Theorem oeoelem 8218

Description: Lemma for oeoe 8219. (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 7158	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o ∅))
2		oveq2 7158	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
3	2	oveq2d 7166	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o ∅)))
4	1, 3	eqeq12d 2837	. . 3 ⊢ (𝑥 = ∅ → (((𝐴 ↑o 𝐵) ↑o 𝑥) = (𝐴 ↑o (𝐵 ·o 𝑥)) ↔ ((𝐴 ↑o 𝐵) ↑o ∅) = (𝐴 ↑o (𝐵 ·o ∅))))
5		oveq2 7158	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o 𝑦))
6		oveq2 7158	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
7	6	oveq2d 7166	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o 𝑦)))
8	5, 7	eqeq12d 2837	. . 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 7158	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o suc 𝑦))
10		oveq2 7158	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
11	10	oveq2d 7166	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o suc 𝑦)))
12	9, 11	eqeq12d 2837	. . 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 7158	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o 𝐶))
14		oveq2 7158	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
15	14	oveq2d 7166	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o 𝐶)))
16	13, 15	eqeq12d 2837	. . 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 8156	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
19	17, 18	mpan 688	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ↑o 𝐵) ∈ On)
20		oe0 8141	. . . . 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 8136	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐵 ·o ∅) = ∅)
23	22	oveq2d 7166	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ↑o (𝐵 ·o ∅)) = (𝐴 ↑o ∅))
24		oe0 8141	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
25	17, 24	ax-mp 5	. . . . 5 ⊢ (𝐴 ↑o ∅) = 1o
26	23, 25	syl6eq 2872	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ↑o (𝐵 ·o ∅)) = 1o)
27	21, 26	eqtr4d 2859	. . 3 ⊢ (𝐵 ∈ On → ((𝐴 ↑o 𝐵) ↑o ∅) = (𝐴 ↑o (𝐵 ·o ∅)))
28		oveq1 7157	. . . . 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 8146	. . . . . . 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 582	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝐵) ↑o suc 𝑦) = (((𝐴 ↑o 𝐵) ↑o 𝑦) ·o (𝐴 ↑o 𝐵)))
31		omsuc 8145	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
32	31	oveq2d 7166	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o (𝐵 ·o suc 𝑦)) = (𝐴 ↑o ((𝐵 ·o 𝑦) +o 𝐵)))
33		omcl 8155	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
34		oeoa 8217	. . . . . . . . . 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 1444	. . . . . . . . 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 582	. . . . . . . 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 2856	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o (𝐵 ·o suc 𝑦)) = ((𝐴 ↑o (𝐵 ·o 𝑦)) ·o (𝐴 ↑o 𝐵)))
39	30, 38	eqeq12d 2837	. . . . 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 248	. . . 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 416	. . 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 4931	. . . . 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 3498	. . . . . . 7 ⊢ 𝑥 ∈ V
44		oeoelem.2	. . . . . . . . . 10 ⊢ ∅ ∈ 𝐴
45		oen0 8206	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))
46	44, 45	mpan2 689	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∅ ∈ (𝐴 ↑o 𝐵))
47		oelim 8153	. . . . . . . . . 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 8152	. . . . . . . . 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 7166	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦)))
55		limord 6245	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥)
56		ordelon 6210	. . . . . . . . . . . 12 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
57	55, 56	sylan 582	. . . . . . . . . . 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 470	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝐵 ·o 𝑦) ∈ On)
60	59	ralrimiva 3182	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → ∀𝑦 ∈ 𝑥 (𝐵 ·o 𝑦) ∈ On)
61		0ellim 6248	. . . . . . . . . 10 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
62	61	ne0d 4301	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 ≠ ∅)
63	62	adantl 484	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → 𝑥 ≠ ∅)
64		vex 3498	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
65		oelim 8153	. . . . . . . . . . . 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 8211	. . . . . . . . . . . 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 1446	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
72	71	3impia 1113	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤))
73	68, 72	onoviun 7974	. . . . . . . 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 1462	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 ·o 𝑦)))
75	54, 74	eqtrd 2856	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o (𝐵 ·o 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 ·o 𝑦)))
76	51, 75	eqeq12d 2837	. . . . 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 248	. . . 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 416	. . 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 7573	. 2 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))))
80	79	impcom 410	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  Vcvv 3495   ⊆ wss 3936  ∅c0 4291  ∪ ciun 4912  Ord word 6185  Oncon0 6186  Lim wlim 6187  suc csuc 6188  (class class class)co 7150  1_oc1o 8089   +_o coa 8093   ·_o comu 8094   ↑_o coe 8095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-omul 8101  df-oexp 8102

This theorem is referenced by:  oeoe  8219