Theorem cvrat4 40766

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 33056 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 40683	. . . . . . . . . 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 40639	. . . . . . . . . 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 40698	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑟))
17	12, 13, 14, 16	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑟))
18		breq1 5111	. . . . . . . . . . . . 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 3177	. . . . . . 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 40686	. . . . . . . . . . . . . 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 40612	. . . . . . . . . . . . . 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 18676	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ≤ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
36	31, 34, 3, 35	syl3anc 1399	. . . . . . . . . . . 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 5120	. . . . . . . . . 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 40699	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → 𝑃 ≤ (𝑄 ∨ 𝑃))
44	41, 32, 42, 43	syl3anc 1399	. . . . . . . 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 5111	. . . . . . 7 ⊢ (𝑟 = 𝑃 → (𝑟 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
49		oveq2 7459	. . . . . . . 8 ⊢ (𝑟 = 𝑃 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑃))
50	49	breq2d 5120	. . . . . . 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 3580	. . . . 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 2960	. . . . . 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 2763	. . . . . . . . . 10 ⊢ (meet‘𝐾) = (meet‘𝐾)
62	4, 5, 15, 61, 7	cvrat3 40765	. . . . . . . . 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 40612	. . . . . . . . . . . . 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 18663	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
68	31, 66, 34, 67	syl3anc 1399	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
69	4, 5, 61	latmle1 18688	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
70	31, 3, 68, 69	syl3anc 1399	. . . . . . . . . 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 40640	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
79	2, 32, 3, 78	syl3anc 1399	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
80	4, 61	latmcom 18687	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
81	31, 34, 3, 80	syl3anc 1399	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
82	81	eqeq1d 2765	. . . . . . . . . . . . . . 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 18664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
85	31, 3, 68, 84	syl3anc 1399	. . . . . . . . . . . . . . . . . . . 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 18694	. . . . . . . . . . . . . . . . . 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 5137	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑋(meet‘𝐾)𝑄))
91		breq2 5112	. . . . . . . . . . . . . . . 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 40685	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
94	93	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ OP)
95	4, 61	latmcl 18664	. . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ∈ 𝐵)
97	4, 5, 6	ople0 40510	. . . . . . . . . . . . . . . 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 18689	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
105	31, 3, 68, 104	syl3anc 1399	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
106	4, 15	latjcom 18671	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
107	31, 66, 34, 106	syl3anc 1399	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
108	105, 107	breqtrd 5137	. . . . . . . . . . 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 40612	. . . . . . . . . . . . . 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 18687	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 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 2765	. . . . . . . . . . 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 40714	. . . . . . . . . . . 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 5111	. . . . . . . 8 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑟 ≤ 𝑋 ↔ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋))
126		oveq2 7459	. . . . . . . . 9 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑄 ∨ 𝑟) = (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))
127	126	breq2d 5120	. . . . . . . 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 3580	. . . . . 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 2147   ≠ wne 2959  ∃wrex 3090   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  lecple 17479  joincjn 18535  meetcmee 18536  0.cp0 18645  Latclat 18655  OPcops 40495  Atomscatm 40586  AtLatcal 40587  HLchlt 40673

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-proset 18518  df-poset 18537  df-plt 18552  df-lub 18568  df-glb 18569  df-join 18570  df-meet 18571  df-p0 18647  df-lat 18656  df-clat 18723  df-oposet 40499  df-ol 40501  df-oml 40502  df-covers 40589  df-ats 40590  df-atl 40621  df-cvlat 40645  df-hlat 40674

This theorem is used by:  cvrat42  40767  ps-2  40801