Theorem oeoelem 8534

Description: Lemma for oeoe 8535. (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 7375	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o ∅))
2		oveq2 7375	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
3	2	oveq2d 7383	. . . 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 7375	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o 𝑦))
6		oveq2 7375	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
7	6	oveq2d 7383	. . . 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 7375	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o suc 𝑦))
10		oveq2 7375	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
11	10	oveq2d 7383	. . . 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 7375	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o 𝐶))
14		oveq2 7375	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
15	14	oveq2d 7383	. . . 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 8472	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
19	17, 18	mpan 691	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ↑o 𝐵) ∈ On)
20		oe0 8457	. . . . 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 8452	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐵 ·o ∅) = ∅)
23	22	oveq2d 7383	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ↑o (𝐵 ·o ∅)) = (𝐴 ↑o ∅))
24		oe0 8457	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
25	17, 24	ax-mp 5	. . . . 5 ⊢ (𝐴 ↑o ∅) = 1o
26	23, 25	eqtrdi 2787	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ↑o (𝐵 ·o ∅)) = 1o)
27	21, 26	eqtr4d 2774	. . 3 ⊢ (𝐵 ∈ On → ((𝐴 ↑o 𝐵) ↑o ∅) = (𝐴 ↑o (𝐵 ·o ∅)))
28		oveq1 7374	. . . . 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 8462	. . . . . . 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 581	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝐵) ↑o suc 𝑦) = (((𝐴 ↑o 𝐵) ↑o 𝑦) ·o (𝐴 ↑o 𝐵)))
31		omsuc 8461	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
32	31	oveq2d 7383	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o (𝐵 ·o suc 𝑦)) = (𝐴 ↑o ((𝐵 ·o 𝑦) +o 𝐵)))
33		omcl 8471	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
34		oeoa 8533	. . . . . . . . . 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 1451	. . . . . . . . 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 581	. . . . . . . 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 667	. . . . . . 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 2771	. . . . . 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	imbitrrid 246	. . . 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 413	. . 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 4953	. . . . 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 3433	. . . . . . 7 ⊢ 𝑥 ∈ V
44		oeoelem.2	. . . . . . . . . 10 ⊢ ∅ ∈ 𝐴
45		oen0 8522	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))
46	44, 45	mpan2 692	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∅ ∈ (𝐴 ↑o 𝐵))
47		oelim 8469	. . . . . . . . . 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 681	. . . . . . . . 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 690	. . . . . . . 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 701	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦))
51	43, 50	mpanr1 704	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦))
52		omlim 8468	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
53	43, 52	mpanr1 704	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
54	53	oveq2d 7383	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦)))
55		limord 6384	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥)
56		ordelon 6347	. . . . . . . . . . . 12 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
57	55, 56	sylan 581	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
58	57, 33	sylan2 594	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ (Lim 𝑥 ∧ 𝑦 ∈ 𝑥)) → (𝐵 ·o 𝑦) ∈ On)
59	58	anassrs 467	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝐵 ·o 𝑦) ∈ On)
60	59	ralrimiva 3129	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → ∀𝑦 ∈ 𝑥 (𝐵 ·o 𝑦) ∈ On)
61		0ellim 6387	. . . . . . . . . 10 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
62	61	ne0d 4282	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 ≠ ∅)
63	62	adantl 481	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → 𝑥 ≠ ∅)
64		vex 3433	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
65		oelim 8469	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
66	44, 65	mpan2 692	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
67	17, 66	mpan 691	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
68	64, 67	mpan 691	. . . . . . . . 9 ⊢ (Lim 𝑤 → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
69		oewordi 8527	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
70	44, 69	mpan2 692	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
71	17, 70	mp3an3 1453	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
72	71	3impia 1118	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤))
73	68, 72	onoviun 8283	. . . . . . . 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 1469	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 ·o 𝑦)))
75	54, 74	eqtrd 2771	. . . . . 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	imbitrrid 246	. . . 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 413	. . 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 7816	. 2 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))))
80	79	impcom 407	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  ∪ ciun 4933  Ord word 6322  Oncon0 6323  Lim wlim 6324  suc csuc 6325  (class class class)co 7367  1_oc1o 8398   +_o coa 8402   ·_o comu 8403   ↑_o coe 8404

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-omul 8410  df-oexp 8411

This theorem is referenced by:  oeoe  8535