Theorem oeoelem 8545

Description: Lemma for oeoe 8546. (Contributed by Eric Schmidt, 26-May-2009.)

Hypotheses

Ref	Expression
oeoelem.1	⊢ 𝐴 ∈ On
oeoelem.2	⊢ ∅ ∈ 𝐴

Assertion

Ref	Expression
oeoelem	⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))

Proof of Theorem oeoelem

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7365	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o ∅))
2		oveq2 7365	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
3	2	oveq2d 7373	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o ∅)))
4	1, 3	eqeq12d 2752	. . 3 ⊢ (𝑥 = ∅ → (((𝐴 ↑o 𝐵) ↑o 𝑥) = (𝐴 ↑o (𝐵 ·o 𝑥)) ↔ ((𝐴 ↑o 𝐵) ↑o ∅) = (𝐴 ↑o (𝐵 ·o ∅))))
5		oveq2 7365	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o 𝑦))
6		oveq2 7365	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
7	6	oveq2d 7373	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o 𝑦)))
8	5, 7	eqeq12d 2752	. . 3 ⊢ (𝑥 = 𝑦 → (((𝐴 ↑o 𝐵) ↑o 𝑥) = (𝐴 ↑o (𝐵 ·o 𝑥)) ↔ ((𝐴 ↑o 𝐵) ↑o 𝑦) = (𝐴 ↑o (𝐵 ·o 𝑦))))
9		oveq2 7365	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o suc 𝑦))
10		oveq2 7365	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
11	10	oveq2d 7373	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o suc 𝑦)))
12	9, 11	eqeq12d 2752	. . 3 ⊢ (𝑥 = suc 𝑦 → (((𝐴 ↑o 𝐵) ↑o 𝑥) = (𝐴 ↑o (𝐵 ·o 𝑥)) ↔ ((𝐴 ↑o 𝐵) ↑o suc 𝑦) = (𝐴 ↑o (𝐵 ·o suc 𝑦))))
13		oveq2 7365	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o 𝐶))
14		oveq2 7365	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
15	14	oveq2d 7373	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o 𝐶)))
16	13, 15	eqeq12d 2752	. . 3 ⊢ (𝑥 = 𝐶 → (((𝐴 ↑o 𝐵) ↑o 𝑥) = (𝐴 ↑o (𝐵 ·o 𝑥)) ↔ ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))))
17		oeoelem.1	. . . . . 6 ⊢ 𝐴 ∈ On
18		oecl 8483	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
19	17, 18	mpan 688	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ↑o 𝐵) ∈ On)
20		oe0 8468	. . . . 5 ⊢ ((𝐴 ↑o 𝐵) ∈ On → ((𝐴 ↑o 𝐵) ↑o ∅) = 1o)
21	19, 20	syl 17	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ↑o 𝐵) ↑o ∅) = 1o)
22		om0 8463	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐵 ·o ∅) = ∅)
23	22	oveq2d 7373	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ↑o (𝐵 ·o ∅)) = (𝐴 ↑o ∅))
24		oe0 8468	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
25	17, 24	ax-mp 5	. . . . 5 ⊢ (𝐴 ↑o ∅) = 1o
26	23, 25	eqtrdi 2792	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ↑o (𝐵 ·o ∅)) = 1o)
27	21, 26	eqtr4d 2779	. . 3 ⊢ (𝐵 ∈ On → ((𝐴 ↑o 𝐵) ↑o ∅) = (𝐴 ↑o (𝐵 ·o ∅)))
28		oveq1 7364	. . . . 5 ⊢ (((𝐴 ↑o 𝐵) ↑o 𝑦) = (𝐴 ↑o (𝐵 ·o 𝑦)) → (((𝐴 ↑o 𝐵) ↑o 𝑦) ·o (𝐴 ↑o 𝐵)) = ((𝐴 ↑o (𝐵 ·o 𝑦)) ·o (𝐴 ↑o 𝐵)))
29		oesuc 8473	. . . . . . 7 ⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝐵) ↑o suc 𝑦) = (((𝐴 ↑o 𝐵) ↑o 𝑦) ·o (𝐴 ↑o 𝐵)))
30	19, 29	sylan 580	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝐵) ↑o suc 𝑦) = (((𝐴 ↑o 𝐵) ↑o 𝑦) ·o (𝐴 ↑o 𝐵)))
31		omsuc 8472	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
32	31	oveq2d 7373	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o (𝐵 ·o suc 𝑦)) = (𝐴 ↑o ((𝐵 ·o 𝑦) +o 𝐵)))
33		omcl 8482	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
34		oeoa 8544	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (𝐵 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ↑o (𝐵 ·o 𝑦)) ·o (𝐴 ↑o 𝐵)))
35	17, 34	mp3an1 1448	. . . . . . . . 9 ⊢ (((𝐵 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ↑o (𝐵 ·o 𝑦)) ·o (𝐴 ↑o 𝐵)))
36	33, 35	sylan 580	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐵 ∈ On) → (𝐴 ↑o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ↑o (𝐵 ·o 𝑦)) ·o (𝐴 ↑o 𝐵)))
37	36	anabss1 664	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ↑o (𝐵 ·o 𝑦)) ·o (𝐴 ↑o 𝐵)))
38	32, 37	eqtrd 2776	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o (𝐵 ·o suc 𝑦)) = ((𝐴 ↑o (𝐵 ·o 𝑦)) ·o (𝐴 ↑o 𝐵)))
39	30, 38	eqeq12d 2752	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 ↑o 𝐵) ↑o suc 𝑦) = (𝐴 ↑o (𝐵 ·o suc 𝑦)) ↔ (((𝐴 ↑o 𝐵) ↑o 𝑦) ·o (𝐴 ↑o 𝐵)) = ((𝐴 ↑o (𝐵 ·o 𝑦)) ·o (𝐴 ↑o 𝐵))))
40	28, 39	syl5ibr 245	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 ↑o 𝐵) ↑o 𝑦) = (𝐴 ↑o (𝐵 ·o 𝑦)) → ((𝐴 ↑o 𝐵) ↑o suc 𝑦) = (𝐴 ↑o (𝐵 ·o suc 𝑦))))
