Theorem oelim2 7906

Description: Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)

Assertion

Ref	Expression
oelim2	⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem oelim2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limelon 5995	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		0ellim 5994	. . . . . . 7 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
3	2	adantl 469	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
4		oe0m1 7832	. . . . . . 7 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
5	4	biimpa 464	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
6	1, 3, 5	syl2anc 575	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
7		eldif 3773	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 1𝑜) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 1𝑜))
8		limord 5991	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → Ord 𝐵)
9		ordelon 5954	. . . . . . . . . . . 12 ⊢ ((Ord 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
10	8, 9	sylan 571	. . . . . . . . . . 11 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
11		on0eln0 5987	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ 𝑥 ≠ ∅))
12		el1o 7810	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 1𝑜 ↔ 𝑥 = ∅)
13	12	necon3bbii 3021	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ 1𝑜 ↔ 𝑥 ≠ ∅)
14	11, 13	syl6bbr 280	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ ¬ 𝑥 ∈ 1𝑜))
15		oe0m1 7832	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑𝑜 𝑥) = ∅))
16	15	biimpd 220	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 → (∅ ↑𝑜 𝑥) = ∅))
17	14, 16	sylbird 251	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (¬ 𝑥 ∈ 1𝑜 → (∅ ↑𝑜 𝑥) = ∅))
18	10, 17	syl 17	. . . . . . . . . 10 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ 𝐵) → (¬ 𝑥 ∈ 1𝑜 → (∅ ↑𝑜 𝑥) = ∅))
19	18	impr 444	. . . . . . . . 9 ⊢ ((Lim 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 1𝑜)) → (∅ ↑𝑜 𝑥) = ∅)
20	7, 19	sylan2b 583	. . . . . . . 8 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ (𝐵 ∖ 1𝑜)) → (∅ ↑𝑜 𝑥) = ∅)
21	20	iuneq2dv 4727	. . . . . . 7 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)∅)
22		df-1o 7790	. . . . . . . . . . 11 ⊢ 1𝑜 = suc ∅
23		limsuc 7273	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → (∅ ∈ 𝐵 ↔ suc ∅ ∈ 𝐵))
24	2, 23	mpbid 223	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → suc ∅ ∈ 𝐵)
25	22, 24	syl5eqel 2885	. . . . . . . . . 10 ⊢ (Lim 𝐵 → 1𝑜 ∈ 𝐵)
26		1on 7797	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
27	26	onirri 6041	. . . . . . . . . 10 ⊢ ¬ 1𝑜 ∈ 1𝑜
28	25, 27	jctir 512	. . . . . . . . 9 ⊢ (Lim 𝐵 → (1𝑜 ∈ 𝐵 ∧ ¬ 1𝑜 ∈ 1𝑜))
29		eldif 3773	. . . . . . . . 9 ⊢ (1𝑜 ∈ (𝐵 ∖ 1𝑜) ↔ (1𝑜 ∈ 𝐵 ∧ ¬ 1𝑜 ∈ 1𝑜))
30	28, 29	sylibr 225	. . . . . . . 8 ⊢ (Lim 𝐵 → 1𝑜 ∈ (𝐵 ∖ 1𝑜))
31		ne0i 4116	. . . . . . . 8 ⊢ (1𝑜 ∈ (𝐵 ∖ 1𝑜) → (𝐵 ∖ 1𝑜) ≠ ∅)
32		iunconst 4714	. . . . . . . 8 ⊢ ((𝐵 ∖ 1𝑜) ≠ ∅ → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)∅ = ∅)
33	30, 31, 32	3syl 18	. . . . . . 7 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)∅ = ∅)
34	21, 33	eqtrd 2836	. . . . . 6 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∅)
35	34	adantl 469	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∅)
36	6, 35	eqtr4d 2839	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (∅ ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥))
37		oveq1 6875	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
38		oveq1 6875	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝑥) = (∅ ↑𝑜 𝑥))
39	38	iuneq2d 4732	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥))
40	37, 39	eqeq12d 2817	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥) ↔ (∅ ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥)))
41	36, 40	syl5ibr 237	. . 3 ⊢ (𝐴 = ∅ → ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥)))
42	41	impcom 396	. 2 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))
43		oelim 7845	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦))
44		limsuc 7273	. . . . . . . . . . . . 13 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 ↔ suc 𝑦 ∈ 𝐵))
45	44	biimpa 464	. . . . . . . . . . . 12 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ∈ 𝐵)
46		nsuceq0 6012	. . . . . . . . . . . . 13 ⊢ suc 𝑦 ≠ ∅
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ≠ ∅)
48		dif1o 7811	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ↔ (suc 𝑦 ∈ 𝐵 ∧ suc 𝑦 ≠ ∅))
49	45, 47, 48	sylanbrc 574	. . . . . . . . . . 11 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ∈ (𝐵 ∖ 1𝑜))
50	49	ex 399	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1𝑜)))
51	50	ad2antlr 709	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1𝑜)))
52		sssucid 6009	. . . . . . . . . . 11 ⊢ 𝑦 ⊆ suc 𝑦
53		ordelon 5954	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
54	8, 53	sylan 571	. . . . . . . . . . . . . . . 16 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
55		suceloni 7237	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
56	54, 55	jccir 513	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On))
57		id 22	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
58	57	3expa 1140	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ On ∧ suc 𝑦 ∈ On) ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
59	58	ancoms 448	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ (𝑦 ∈ On ∧ suc 𝑦 ∈ On)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
60	56, 59	sylan2 582	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ (Lim 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
61	60	anassrs 455	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
62		oewordi 7902	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)))
63	61, 62	sylan 571	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)))
64	63	an32s 634	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)))
65	52, 64	mpi 20	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦))
66	65	ex 399	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)))
67	51, 66	jcad 504	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → (suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ∧ (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦))))
68		oveq2 6876	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
69	68	sseq2d 3824	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 𝑥) ↔ (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)))
70	69	rspcev 3498	. . . . . . . 8 ⊢ ((suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ∧ (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)) → ∃𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 𝑥))
71	67, 70	syl6 35	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 𝑥)))
72	71	ralrimiv 3149	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 𝑥))
73		iunss2 4750	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 𝑥) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦) ⊆ ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))
74	72, 73	syl 17	. . . . 5 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦) ⊆ ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))
75		difss 3930	. . . . . . . 8 ⊢ (𝐵 ∖ 1𝑜) ⊆ 𝐵
76		iunss1 4717	. . . . . . . 8 ⊢ ((𝐵 ∖ 1𝑜) ⊆ 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥) ⊆ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥))
77	75, 76	ax-mp 5	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥) ⊆ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥)
78		oveq2 6876	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
79	78	cbviunv 4744	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦)
80	77, 79	sseqtri 3828	. . . . . 6 ⊢ ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦)
81	80	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥) ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦))
82	74, 81	eqssd 3809	. . . 4 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))
83	82	adantlrl 702	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))
84	43, 83	eqtrd 2836	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))
85	42, 84	oe0lem 7824	1 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1100   = wceq 1637   ∈ wcel 2155   ≠ wne 2974  ∀wral 3092  ∃wrex 3093   ∖ cdif 3760   ⊆ wss 3763  ∅c0 4110  ∪ ciun 4705  Ord word 5929  Oncon0 5930  Lim wlim 5931  suc csuc 5932  (class class class)co 6868  1_𝑜c1o 7783   ↑_𝑜 coe 7789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-2o 7791  df-oadd 7794  df-omul 7795  df-oexp 7796

This theorem is referenced by:  oelimcl  7911  oaabs2  7956  omabs  7958