hlrelat2 - Mathbox for Norm Megill

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Theorem hlrelat2 40218

Description: A consequence of relative atomicity. (chrelat2i 32754 analog.) (Contributed by NM, 5-Feb-2012.)

**Hypotheses**
Ref	Expression
hlrelat2.b	⊢ 𝐵 = (Base‘𝐾)
hlrelat2.l	⊢ ≤ = (le‘𝐾)
hlrelat2.a	⊢ 𝐴 = (Atoms‘𝐾)

**Assertion**
Ref	Expression
hlrelat2	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑌 ↔ ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌)))

Distinct variable groups: 𝐴,𝑝 𝐵,𝑝 𝐾,𝑝 ≤ ,𝑝 𝑋,𝑝 𝑌,𝑝

**Proof of Theorem hlrelat2**
Step	Hyp	Ref	Expression
1		hllat 40178	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2		hlrelat2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		hlrelat2.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		eqid 2766	. . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾)
5		eqid 2766	. . . . 5 ⊢ (meet‘𝐾) = (meet‘𝐾)
6	2, 3, 4, 5	latnlemlt 18553	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑌 ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋))
7	1, 6	syl3an1 1181	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑌 ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋))
8		simp1 1154	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ HL)
9	2, 5	latmcl 18521	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
10	1, 9	syl3an1 1181	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
11		simp2 1155	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
12		eqid 2766	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
13		hlrelat2.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
14	2, 3, 4, 12, 13	hlrelat 40217	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋) → ∃𝑝 ∈ 𝐴 ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋))
15	14	ex 418	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋 → ∃𝑝 ∈ 𝐴 ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋)))
16	8, 10, 11, 15	syl3anc 1398	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋 → ∃𝑝 ∈ 𝐴 ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋)))
17		simpl1 1210	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝐾 ∈ HL)
18	17	hllatd 40179	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝐾 ∈ Lat)
19	10	adantr 486	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
20	2, 13	atbase 40104	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
21	20	adantl 487	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐵)
22		simpl2 1211	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑋 ∈ 𝐵)
23	2, 3, 12	latjle12 18531	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)𝑌) ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (((𝑋(meet‘𝐾)𝑌) ≤ 𝑋 ∧ 𝑝 ≤ 𝑋) ↔ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋))
24	18, 19, 21, 22, 23	syl13anc 1399	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (((𝑋(meet‘𝐾)𝑌) ≤ 𝑋 ∧ 𝑝 ≤ 𝑋) ↔ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋))
25		simpr 490	. . . . . . . 8 ⊢ (((𝑋(meet‘𝐾)𝑌) ≤ 𝑋 ∧ 𝑝 ≤ 𝑋) → 𝑝 ≤ 𝑋)
26	24, 25	biimtrrdi 257	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋 → 𝑝 ≤ 𝑋))
27	26	adantld 496	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋) → 𝑝 ≤ 𝑋))
28		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑌 ∈ 𝐵)
29	2, 3, 5	latlem12 18547	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ↔ 𝑝 ≤ (𝑋(meet‘𝐾)𝑌)))
30	18, 21, 22, 28, 29	syl13anc 1399	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → ((𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ↔ 𝑝 ≤ (𝑋(meet‘𝐾)𝑌)))
31	30	notbid 321	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (¬ (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ↔ ¬ 𝑝 ≤ (𝑋(meet‘𝐾)𝑌)))
32	2, 3, 4, 12	latnle 18554	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → (¬ 𝑝 ≤ (𝑋(meet‘𝐾)𝑌) ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝)))
33	18, 19, 21, 32	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (¬ 𝑝 ≤ (𝑋(meet‘𝐾)𝑌) ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝)))
34	31, 33	bitrd 282	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (¬ (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝)))
35	34, 24	anbi12d 644	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → ((¬ (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ∧ ((𝑋(meet‘𝐾)𝑌) ≤ 𝑋 ∧ 𝑝 ≤ 𝑋)) ↔ ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋)))