41	40	expcom 414	. . 3 ⊢ (𝑦 ∈ On → (𝐵 ∈ On → (((𝐴 ↑o 𝐵) ↑o 𝑦) = (𝐴 ↑o (𝐵 ·o 𝑦)) → ((𝐴 ↑o 𝐵) ↑o suc 𝑦) = (𝐴 ↑o (𝐵 ·o suc 𝑦)))))
42		iuneq2 4973	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦) = (𝐴 ↑o (𝐵 ·o 𝑦)) → ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 ·o 𝑦)))
43		vex 3449	. . . . . . 7 ⊢ 𝑥 ∈ V
44		oeoelem.2	. . . . . . . . . 10 ⊢ ∅ ∈ 𝐴
45		oen0 8533	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))
46	44, 45	mpan2 689	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∅ ∈ (𝐴 ↑o 𝐵))
47		oelim 8480	. . . . . . . . . 10 ⊢ ((((𝐴 ↑o 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ (𝐴 ↑o 𝐵)) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦))
48	18, 47	sylanl1 678	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ (𝐴 ↑o 𝐵)) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦))
49	46, 48	mpidan 687	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦))
50	17, 49	mpanl1 698	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦))
51	43, 50	mpanr1 701	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦))
52		omlim 8479	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
53	43, 52	mpanr1 701	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
54	53	oveq2d 7373	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦)))
55		limord 6377	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥)
56		ordelon 6341	. . . . . . . . . . . 12 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
57	55, 56	sylan 580	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
58	57, 33	sylan2 593	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ (Lim 𝑥 ∧ 𝑦 ∈ 𝑥)) → (𝐵 ·o 𝑦) ∈ On)
59	58	anassrs 468	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝐵 ·o 𝑦) ∈ On)
60	59	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → ∀𝑦 ∈ 𝑥 (𝐵 ·o 𝑦) ∈ On)
61		0ellim 6380	. . . . . . . . . 10 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
62	61	ne0d 4295	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 ≠ ∅)
63	62	adantl 482	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → 𝑥 ≠ ∅)
64		vex 3449	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
65		oelim 8480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
66	44, 65	mpan2 689	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
67	17, 66	mpan 688	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
68	64, 67	mpan 688	. . . . . . . . 9 ⊢ (Lim 𝑤 → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
69		oewordi 8538	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
70	44, 69	mpan2 689	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
71	17, 70	mp3an3 1450	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
72	71	3impia 1117	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤))
73	68, 72	onoviun 8289	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐵 ·o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 ·o 𝑦)))
74	43, 60, 63, 73	mp3an2i 1466	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 ·o 𝑦)))
75	54, 74	eqtrd 2776	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o (𝐵 ·o 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 ·o 𝑦)))
76	51, 75	eqeq12d 2752	. . . . 5 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (((𝐴 ↑o 𝐵) ↑o 𝑥) = (𝐴 ↑o (𝐵 ·o 𝑥)) ↔ ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 ·o 𝑦))))
77	42, 76	syl5ibr 245	. . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦) = (𝐴 ↑o (𝐵 ·o 𝑦)) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = (𝐴 ↑o (𝐵 ·o 𝑥))))
78	77	expcom 414	. . 3 ⊢ (Lim 𝑥 → (𝐵 ∈ On → (∀𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦) = (𝐴 ↑o (𝐵 ·o 𝑦)) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = (𝐴 ↑o (𝐵 ·o 𝑥)))))
79	4, 8, 12, 16, 27, 41, 78	tfinds3 7801	. 2 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))))
80	79	impcom 408	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  ∪ ciun 4954  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7357  1_oc1o 8405   +_o coa 8409   ·_o comu 8410   ↑_o coe 8411

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-oexp 8418

This theorem is referenced by:  oeoe  8546