Theorem oeoe 7679

Description: Product of exponents law for ordinal exponentiation. Theorem 8S of [Enderton] p. 238. Also Proposition 8.42 of [TakeutiZaring] p. 70. (Contributed by Eric Schmidt, 26-May-2009.)

Assertion

Ref	Expression
oeoe	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))

Proof of Theorem oeoe

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2		oe0m0 7600	. . . . . . . . . . . 12 ⊢ (∅ ↑𝑜 ∅) = 1𝑜
3	1, 2	syl6eq 2672	. . . . . . . . . . 11 ⊢ (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = 1𝑜)
4	3	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (1𝑜 ↑𝑜 𝐶))
5		oe1m 7625	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (1𝑜 ↑𝑜 𝐶) = 1𝑜)
6	4, 5	sylan9eqr 2678	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐵 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
7	6	adantll 750	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
8		oveq2 6658	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ((∅ ↑𝑜 𝐵) ↑𝑜 ∅))
9		0elon 5778	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
10		oecl 7617	. . . . . . . . . . . 12 ⊢ ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
11	9, 10	mpan 706	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
12		oe0 7602	. . . . . . . . . . 11 ⊢ ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → ((∅ ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
14	8, 13	sylan9eqr 2678	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
15	14	adantlr 751	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
16	7, 15	jaodan 826	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
17		om00 7655	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ·𝑜 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅)))
18	17	biimpar 502	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (𝐵 ·𝑜 𝐶) = ∅)
19	18	oveq2d 6666	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = (∅ ↑𝑜 ∅))
20	19, 2	syl6eq 2672	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = 1𝑜)
21	16, 20	eqtr4d 2659	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
22		on0eln0 5780	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
23		on0eln0 5780	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
24	22, 23	bi2anan9 917	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅)))
25		neanior 2886	. . . . . . . . 9 ⊢ ((𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅))
26	24, 25	syl6bb 276	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)))
27		oe0m1 7601	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
28	27	biimpa 501	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
29	28	oveq1d 6665	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 𝐶))
30		oe0m1 7601	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑𝑜 𝐶) = ∅))
31	30	biimpa 501	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑𝑜 𝐶) = ∅)
32	29, 31	sylan9eq 2676	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ∅)
33	32	an4s 869	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ∅)
34		om00el 7656	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·𝑜 𝐶) ↔ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
35		omcl 7616	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ·𝑜 𝐶) ∈ On)
36		oe0m1 7601	. . . . . . . . . . . . 13 ⊢ ((𝐵 ·𝑜 𝐶) ∈ On → (∅ ∈ (𝐵 ·𝑜 𝐶) ↔ (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅))
37	35, 36	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·𝑜 𝐶) ↔ (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅))
38	34, 37	bitr3d 270	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅))
39	38	biimpa 501	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅)
40	33, 39	eqtr4d 2659	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
41	40	ex 450	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
42	26, 41	sylbird 250	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∨ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
43	42	imp 445	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
44	21, 43	pm2.61dan 832	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
45		oveq1 6657	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
46	45	oveq1d 6665	. . . . . 6 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶))
47		oveq1 6657	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
48	46, 47	eqeq12d 2637	. . . . 5 ⊢ (𝐴 = ∅ → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)) ↔ ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
49	44, 48	syl5ibr 236	. . . 4 ⊢ (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶))))
50	49	impcom 446	. . 3 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))
51		oveq1 6657	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ↑𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵))
52	51	oveq1d 6665	. . . . . . . 8 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ↑𝑜 𝐶))
53		oveq1 6657	. . . . . . . 8 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 ·𝑜 𝐶)))
54	52, 53	eqeq12d 2637	. . . . . . 7 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)) ↔ ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ↑𝑜 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 ·𝑜 𝐶))))
55	54	imbi2d 330	. . . . . 6 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶))) ↔ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ↑𝑜 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 ·𝑜 𝐶)))))
56		eleq1 2689	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
57		eleq2 2690	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
58	56, 57	anbi12d 747	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ↔ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))))
59		eleq1 2689	. . . . . . . . . 10 ⊢ (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (1𝑜 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
60		eleq2 2690	. . . . . . . . . 10 ⊢ (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 1𝑜 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
61	59, 60	anbi12d 747	. . . . . . . . 9 ⊢ (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((1𝑜 ∈ On ∧ ∅ ∈ 1𝑜) ↔ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))))
62		1on 7567	. . . . . . . . . 10 ⊢ 1𝑜 ∈ On
63		0lt1o 7584	. . . . . . . . . 10 ⊢ ∅ ∈ 1𝑜
64	62, 63	pm3.2i 471	. . . . . . . . 9 ⊢ (1𝑜 ∈ On ∧ ∅ ∈ 1𝑜)
65	58, 61, 64	elimhyp 4146	. . . . . . . 8 ⊢ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))
66	65	simpli 474	. . . . . . 7 ⊢ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On
67	65	simpri 478	. . . . . . 7 ⊢ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)
68	66, 67	oeoelem 7678	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ↑𝑜 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 ·𝑜 𝐶)))
69	55, 68	dedth 4139	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶))))
70	69	imp 445	. . . 4 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))
71	70	an32s 846	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))
72	50, 71	oe0lem 7593	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))
73	72	3impb 1260	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∅c0 3915  ifcif 4086  Oncon0 5723  (class class class)co 6650  1_𝑜c1o 7553   ·_𝑜 comu 7558   ↑_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-rmo 2920  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-int 4476  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-1st 7168  df-2nd 7169  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:  infxpenc  8841