Theorem paddasslem13 40692

Description: Lemma for paddass 40698. The case when 𝑟 ≤ (𝑥 ∨ 𝑦). (Unlike the proof in Maeda and Maeda, we don't need 𝑥 ≠ 𝑦.) (Contributed by NM, 11-Jan-2012.)

Hypotheses

Ref	Expression
paddasslem.l	⊢ ≤ = (le‘𝐾)
paddasslem.j	⊢ ∨ = (join‘𝐾)
paddasslem.a	⊢ 𝐴 = (Atoms‘𝐾)
paddasslem.p	⊢ + = (+𝑃‘𝐾)

Assertion

Ref	Expression
paddasslem13	⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

Proof of Theorem paddasslem13

Step	Hyp	Ref	Expression
1		simpl1l 1243	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝐾 ∈ HL)
2		simpl21 1270	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑋 ⊆ 𝐴)
3		simpl22 1271	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑌 ⊆ 𝐴)
4		paddasslem.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
5		paddasslem.p	. . . . 5 ⊢ + = (+𝑃‘𝐾)
6	4, 5	paddssat 40674	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
7	1, 2, 3, 6	syl3anc 1398	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → (𝑋 + 𝑌) ⊆ 𝐴)
8		simpl23 1272	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑍 ⊆ 𝐴)
9	4, 5	sspadd1 40675	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ ((𝑋 + 𝑌) + 𝑍))
10	1, 7, 8, 9	syl3anc 1398	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → (𝑋 + 𝑌) ⊆ ((𝑋 + 𝑌) + 𝑍))
11	1	hllatd 40224	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝐾 ∈ Lat)
12		simprll 791	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑥 ∈ 𝑋)
13		simprlr 792	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑦 ∈ 𝑌)
14		simpl3l 1247	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ 𝐴)
15		eqid 2762	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		paddasslem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
17	15, 4	atbase 40149	. . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾))
18	14, 17	syl 18	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ (Base‘𝐾))
19	2, 12	sseldd 3935	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑥 ∈ 𝐴)
20	15, 4	atbase 40149	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (Base‘𝐾))
21	19, 20	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑥 ∈ (Base‘𝐾))
22		simpl3r 1248	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑟 ∈ 𝐴)
23	15, 4	atbase 40149	. . . . . 6 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾))
24	22, 23	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑟 ∈ (Base‘𝐾))
25		paddasslem.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
26	15, 25	latjcl 18531	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → (𝑥 ∨ 𝑟) ∈ (Base‘𝐾))
27	11, 21, 24, 26	syl3anc 1398	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → (𝑥 ∨ 𝑟) ∈ (Base‘𝐾))
28	3, 13	sseldd 3935	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑦 ∈ 𝐴)
29	15, 4	atbase 40149	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (Base‘𝐾))
30	28, 29	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑦 ∈ (Base‘𝐾))
31	15, 25	latjcl 18531	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥 ∨ 𝑦) ∈ (Base‘𝐾))
32	11, 21, 30, 31	syl3anc 1398	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → (𝑥 ∨ 𝑦) ∈ (Base‘𝐾))
33		simprrr 794	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ≤ (𝑥 ∨ 𝑟))
34	15, 16, 25	latlej1 18540	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 ≤ (𝑥 ∨ 𝑦))
35	11, 21, 30, 34	syl3anc 1398	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑥 ≤ (𝑥 ∨ 𝑦))
36		simprrl 793	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑟 ≤ (𝑥 ∨ 𝑦))
37	15, 16, 25	latjle12 18542	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾) ∧ (𝑥 ∨ 𝑦) ∈ (Base‘𝐾))) → ((𝑥 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑥 ∨ 𝑦)) ↔ (𝑥 ∨ 𝑟) ≤ (𝑥 ∨ 𝑦)))
38	11, 21, 24, 32, 37	syl13anc 1399	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → ((𝑥 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑥 ∨ 𝑦)) ↔ (𝑥 ∨ 𝑟) ≤ (𝑥 ∨ 𝑦)))
39	35, 36, 38	mpbi2and 725	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → (𝑥 ∨ 𝑟) ≤ (𝑥 ∨ 𝑦))
40	15, 16, 11, 18, 27, 32, 33, 39	lattrd 18538	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ≤ (𝑥 ∨ 𝑦))
41	16, 25, 4, 5	elpaddri 40662	. . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑝 ∈ 𝐴 ∧ 𝑝 ≤ (𝑥 ∨ 𝑦))) → 𝑝 ∈ (𝑋 + 𝑌))
42	11, 2, 3, 12, 13, 14, 40, 41	syl322anc 1425	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ (𝑋 + 𝑌))
43	10, 42	sseldd 3935	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957   ⊆ wss 3902   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  lecple 17353  joincjn 18403  Latclat 18523  Atomscatm 40123  HLchlt 40210  +_𝑃cpadd 40655

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-poset 18405  df-lub 18436  df-glb 18437  df-join 18438  df-meet 18439  df-lat 18524  df-ats 40127  df-atl 40158  df-cvlat 40182  df-hlat 40211  df-padd 40656

This theorem is used by:  paddasslem14  40693