36		pm3.21 477	. . . . . . . . . 10 ⊢ (𝑝 ≤ 𝑌 → (𝑝 ≤ 𝑋 → (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)))
37		orcom 884	. . . . . . . . . . 11 ⊢ (((𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ∨ ¬ 𝑝 ≤ 𝑋) ↔ (¬ 𝑝 ≤ 𝑋 ∨ (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)))
38		pm4.55 1003	. . . . . . . . . . 11 ⊢ (¬ (¬ (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ∧ 𝑝 ≤ 𝑋) ↔ ((𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ∨ ¬ 𝑝 ≤ 𝑋))
39		imor 867	. . . . . . . . . . 11 ⊢ ((𝑝 ≤ 𝑋 → (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)) ↔ (¬ 𝑝 ≤ 𝑋 ∨ (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)))
40	37, 38, 39	3bitr4ri 307	. . . . . . . . . 10 ⊢ ((𝑝 ≤ 𝑋 → (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)) ↔ ¬ (¬ (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ∧ 𝑝 ≤ 𝑋))
41	36, 40	sylib 221	. . . . . . . . 9 ⊢ (𝑝 ≤ 𝑌 → ¬ (¬ (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ∧ 𝑝 ≤ 𝑋))
42	41	con2i 140	. . . . . . . 8 ⊢ ((¬ (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ∧ 𝑝 ≤ 𝑋) → ¬ 𝑝 ≤ 𝑌)
43	42	adantrl 729	. . . . . . 7 ⊢ ((¬ (𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ∧ ((𝑋(meet‘𝐾)𝑌) ≤ 𝑋 ∧ 𝑝 ≤ 𝑋)) → ¬ 𝑝 ≤ 𝑌)
44	35, 43	biimtrrdi 257	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋) → ¬ 𝑝 ≤ 𝑌))
45	27, 44	jcad 522	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋) → (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌)))
46	45	reximdva 3181	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∃𝑝 ∈ 𝐴 ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ≤ 𝑋) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌)))
47	16, 46	syld 48	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋 → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌)))
48	7, 47	sylbid 243	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑌 → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌)))
49	2, 3	lattr 18525	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑝 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑝 ≤ 𝑌))
50	18, 21, 22, 28, 49	syl13anc 1399	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → ((𝑝 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑝 ≤ 𝑌))
51	50	exp4b 436	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ 𝐴 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))
52	51	com34 92	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ 𝐴 → (𝑋 ≤ 𝑌 → (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))))
53	52	com23 87	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑝 ∈ 𝐴 → (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))))
54	53	ralrimdv 3166	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌)))
55		iman 407	. . . . . 6 ⊢ ((𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌) ↔ ¬ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌))
56	55	ralbii 3114	. . . . 5 ⊢ (∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌) ↔ ∀𝑝 ∈ 𝐴 ¬ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌))
57		ralnex 3094	. . . . 5 ⊢ (∀𝑝 ∈ 𝐴 ¬ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌) ↔ ¬ ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌))
58	56, 57	bitri 278	. . . 4 ⊢ (∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌) ↔ ¬ ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌))
59	54, 58	imbitrdi 254	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ¬ ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌)))
60	59	con2d 135	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌) → ¬ 𝑋 ≤ 𝑌))
61	48, 60	impbid 215	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑌 ↔ ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑌)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3082 ∃wrex 3092 class class class wbr 5114 ‘cfv 6543 (class class class)co 7423 Basecbs 17294 lecple 17342 ltcplt 18389 joincjn 18392 meetcmee 18393 Latclat 18512 Atomscatm 40078 HLchlt 40165

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 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-proset 18375 df-poset 18394 df-plt 18409 df-lub 18425 df-glb 18426 df-join 18427 df-meet 18428 df-p0 18504 df-lat 18513 df-clat 18580 df-oposet 39991 df-ol 39993 df-oml 39994 df-covers 40081 df-ats 40082 df-atl 40113 df-cvlat 40137 df-hlat 40166

This theorem is used by: lhpj1 40837

W3C validator