Theorem oeoelem 8621

Description: Lemma for oeoe 8622. (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 7438	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o ∅))
2		oveq2 7438	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
3	2	oveq2d 7446	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o ∅)))
4	1, 3	eqeq12d 2779	. . 3 ⊢ (𝑥 = ∅ → (((𝐴 ↑o 𝐵) ↑o 𝑥) = (𝐴 ↑o (𝐵 ·o 𝑥)) ↔ ((𝐴 ↑o 𝐵) ↑o ∅) = (𝐴 ↑o (𝐵 ·o ∅))))
5		oveq2 7438	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o 𝑦))
6		oveq2 7438	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
7	6	oveq2d 7446	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o 𝑦)))
8	5, 7	eqeq12d 2779	. . 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 7438	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o suc 𝑦))
10		oveq2 7438	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
11	10	oveq2d 7446	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o suc 𝑦)))
12	9, 11	eqeq12d 2779	. . 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 7438	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ((𝐴 ↑o 𝐵) ↑o 𝐶))
14		oveq2 7438	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
15	14	oveq2d 7446	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o (𝐵 ·o 𝐶)))
16	13, 15	eqeq12d 2779	. . 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 8559	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
19	17, 18	mpan 703	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ↑o 𝐵) ∈ On)
20		oe0 8544	. . . . 5 ⊢ ((𝐴 ↑o 𝐵) ∈ On → ((𝐴 ↑o 𝐵) ↑o ∅) = 1o)
21	19, 20	syl 18	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ↑o 𝐵) ↑o ∅) = 1o)
22		om0 8539	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐵 ·o ∅) = ∅)
23	22	oveq2d 7446	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ↑o (𝐵 ·o ∅)) = (𝐴 ↑o ∅))
24		oe0 8544	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
25	17, 24	ax-mp 5	. . . . 5 ⊢ (𝐴 ↑o ∅) = 1o
26	23, 25	eqtrdi 2814	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ↑o (𝐵 ·o ∅)) = 1o)
27	21, 26	eqtr4d 2801	. . 3 ⊢ (𝐵 ∈ On → ((𝐴 ↑o 𝐵) ↑o ∅) = (𝐴 ↑o (𝐵 ·o ∅)))
28		oveq1 7437	. . . . 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 8549	. . . . . . 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 592	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ↑o 𝐵) ↑o suc 𝑦) = (((𝐴 ↑o 𝐵) ↑o 𝑦) ·o (𝐴 ↑o 𝐵)))
31		omsuc 8548	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
32	31	oveq2d 7446	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o (𝐵 ·o suc 𝑦)) = (𝐴 ↑o ((𝐵 ·o 𝑦) +o 𝐵)))
33		omcl 8558	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
34		oeoa 8620	. . . . . . . . . 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 1477	. . . . . . . . 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 592	. . . . . . . 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 679	. . . . . . 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 2798	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o (𝐵 ·o suc 𝑦)) = ((𝐴 ↑o (𝐵 ·o 𝑦)) ·o (𝐴 ↑o 𝐵)))
39	30, 38	eqeq12d 2779	. . . . 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 249	. . . 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 419	. . 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 4974	. . . . 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 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
44		oeoelem.2	. . . . . . . . . 10 ⊢ ∅ ∈ 𝐴
45		oen0 8609	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))
46	44, 45	mpan2 704	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∅ ∈ (𝐴 ↑o 𝐵))
47		oelim 8556	. . . . . . . . . 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 693	. . . . . . . . 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 702	. . . . . . . 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 713	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦))
51	43, 50	mpanr1 716	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐴 ↑o 𝐵) ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ↑o 𝐵) ↑o 𝑦))
52		omlim 8555	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
53	43, 52	mpanr1 716	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦))
54	53	oveq2d 7446	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o (𝐵 ·o 𝑥)) = (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦)))
55		limord 6431	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥)
56		ordelon 6393	. . . . . . . . . . . 12 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
57	55, 56	sylan 592	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
58	57, 33	sylan2 605	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ (Lim 𝑥 ∧ 𝑦 ∈ 𝑥)) → (𝐵 ·o 𝑦) ∈ On)
59	58	anassrs 473	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝐵 ·o 𝑦) ∈ On)
60	59	ralrimiva 3158	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → ∀𝑦 ∈ 𝑥 (𝐵 ·o 𝑦) ∈ On)
61		0ellim 6434	. . . . . . . . . 10 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
62	61	ne0d 4291	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 ≠ ∅)
63	62	adantl 487	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → 𝑥 ≠ ∅)
64		vex 3458	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
65		oelim 8556	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
66	44, 65	mpan2 704	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
67	17, 66	mpan 703	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
68	64, 67	mpan 703	. . . . . . . . 9 ⊢ (Lim 𝑤 → (𝐴 ↑o 𝑤) = ∪ 𝑧 ∈ 𝑤 (𝐴 ↑o 𝑧))
69		oewordi 8614	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
70	44, 69	mpan2 704	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
71	17, 70	mp3an3 1479	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧 ⊆ 𝑤 → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤)))
72	71	3impia 1135	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧 ⊆ 𝑤) → (𝐴 ↑o 𝑧) ⊆ (𝐴 ↑o 𝑤))
73	68, 72	onoviun 8365	. . . . . . . 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 1495	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o ∪ 𝑦 ∈ 𝑥 (𝐵 ·o 𝑦)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 ·o 𝑦)))
75	54, 74	eqtrd 2798	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ↑o (𝐵 ·o 𝑥)) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o (𝐵 ·o 𝑦)))
76	51, 75	eqeq12d 2779	. . . . 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 249	. . . 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 419	. . 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 7886	. 2 ⊢ (𝐶 ∈ On → (𝐵 ∈ On → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))))
80	79	impcom 413	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146   ≠ wne 2959  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ∪ ciun 4954  Ord word 6368  Oncon0 6369  Lim wlim 6370  suc csuc 6371  (class class class)co 7430  1_oc1o 8483   +_o coa 8487   ·_o comu 8488   ↑_o coe 8489

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-2o 8491  df-oadd 8494  df-omul 8495  df-oexp 8496

This theorem is used by:  oeoe  8622