
18		breq1 5089	. . . . . . . . . . . . 13 ⊢ (𝑃 = 𝑄 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑄 ≤ (𝑄 ∨ 𝑟)))
19	17, 18	imbitrrid 246	. . . . . . . . . . . 12 ⊢ (𝑃 = 𝑄 → (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑃 ≤ (𝑄 ∨ 𝑟)))
20	19	expd 415	. . . . . . . . . . 11 ⊢ (𝑃 = 𝑄 → ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑟 ∈ 𝐴 → 𝑃 ≤ (𝑄 ∨ 𝑟))))
21	20	impcom 407	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑟 ∈ 𝐴 → 𝑃 ≤ (𝑄 ∨ 𝑟)))
22	21	anim2d 613	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → ((𝑟 ≤ 𝑋 ∧ 𝑟 ∈ 𝐴) → (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
23	22	expcomd 416	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑟 ∈ 𝐴 → (𝑟 ≤ 𝑋 → (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
24	23	reximdvai 3149	. . . . . . 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 412	. . . . 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 422	. 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 39823	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
31	30	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat)
32		simpr3 1198	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
33	4, 7	atbase 39749	. . . . . . . . . . . . . 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 18409	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ≤ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
36	31, 34, 3, 35	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ≤ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
37	36	biimpa 476	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) = 𝑋)
38	37	breq2d 5098	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ 𝑃 ≤ 𝑋))
39	38	biimpa 476	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ 𝑋)
40	39	expl 457	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ 𝑋))
41		simpl 482	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ HL)
42		simpr2 1197	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
43	5, 15, 7	hlatlej2 39836	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → 𝑃 ≤ (𝑄 ∨ 𝑃))
44	41, 32, 42, 43	syl3anc 1374	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ≤ (𝑄 ∨ 𝑃))
45	40, 44	jctird 526	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
46	45, 42	jctild 525	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃)))))
47	46	impl 455	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
48		breq1 5089	. . . . . . 7 ⊢ (𝑟 = 𝑃 → (𝑟 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
49		oveq2 7368	. . . . . . . 8 ⊢ (𝑟 = 𝑃 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑃))
50	49	breq2d 5098	. . . . . . 7 ⊢ (𝑟 = 𝑃 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑃)))
51	48, 50	anbi12d 633	. . . . . 6 ⊢ (𝑟 = 𝑃 → ((𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)) ↔ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
52	51	rspcev 3565	. . . . 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 717	. . 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 419	. 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 484	. . 3 ⊢ ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ (𝑋 ∨ 𝑄))
57		ioran 986	. . . . 5 ⊢ (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
58		df-ne 2934	. . . . . 6 ⊢ (𝑃 ≠ 𝑄 ↔ ¬ 𝑃 = 𝑄)
59	58	anbi1i 625	. . . . 5 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
60	57, 59	bitr4i 278	. . . 4 ⊢ (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) ↔ (𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
61		eqid 2737	. . . . . . . . . 10 ⊢ (meet‘𝐾) = (meet‘𝐾)
62	4, 5, 15, 61, 7	cvrat3 39902	. . . . . . . . 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 1355	. . . . . . . 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 423	. . . . . . 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 39749	. . . . . . . . . . . . 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 18396	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
68	31, 66, 34, 67	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
69	4, 5, 61	latmle1 18421	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
70	31, 3, 68, 69	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
71	70	adantr 480	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
72		simpll 767	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝐾 ∈ HL)
73	63	imp44 428	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴)
74		simplr2 1218	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
75	34	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑄 ∈ 𝐵)
76	73, 74, 75	3jca 1129	. . . . . . . . . . 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 511	. . . . . . . . . 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 39777	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
79	2, 32, 3, 78	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
80	4, 61	latmcom 18420	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
81	31, 34, 3, 80	syl3anc 1374	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
82	81	eqeq1d 2739	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄(meet‘𝐾)𝑋) = 0 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
83	79, 82	bitrd 279	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
84	4, 61	latmcl 18397	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
85	31, 3, 68, 84	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
86	85, 3, 34	3jca 1129	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵))
87	31, 86	jca 511	. . . . . . . . . . . . . . . . . 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 18427	. . . . . . . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑋(meet‘𝐾)𝑄))
91		breq2 5090	. . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . 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 39822	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
94	93	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ OP)
95	4, 61	latmcl 18397	. . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ∈ 𝐵)
97	4, 5, 6	ople0 39647	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . 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 240	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ))
101	100	imp 406	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 )
102	101	adantrl 717	. . . . . . . . . . 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 718	. . . . . . . . . 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 18422	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
105	31, 3, 68, 104	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
106	4, 15	latjcom 18404	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
107	31, 66, 34, 106	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
108	105, 107	breqtrd 5112	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃))
109	108	adantr 480	. . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → 𝐾 ∈ Lat)
111		simpr3 1198	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → 𝑄 ∈ 𝐵)
112		simpr1 1196	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴)
113	4, 7	atbase 39749	. . . . . . . . . . . . . 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 18420	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 2739	. . . . . . . . . . 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 39851	. . . . . . . . . . . 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 1122	. . . . . . . . . . 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 240	. . . . . . . . . 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 511	. . . . . . . 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 412	. . . . . . 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 512	. . . . . 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 5089	. . . . . . . 8 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑟 ≤ 𝑋 ↔ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋))
126		oveq2 7368	. . . . . . . . 9 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑄 ∨ 𝑟) = (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))
127	126	breq2d 5098	. . . . . . . 8 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)))))
128	125, 127	anbi12d 633	. . . . . . 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 3565	. . . . . 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 415	. . . 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 242	. . 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 1035	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 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  lecple 17218  joincjn 18268  meetcmee 18269  0.cp0 18378  Latclat 18388  OPcops 39632  Atomscatm 39723  AtLatcal 39724  HLchlt 39810

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18389  df-clat 18456  df-oposet 39636  df-ol 39638  df-oml 39639  df-covers 39726  df-ats 39727  df-atl 39758  df-cvlat 39782  df-hlat 39811

This theorem is referenced by:  cvrat42  39904  ps-2  39938