Theorem oelim2 7675

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 5788	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2		0ellim 5787	. . . . . . 7 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
3	2	adantl 482	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
4		oe0m1 7601	. . . . . . 7 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
5	4	biimpa 501	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
6	1, 3, 5	syl2anc 693	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
7		eldif 3584	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 1𝑜) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 1𝑜))
8		limord 5784	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → Ord 𝐵)
9		ordelon 5747	. . . . . . . . . . . 12 ⊢ ((Ord 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
10	8, 9	sylan 488	. . . . . . . . . . 11 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
11		on0eln0 5780	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ 𝑥 ≠ ∅))
12		el1o 7579	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 1𝑜 ↔ 𝑥 = ∅)
13	12	necon3bbii 2841	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ 1𝑜 ↔ 𝑥 ≠ ∅)
14	11, 13	syl6bbr 278	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ ¬ 𝑥 ∈ 1𝑜))
15		oe0m1 7601	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑𝑜 𝑥) = ∅))
16	15	biimpd 219	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (∅ ∈ 𝑥 → (∅ ↑𝑜 𝑥) = ∅))
17	14, 16	sylbird 250	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (¬ 𝑥 ∈ 1𝑜 → (∅ ↑𝑜 𝑥) = ∅))
18	10, 17	syl 17	. . . . . . . . . 10 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ 𝐵) → (¬ 𝑥 ∈ 1𝑜 → (∅ ↑𝑜 𝑥) = ∅))
19	18	impr 649	. . . . . . . . 9 ⊢ ((Lim 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 1𝑜)) → (∅ ↑𝑜 𝑥) = ∅)
20	7, 19	sylan2b 492	. . . . . . . 8 ⊢ ((Lim 𝐵 ∧ 𝑥 ∈ (𝐵 ∖ 1𝑜)) → (∅ ↑𝑜 𝑥) = ∅)
21	20	iuneq2dv 4542	. . . . . . 7 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)∅)
22		df-1o 7560	. . . . . . . . . . 11 ⊢ 1𝑜 = suc ∅
23		limsuc 7049	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → (∅ ∈ 𝐵 ↔ suc ∅ ∈ 𝐵))
24	2, 23	mpbid 222	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → suc ∅ ∈ 𝐵)
25	22, 24	syl5eqel 2705	. . . . . . . . . 10 ⊢ (Lim 𝐵 → 1𝑜 ∈ 𝐵)
26		1on 7567	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
27	26	onirri 5834	. . . . . . . . . 10 ⊢ ¬ 1𝑜 ∈ 1𝑜
28	25, 27	jctir 561	. . . . . . . . 9 ⊢ (Lim 𝐵 → (1𝑜 ∈ 𝐵 ∧ ¬ 1𝑜 ∈ 1𝑜))
29		eldif 3584	. . . . . . . . 9 ⊢ (1𝑜 ∈ (𝐵 ∖ 1𝑜) ↔ (1𝑜 ∈ 𝐵 ∧ ¬ 1𝑜 ∈ 1𝑜))
30	28, 29	sylibr 224	. . . . . . . 8 ⊢ (Lim 𝐵 → 1𝑜 ∈ (𝐵 ∖ 1𝑜))
31		ne0i 3921	. . . . . . . 8 ⊢ (1𝑜 ∈ (𝐵 ∖ 1𝑜) → (𝐵 ∖ 1𝑜) ≠ ∅)
32		iunconst 4529	. . . . . . . 8 ⊢ ((𝐵 ∖ 1𝑜) ≠ ∅ → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)∅ = ∅)
33	30, 31, 32	3syl 18	. . . . . . 7 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)∅ = ∅)
34	21, 33	eqtrd 2656	. . . . . 6 ⊢ (Lim 𝐵 → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∅)
35	34	adantl 482	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥) = ∅)
36	6, 35	eqtr4d 2659	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (∅ ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥))
37		oveq1 6657	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
38		oveq1 6657	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝑥) = (∅ ↑𝑜 𝑥))
39	38	iuneq2d 4547	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥))
40	37, 39	eqeq12d 2637	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥) ↔ (∅ ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(∅ ↑𝑜 𝑥)))
41	36, 40	syl5ibr 236	. . 3 ⊢ (𝐴 = ∅ → ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥)))
42	41	impcom 446	. 2 ⊢ (((𝐵 ∈ 𝐶 ∧ Lim 𝐵) ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))
43		oelim 7614	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦))
44		limsuc 7049	. . . . . . . . . . . . 13 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 ↔ suc 𝑦 ∈ 𝐵))
45	44	biimpa 501	. . . . . . . . . . . 12 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ∈ 𝐵)
46		nsuceq0 5805	. . . . . . . . . . . . 13 ⊢ suc 𝑦 ≠ ∅
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ≠ ∅)
48		dif1o 7580	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ↔ (suc 𝑦 ∈ 𝐵 ∧ suc 𝑦 ≠ ∅))
49	45, 47, 48	sylanbrc 698	. . . . . . . . . . 11 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → suc 𝑦 ∈ (𝐵 ∖ 1𝑜))
50	49	ex 450	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1𝑜)))
51	50	ad2antlr 763	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → suc 𝑦 ∈ (𝐵 ∖ 1𝑜)))
52		sssucid 5802	. . . . . . . . . . 11 ⊢ 𝑦 ⊆ suc 𝑦
53		ordelon 5747	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
54	8, 53	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ ((Lim 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
55		suceloni 7013	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
56	54, 55	jccir 562	. . . . . . . . . . . . . . 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 1265	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ On ∧ suc 𝑦 ∈ On) ∧ 𝐴 ∈ On) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
59	58	ancoms 469	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ (𝑦 ∈ On ∧ suc 𝑦 ∈ On)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
60	56, 59	sylan2 491	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ (Lim 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
61	60	anassrs 680	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On))
62		oewordi 7671	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ On ∧ suc 𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)))
63	61, 62	sylan 488	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ On ∧ Lim 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ⊆ suc 𝑦 → (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)))
64	63	an32s 846	. . . . . . . . . . 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 450	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)))
67	51, 66	jcad 555	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ 𝐵 → (suc 𝑦 ∈ (𝐵 ∖ 1𝑜) ∧ (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦))))
68		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
69	68	sseq2d 3633	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 𝑥) ↔ (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)))
70	69	rspcev 3309	. . . . . . . 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 2965	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 𝑥))
73		iunss2 4565	. . . . . 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 3737	. . . . . . . 8 ⊢ (𝐵 ∖ 1𝑜) ⊆ 𝐵
76		iunss1 4532	. . . . . . . 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 6658	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
79	78	cbviunv 4559	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦)
80	77, 79	sseqtri 3637	. . . . . 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 3620	. . . 4 ⊢ (((𝐴 ∈ On ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))
83	82	adantlrl 756	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ∪ 𝑦 ∈ 𝐵 (𝐴 ↑𝑜 𝑦) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))
84	43, 83	eqtrd 2656	. 2 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑥))
85	42, 84	oe0lem 7593	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 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650  1_𝑜c1o 7553   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oelimcl  7680  oaabs2  7725  omabs  7727