Theorem oeoe 7566

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 6557	. . . . . . . . . . . 12 ⊢ (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2		oe0m0 7487	. . . . . . . . . . . 12 ⊢ (∅ ↑𝑜 ∅) = 1𝑜
3	1, 2	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = 1𝑜)
4	3	oveq1d 6564	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (1𝑜 ↑𝑜 𝐶))
5		oe1m 7512	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (1𝑜 ↑𝑜 𝐶) = 1𝑜)
6	4, 5	sylan9eqr 2666	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐵 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
7	6	adantll 746	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
8		oveq2 6557	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ((∅ ↑𝑜 𝐵) ↑𝑜 ∅))
9		0elon 5695	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
10		oecl 7504	. . . . . . . . . . . 12 ⊢ ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
11	9, 10	mpan 702	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
12		oe0 7489	. . . . . . . . . . 11 ⊢ ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → ((∅ ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
14	8, 13	sylan9eqr 2666	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
15	14	adantlr 747	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
16	7, 15	jaodan 822	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
17		om00 7542	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ·𝑜 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅)))
18	17	biimpar 501	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (𝐵 ·𝑜 𝐶) = ∅)
19	18	oveq2d 6565	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = (∅ ↑𝑜 ∅))
20	19, 2	syl6eq 2660	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = 1𝑜)
21	16, 20	eqtr4d 2647	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
22		on0eln0 5697	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
23		on0eln0 5697	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
24	22, 23	bi2anan9 913	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅)))
25		neanior 2874	. . . . . . . . 9 ⊢ ((𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅))
26	24, 25	syl6bb 275	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)))
27		oe0m1 7488	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
28	27	biimpa 500	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
29	28	oveq1d 6564	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 𝐶))
30		oe0m1 7488	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑𝑜 𝐶) = ∅))
31	30	biimpa 500	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑𝑜 𝐶) = ∅)
32	29, 31	sylan9eq 2664	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ∅)
33	32	an4s 865	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ∅)
34		om00el 7543	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·𝑜 𝐶) ↔ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
35		omcl 7503	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ·𝑜 𝐶) ∈ On)
36		oe0m1 7488	. . . . . . . . . . . . 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 269	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅))
39	38	biimpa 500	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅)
40	33, 39	eqtr4d 2647	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
41	40	ex 449	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
42	26, 41	sylbird 249	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∨ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
43	42	imp 444	. . . . . 6 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
44	21, 43	pm2.61dan 828	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
45		oveq1 6556	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
46	45	oveq1d 6564	. . . . . 6 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶))
47		oveq1 6556	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
48	46, 47	eqeq12d 2625	. . . . 5 ⊢ (𝐴 = ∅ → (((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)) ↔ ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
49	44, 48	syl5ibr 235	. . . 4 ⊢ (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶))))
50	49	impcom 445	. . 3 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))
51		oveq1 6556	. . . . . . . . 9 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ↑𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵))
52	51	oveq1d 6564	. . . . . . . 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 6556	. . . . . . . 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 2625	. . . . . . 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 329	. . . . . 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 2676	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
57		eleq2 2677	. . . . . . . . . 10 ⊢ (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
58	56, 57	anbi12d 743	. . . . . . . . 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 2676	. . . . . . . . . 10 ⊢ (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (1𝑜 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
60		eleq2 2677	. . . . . . . . . 10 ⊢ (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 1𝑜 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
61	59, 60	anbi12d 743	. . . . . . . . 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 7454	. . . . . . . . . 10 ⊢ 1𝑜 ∈ On
63		0lt1o 7471	. . . . . . . . . 10 ⊢ ∅ ∈ 1𝑜
64	62, 63	pm3.2i 470	. . . . . . . . 9 ⊢ (1𝑜 ∈ On ∧ ∅ ∈ 1𝑜)
65	58, 61, 64	elimhyp 4096	. . . . . . . 8 ⊢ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))
66	65	simpli 473	. . . . . . 7 ⊢ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On
67	65	simpri 477	. . . . . . 7 ⊢ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)
68	66, 67	oeoelem 7565	. . . . . 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 4089	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶))))
70	69	imp 444	. . . 4 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))
71	70	an32s 842	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))
72	50, 71	oe0lem 7480	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))
73	72	3impb 1252	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) ↑𝑜 𝐶) = (𝐴 ↑𝑜 (𝐵 ·𝑜 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ifcif 4036  Oncon0 5640  (class class class)co 6549  1_𝑜c1o 7440   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by:  infxpenc  8724